AN HONORS eBOOK FROM MASTER POINT PRESS

Patrick Darricades

Optimal
Hand
Evaluation
 Text © 2019 Patrick Darricades
Cover image: 123RF/dolgachov

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 OPTIMAL HAND EVALUATION
______________________________________________________________________

TABLE OF CONTENTS

FOREWORD
CHAPTER 1 : GOREN’s point count : First corrections

Ø Assessing current hand evaluation practices Page 1

Ø Corrections to the 4 3 2 1 Honor point count Page 2
Ø Corrections to the 5 3 1 Distribution-Fit count Page 4
Ø The true value of Aces, Queens and Jacks Page 6
Ø Year 1995 : Fit points for 9 or more trumps Page 15
CHAPTER 2 : 2015 : New findings and further corrections

Ø Points for suit length and for short suits Page 19

Ø Combination of honors and the true value of 10s Page 22
Ø The importance of Qs and Ks : point corrections Page 33
Ø 9 + card Fit points for All suits and All contracts ! Page 39
Ø 8-card Fit points for All suits and All contracts ! Page 45
Ø Optimal Hand Evaluation : Illustrations Page 57
CHAPTER 3 : “Wasted honor points” count

Ø “Wasted honor points” opposite short suits Page 61

Ø Misfit points opposite long suits ………..…… Page 66
Ø Misfit points in No Trump ………………...…. Page 71
CHAPTER 4 : Point count in response to partner’s opening

Ø Semi-Fit points ……….………………………… Page 73

Ø Point count in response to a No Trump bid Page 76
 OPTIMAL HAND EVALUATION : TABLE OF CONTENTS

TABLE OF CONTENTS – CONT’D

CHAPTER 5 : Point count : summary and conversion table

Ø Conclusion ……………………………………… Page 83

Ø The optimal point count vs « Trick » counts Page 84
Ø The optimal point count : summary chart Page 90
Ø Converting points into corresponding tricks Page 93
Ø Exercises / Illustrations ……………………….. Page 97
CHAPTER 6 : Other point counts

Ø Other point counts : Overview ………….…... Page 105

CHAPTER 7 : Which point count for opening hands ?

Ø H or HL or HD points ? ……………………… Page 121

CHAPTER 8 : Summary and Conclusion Page 131

EPILOGUE : Optimal pt count implications on Bidding Page 139

INDEX / Complementary notes Page 145

 OPTIMAL HAND EVALUATION

FOREWORD

In his 1968 Bridge Dictionary (« Dictionnaire du Bridge », G. Versini),

the author illustrates perfectly our topic with the following
example of these two hands of the same side :
ª KQxx ª AJxx
© x © xxxx
¨ xxx ¨ xx
§ AQxxx § Kxx
Which he comments on as follows : « One can see that a 4 ª contract
is virtually assured with these two hands, even though they total only
19 honor points and 23 HDS points (Honor, Distribution, trump
Support) – while it is generally considered that 27 HDS points are needed
to bid and make a major-suit game. In this case, it must be acknowledged
that the Goren * point count, universally used today, does not adequately
translate this perfect Fit ».
* Goren point count : Refers to the great American champion Charles
Goren who popularized the addition of distribution points for short suits
(3 2 1 for void, singleton and doubleton) to Milton Work’s 4 3 2 1 honor
point count (for Ace, K, Q, J) to reflect the total value of a hand.
About 50 years after this very pertinent observation, we should
have found a more accurate hand evaluation method that correctly
assesses the combined value of these two hands at 29 total points –
Honors, Length, Distribution and Fit points – rather than only 23
HDS points.
Surprisingly, this is not the case, despite several advances made in
hand evaluation since 1968, many of which have yet to be taught
today or systematically integrated by a majority of champions and
experts – not even by the very pioneers of several of these
advances, essentially to avoid counting in half-points in order to
keep the point count simple…
A significant error of judgment which has unfortunately resulted
in delaying for several decades the correct point count and
therefore an appropriate and accurate hand evaluation.
 OPTIMAL HAND EVALUATION – FOREWORD

However, most experts agree that we can generally rely on the

widely acknowledged accuracy of M. Work’s 4 3 2 1 honor point
count to evaluate balanced hands for a No Trump contract.
But then, how do we account for the 6 NT contract which should
be bid with the hands below totaling only 27 honor points, when
it is generally considered that a minimum of 33 HCP points (High
Card Points) are needed for a 6 NT slam :
ª Axx ª xx
© xx © Axx
¨ Kxx ¨ AQJxx
§ AQxxx § Kxx
Even adding 1 point for the 5 clubs and another for the 5 diamonds
only brings the total up to 29 HL points – far below 33 points !
Here again, we should have found by now a more accurate hand
evaluation method that correctly assesses the combined value of
these two hands at 35 total pts – Honor, Length and Fit points.
And then there is the matter of the value of Tens – generally
ignored, particularly for suit contracts. As an example of this, take
the deal below, from a tournament where not a single pair found
the 4 ª game :
ª A J 10 x x ª Kxx
© KQx © Jx
¨ Qxxx ¨ A J 10 x
§ x § xxxx
The bidding was invariably :
1ª 2ª
Pass
But an optimal point count will show that East’s hand is worth
12 HLD Fit points, not 9 or 10. The bidding could then be :
1ª 2¨
4 § (¨ Fit, § singleton) 4ª
The above is clearly a case where each ten is of significant value –
counting the appropriate value of tens is of critical importance to
reaching the right contracts. Here again, the traditional point
count gives these two hands a total of only 23 HDS points when
they actually total 29 HLDFit points.
 OPTIMAL HAND EVALUATION – FOREWORD

And accurately evaluating the combined value of two hands ought

to start with properly assessing the value of a single hand.
Yet, the 4 3 2 1 honor point count gives the two opening hands
below the same count of 13 H pts when the first hand has 2 ½
winning tricks, with only one sure trick, while the second hand has
more than 3 ½ winning tricks, with three sure tricks :
ª Kxxx ª A Q 10 x
© Kxx © xxx
¨ QJx ¨ AKxx
§ A 10 x § xx
A more appropriate and accurate hand evaluation point count
would give the first hand 12 points and the second 15 points.
Yet, to obtain a very precise, optimal hand evaluation, all that was
needed was to apply corrections identified long ago and some new
ones involving just a few more corrections to make to a point
count already very familiar to all.
That is precisely what this book proposes to do and illustrate.
In the process, this optimal hand evaluation also addresses how to
precisely evaluate misfitted hands – a key issue generally ignored
and not precisely quantified. The following deal illustrates this :
in the Final of a 1981 National selection, 3 teams out of four played
3 NT, down 2, after East’s 2 § response to West’s 1 © opening :
ª xx ª Qxx
© A J 10 x x © Q
¨ Kxxx ¨ QJxx
§ Kx § AQxxx
But East’s hand, far from being worth 13 H/14 HL pts, should be
counted for less than 10 HL misfit pts, which makes a 2 § response
inappropriate. A 1 NT response, instead, would have enabled the
teams to settle in 3 ¨.
This book includes many examples of the disastrous consequences
of the inadequate point count universally used today as well as the
significant benefits of a very accurate optimal point count.
Finally, an Optimal hand evaluation has implications that greatly
impact some bidding principles and practices. These implications
are outlined in this book’s Epilogue.

Patrick Darricades
 OPTIMAL HAND EVALUATION

CHAPTER 1

GOREN’S Point Count : First Corrections

ASSESSING CURRENT HAND EVALUATION PRACTICES

Assessing as accurately as possible the combined value of two
hands of the same side is the very foundation of Bridge bidding
and has therefore been, understandably, the subject of substantial
study leading to various suggested hand evaluation methods from
the very beginning of Contract Bridge, over 90 years ago.
But neither E. Culbertson’s honor trick count (assessing the trick-
taking potential of various honor combinations), nor the 3 2 1 ½
honor point count (for the Ace, King, Queen, Jack) advocated by
the formidable « 4 Aces » American team in the mid-‘30s, was
adopted by players; the former, because it was complex,
inaccurate and found ill-suited to evaluate evenly-distributed
hands for a No Trump contract; the latter, because it involves
counting in half-points (the Jack) and dividing by two the total
number of points in order to estimate the probable number of
tricks that can be expected.
Instead, the count which prevailed, from the mid-‘30s in Great
Britain and the mid-‘40s in the United States, has been the 4 3 2 1
honor point count (for the Ace, K, Q, J) pioneered early on by
Milton Work, later coupled with the Distribution point count for
short suits (3, 2, 1 points for void, singleton, doubleton, 5, 3, 1 pts
for the same short suits in a hand with a 4-card trump fit),
popularized by Charles Goren – a count universally adopted
because of its apparent simplicity and its relative overall accuracy.
But this claimed overall accuracy is neither sufficient nor
satisfactory, and not just for competitive tournament Bridge, and
several elements of the M. Work/C. Goren point count have been
challenged for some time – and justifiably so – by several
champions and experts. In particular, they challenged the value it
attributes to Aces (considered under-valued) and to Queens and
Jacks (considered over-valued, particularly when isolated, i.e.
when not in combination with other honors).
Furthemore, it also fails to adequately account for the appropriate
value of Fits and Misfits between two hands of the same side – to
mention just its most important weaknesses.

1
 GOREN’S POINT COUNT : FIRST CORRECTIONS

Fortunately, from now on, we will no longer have to remain

handicapped by the many, serious inaccuracies of the M. Work/
C. Goren point count as we only need to make a few corrections to
this count to obtain optimal accuracy in hand evaluation – as
illustrated hereafter.

1. FIRST CORRECTIONS TO MAKE TO GOREN’S POINT COUNT

Among the champions and experts having advocated, over the
years, several adjustments to make to the M. Work/C. Goren point
count, the French statistician J-R. Vernes has conducted the most
substantial statistical research identifying key specific adjustments
required (his major contribution in this area not being well known
because his work was not translated into English).
It is his statistical findings that will be used throughout this book
as the most scientifically reliable reference concerning hand
evaluation. Many of the adjustments advocated by M. Bergen,
L. Cohen, and D. Kleinman, to name only three prominent
American experts and champions who published books on the
topic of hand evaluation, correspond to J-R. Vernes’ conclusions in
several areas.
It is within the framework of his scientific research on competitive
bidding – which led him to conceive the Law of Total Tricks,
universally known today since the publication of his findings in
English, in 1992, by the American champion Larry Cohen – that
J-R. Vernes indicated, in 1966, in « Bridge Moderne de la Défense » /
Modern competitive bidding (my translation as the book was never
translated into English), that several corrections to the traditional
point count were in order, as follows :
A) Concerning the 4 3 2 1 honor point count :
– Aces, particularly in suit contracts, have a statistical value closer to
4 ½ points than to 4 points and, on the other hand, one point should be
deducted from a hand without a single Ace.
– 10s have some value, particularly at No Trump contracts, which can
be estimated at ½ point for two 10s.
– In suit contracts, secondary honors, Queens and Jacks, in suits other
than the trump suit, are worth half a point less than the 2 points and
1 point attributed to them by the 4 3 2 1 count.

2
 GOREN’S POINT COUNT : FIRST CORRECTIONS

– Any honor located in a short suit – singleton or doubleton – is worth

less than the 4 3 2 1 count attributed to them and should be downgraded
as follows :
A singleton honor is worth one point less than its normal value – and
this applies to Aces as well as they lose, when singletons, two of their
major attributes : enhancing other honors and acting as defensive
control (hand blockage).
Queens and Jacks, when located in a doubleton, are worth half a point
less than their normal value.
1 point should be deducted for two honors doubleton when one of these
honors is not a Jack (as A J is worth more than A x, and K J more than
K x).

B) Concerning the 3 2 1 Distribution point count :

– Long suits of 6 cards are underrated by one point : they are worth 2 pts
for length rather than 1 point only. And balanced 5 3 3 2 and 6 3 2 2
single-suit hands are worth, respectively, 1 point and 2 points for length
whether played in NT or in a suit contract.
Author’s note : In practice, these points for length should only be
counted when the 5 or 6-card suit is headed by at least Q J or K –
at least 3 honor pts – to avoid over-evaluating the suit length when
finding partner short in that suit.
This seems validated by John Thomas’ statistical analysis which
revealed, in 2001, that 68 % of borderline 3 NT contracts (24 H pts)
succeed when the hands have a 5-card suit headed by at least Q J
but only 10 % do when the 5-card suit has less than 3 H pts.
– 5 4 3 1 hands have an average distribution value of 2 pts but this is
only a theoretical statistical average as a value of 2 pts is never found in
any specific case : this distribution is, in fact, worth either 1 point for NT
contracts or 3 pts in suit contracts.

– Statistics show that the distributional value of a 4 4 3 2 hand is nil;

therefore, 1 distribution point should not be counted for a doubleton.
And, relative to a 4 4 3 2 distribution, 4 3 3 3 hands have a negative
distribution value – of at least half a point.
Author’s note : An important correction to this last comment is in
order : it is one whole point, and not just half a point, which must
be deducted for a 4 3 3 3 distribution.
– A singleton is always a weakness when playing No Trump and at least
1 point must be deducted when proposing a NT contract.

3
 GOREN’S POINT COUNT : FIRST CORRECTIONS

2. CORRECTION TO THE 5 3 1 COUNT FOR DISTRIBUTION-FIT POINTS

Last but not least, concerning distribution points in a hand with a

trump fit (8-card fit minimum) in partner’s suit, J-R. Vernes makes
the following comment in his 1966 book Modern competitive
bidding : « the proper count of distribution points by the hand having a
trump fit is probably the bidding issue treated in the most approximate
fashion. Statistics show that the distributional value of a short suit in a
hand with a trump fit varies depending upon the number of trumps held
by that hand as it is dependant on two variables : on one hand, the number
of cards in the short suit (void, singleton or doubleton) and, on the other
hand, the number of trumps (2, 3, 4 or 5) allowing one or several ruffs.
In practice, the Distribution-Fit point count rule should be the following :
the hand having fewer trumps (trump support) should count its
distribution points as the difference between the number of trumps held
by that hand and the number of cards it holds in its shortest suit ».
Applying this rule translates as follows for each specific case :
– In a hand with 2-card trump support : a doubleton counts for
0 point (2 - 2), a singleton for 1 point (2 - 1), a void for 2 pts (2 - 0).
– In a hand with 3-card trump support : a doubleton counts for
1 point (3 - 2), a singleton for 2 pts (3 - 1), a void for 3 pts (3 - 0).
– In a hand with 4-card trump support : a doubleton counts for
2 pts (4 - 2), a singleton for 3 pts (4 - 1), a void for 4 pts (4 - 0).

Caution : This calculation does not apply to a hand having

5 trump cards as these 5 cards would not constitute trump
support – they would be the long side of the suit, whether Partner
has 4 or 5 cards in that suit. Therefore, that hand should only count
the distribution pts that apply to an opening hand i.e. 2 pts for a
singleton, not the difference between 5 trumps and the short suit.
The following table recaps the above mentioned values :

Number of trumps in Fit support ……. 4 3 2

DISTRIBUTION-FIT POINTS Void .… 4 pts 3 pts 2 pts
Singleton .… 3 pts 2 pts 1 pt
Doubleton .… 2 pts 1 pt 0 pt

As the above table shows, the traditional 5 3 1 distribution point

count for, respectively, void, singleton, doubleton in a hand with
a 4-card trump support is not accurate : a doubleton in a hand with
4-card support is worth 2 pts, not 1. A significant difference, as many
examples in this book will show.
4
 GOREN’S POINT COUNT : FIRST CORRECTIONS

The traditional 5 3 1 distribution point count must be replaced by

the 4 3 2/ 3 2 1/ 2 1 0 count shown above. But there is no need to
memorize it : only count the difference between the number of
trumps and the number of cards in the support hand’s shortest suit.
Note : These Distribution-Fit points must be counted in
addition to the points attributed for long suits (1 point for a 5-card
suit, 2 pts for a 6-card suit, etc.), to the points attributed for fits
(covered in the next section) and to the points attributed for honors
in a fit.
Clearly, these conclusions were significant and most of them are
generally acknowledged by a majority of experts who factor them
in, in actual play, as pluses or minuses when evaluating their hand
– yet without quantifying them as precisely as above.
Actually, we continue to observe, far too often still, that more than
50 years after these conclusions were reached, most of them have
yet to be systematically taught or integrated. Perhaps, some may
argue, for “teaching simplicity” with regard to the everyday player
of intermediate skills ?
But this would hardly be credible in view of the ever-increasing
number of new, complex conventions that spring up everyday and
which are taught with abandon to these same players…
In reality, these undeniable conclusions are, generally, still not
reflected in the comments made by experts and champions in
Bridge books or magazines, and neither are they applied by
champions in actual bidding at the bridge table as numerous
examples in this book will show.
But before forging further ahead, we need to bring greater
precision to the first three conclusions reached by J-R. Vernes,
identified above, with regard to the values of Aces, Queens, Jacks
and tens.

3. THE TRUE VALUE OF ACES, QUEENS AND JACKS.

1. The value of Aces. Each Ace must, in fact, be counted in all
cases 4 ½ pts, not 4 pts, and this applies to all contracts, including
No Trump contracts.
That is a fact which has been largely documented and validated
for over 40 years now, starting with B. Charles’ statistical work
published in 1976, later confirmed, in the ‘80s, by an American
computer analysis. Both studies were conducted over thousands
of hands played in international championships and evaluating
tricks actually made at the table by Aces, Kings, Queens and Jacks.
5
 GOREN’S POINT COUNT : FIRST CORRECTIONS

Their findings were very similar and can be combined as follows :

At NT, the Ace makes, on average, 1.78 * tricks and the King 1.24
tricks. This, whatever the point scale used, gives the Ace a value
that is 44 % greater than that of the King. Relative to a King valued
at 3 pts, an Ace is therefore worth, on average, 4.3 pts at NT.
Author’s note : not only is 4.5 closer to 4.3 than 4 pts is, but the
discrepancy grows with more Aces : as 9 pts for 2 Aces is much closer to
8.6 pts (2 x 4.3) than 8 pts is and 13.5 pts for 3 Aces is much closer to
12.9 pts (3 x 4.3) than 12 pts is.
* The Ace and the King generate more than 1 trick each because of their
“control” feature, whether offensive or defensive. One example is the
typical defensive move, at a NT contract, to hold up the Ace until the
third round of a lead in a 5-card suit to cut communications between
opponents and prevent a subsequent run of that suit, thus avoiding the
loss of two tricks. Another is the orientation of a finesse, with, for
example, K 10 x opposite A J x, to prevent a return from the wrong side
in another suit guarded only by a King, thus avoiding losing more tricks.
Furthermore, the Ace has trick-generating value, beyond its trick-
taking value; for example : A x x x x facing K x will likely generate a
third trick – and possibly a fourth trick, as well (whenever the missing
six cards are distributed 3 – 3).
And the same trick-generating value applies to the King, as well;
for example, while Q J 10 x by itself may not generate any trick, facing
K x it will now produce three tricks.
This totally debunks E. Culbertson’s honor trick count which gives the
Ace credit for only one trick-taking value and the King credit for only
½ trick when by itself.
And in a suit contract, the Ace generates, on average (weighted
average between its trump value and its value in suits other than the
trump suit), 1.34 * tricks and the King 0.88 * tricks. the Ace then has
a value that is 54 % greater than that of the King. Thus, relative to
a King valued at 3 pts, an Ace is therefore worth, on average,
4.6 pts in a suit contract.
* In a suit contract, the value of honors is reduced by the distribution
values provided by short suits allowing ruffs, the eventual establishment
of small cards in long suits, etc.
Author’s note : here, the error of counting only 4 pts per Ace becomes
very serious as 9 pts for 2 Aces is much closer to 9.2 pts than 8 pts is, and
13.5 pts for 3 Aces is much closer to 13.8 pts than 12 pts is.

6
 GOREN’S POINT COUNT : FIRST CORRECTIONS

The conclusion is clear : the Ace is worth, on average, 50 % more

than the King and must therefore be counted 4 ½ pts, not 4 pts,
regardless of the hand distribution or the contract played, NT or
in a suit. And that is what the great American « 4 Aces » team had
understood very early and reflected in their 3 2 1 ½ count.
And for those who would rather avoid counting each Ace 4 ½ pts,
an alternative is to add 1 point for a second Ace and another for a
4th Ace i.e. two Aces = 9 pts, four Aces = 18 pts.
2. The value of Queens and Jacks. The conclusion on the lesser
value of Queens and Jacks for suit contracts has not been
translated in practice and what some experts advocate (add 1 point
to hands having no Queen or Jack, deduct 1 point from hands
having four or more Queens and/or Jacks) has generally not been
adopted – and rightly so as this is without statistical foundation.
In fact, the same statistical research referred to above establishes
the King and the Queen average values as, respectively, 3.15 pts
and 1.75 pts, which means that the King’s value is, on average,
70 % greater than that of the Queen. But, just like 2 pts is a purely
theoretical statistical average for 5 4 3 1 distributions, so are the
values of 1.75 and 0.75 pts for Queens and Jacks which, in practice,
are never found in any specific case !
Indeed, unlike Aces, which essentially keep the same value,
offensive or defensive, whether isolated or not, the value of
Queens and Jacks varies depending upon whether or not they are
accompanied by other honors.
And the fact that Queens and Jacks have a lesser value when
isolated has often been pointed out by experts and champions and
can easily be illustrated :
The honor combination : K Q x x x x will produce two tricks
whenever the Ace in that suit is located before K Q – that’s 50 % of
the time. While the combination : K x x Q x x will very rarely
produce more than one single trick.
The same applies to several other combinations, such as :
AQxx xxx compared to : Axxx Qxx
QJxx xxx compared to : Qxxx Jxx
AQJx xxx compared to : AJxx Qxx
AKJx xxx compared to : AKxx Jxx
A Q 10 x xxx compared to : A 10 x x Qxx
K Q 10 x xxx compared to : K 10 x x Qxx

7
 GOREN’S POINT COUNT : FIRST CORRECTIONS

The difference in value between isolated Queens and Jacks and

accompanied Queens and Jacks can be estimated to be, on average,
about about half a point. The Queen’s value therefore varies from
1.5 to 2 pts while the Jack’s value varies from 0.5 to 1 point.
This variation of half a point translates precisely the well known
“synergy” principle : Touching honors are worth more than
separated honors, and, particularly, the Q J x combination in the
same suit, doubles-up on the upgrade by 1 full point.
As a result, giving the Queen and Jack their average statistical
value – of 1.75 pts and 0.75 pts respectively – would not only be
totally impractical, it would be seriously erroneous and misguided.
Just as is giving them a fixed, constant value of 2 pts and 1 point
which makes the King worth only 50 % more than the Queen.
This has considerable practical consequences which many
examples in this book will illustrate.
The proper count is therefore : Queens and Jacks, are indeed worth
2 points and 1 point, respectively, when accompanied by another
honor, but are only worth 1 ½ and ½ points when isolated – and this
holds true whether playing NT or a suit contract !
And, here again, an alternative to counting in half-points is to
count 3 pts for 2 isolated Queens or 1 point for 2 isolated Jacks.

3. The value of tens. To attribute a value of ½ point for two tens

is another serious error – as the value of each ten varies considerably
depending upon whether it is isolated or accompanies a King, a
Queen or a Jack. This is very important and we will establish, later
on, the “true” value of each ten.

A preliminary summary, on the next page, recaps these initial

corrections to make to the traditional point count.

8
 GOREN’S POINT COUNT : FIRST CORRECTIONS

Preliminary summary of first corrections to GOREN’s point count

HONOR POINTS

Ace : 4 ½ pts K : 3 pts Q accompanied : 2 pts Q isolated : 1 ½ pt

J accompanied : 1 pt J isolated : ½ pt
No Ace : - 1 point (opening hands only)
Singleton Honor : - 1 pt
2 Honors doubleton : - 1 pt Q or J doubleton : - ½ pt
(A K, A Q, K Q, Q J) i.e. : Q x : 1 pt J x : 0 pt
Note : Do not deduct 1 point when 2 honors doubleton include
a J with A or K as A J or K J are worth more than A x or K x.

POINTS FOR LENGTH

5-card suit (with Q J or K minimum) : 1 pt 6-card suit : 2 pts

6-card suit without Q J or K : 1 pt

DISTRIBUTION POINTS AND DISTRIBUTION-FIT POINTS

Void : 3 pts Singleton : 2 pts Doubleton : 1 pt

4 4 3 2 distribution : 0 pt 4 3 3 3 distribution : - 1 pt
A singleton playing NT : - 1 pt
Distribution-Fit points with a trump Fit :

number of trumps : 4 3 2
Distribution-Fit points Void : 4 pts 3 pts 2 pts
Singleton : 3 pts 2 pts 1 pt
Doubleton : 2 pts 1 pt 0 pt

Let’s now see this corrected point count at work, with a few
examples from tournaments as reported by experts.

9
 CORRECTIONS TO GOREN’S POINT COUNT : EXAMPLE DEALS

1986 World championships.

In his 1995 book, « Evaluation of Bridge hands », J-R. Vernes makes
the following comment based on his analysis of deals played in
world championships from 1980 to 1989 :
« One is struck by the very high number of bad contracts reached (20 %),
often due to an inadequate hand evaluation ». He gives the following
example (1986), among many others, as illustration :
ª AJx ª xxx
© KJx © Qxxx
¨ AKxx ¨ Qxx
§ Axx § Jxx
At both tables, the bidding was :
2 NT (20/21 H pts) 3 NT
Down one. East bids 3 NT as he counts his hand for 5 points.
But East’s hand is only worth 2 ½ pts, not 5 : 3 pts for its two
isolated Queens, ½ point for its isolated Jack, minus 1 point for its
4 3 3 3 distribution. And another point could be deducted for
having no Ace and no King. East should have passed on 2 NT.

Topic : « Technique ». Le Bridgeur, 2008.

ª Qx ª Kxx
© Axxx © QJx
¨ AKJ ¨ Qxx
§ xxxx § QJxx
The bidding was : 1§ 2 NT (11/12 H)
3 NT
Comment : « A perfectly normal contract – but, still, a very bad one… ».
In fact, far from having 11 pts, East only has 9 ½ pts. The isolated
¨ Q is only worth one and a half point, not 2, and one point must
be deducted for East’s 4 3 3 3 distribution.
The proper count therefore calls for a 1 NT response, limited to
10 pts, not 2 NT. On which West would then pass as he only has
14 pts (only one point for his doubleton ª Q).

10
 CORRECTIONS TO GOREN’S POINT COUNT : EXAMPLE DEALS

National selection, Teams, 2008.

ª Jxx ª Axx
© Qxx © Kxx
¨ AKxx ¨ Qxx
§ KJx § Qxxx
The bidding was often : 1 ¨ or 1 NT (12/14) 2 NT (11/12)
3 NT
East, counting his hand for 11 pts, proposes to play 3 NT to an
opener having a maximum hand of 14 pts.
But, far from having 11 pts, East only has 9 ½ pts, with two isolated
Queens and a 4 3 3 3 distribution. It is 1 NT (or pass on a weak
1 NT opening) which should be bid, not 2 NT. West further
contributes to overreaching to 3 NT by counting his own hand for
14 pts when it only has 12 ½ pts: isolated Queen and Jack and
4 3 3 3 distribution.
Confirmation that 4 3 3 3 distributions are handicaps, in NT or in
suit contracts, and must always be downgraded by one full point.

Regional tournament, (U.K.), 2010.

ª AKx ª Jxx
© Jxx © Axx
¨ Qxx ¨ Axxx
§ Qxxx § Axx
The bidding was often : 1 NT (12/14 H) 3 NT
A perfectly normal 3 NT bid from East with its 13 pts (4 3 3 3),
facing an opener guaranteeing a minimum of 12 H pts.
But West only has 10 pts ! 4 3 3 3 distribution (- 1) and the isolated
Queens and Jack are worth a total of 3 ½ pts, not 5. The bidding
should have been :
Pass 1 NT (12/14 H)
Pass

European Open, 2011.

ª AQx ª K
© AKxx © J 10 x x
¨ 10 x x x ¨ KJx
§ Ax § xxxxx

11
 CORRECTIONS TO GOREN’S POINT COUNT : EXAMPLE DEALS

In both rooms, West opened 1 NT - which was passed by one team.

But West has 18 ½ pts (3 Aces at 4 ½ pts each), not 17, and therefore
should not open 1 NT. On a 1 ¨ (or a strong Club) opening, finding
the game in hearts should not be difficult :
1¨ 1©
3 © (17/19 HDS) 4©
East has 11 HDFit pts (2 pts for the ª K singleton + 3 pts for the
ª singleton with 4 trumps + 1 point for the Jack in the trump suit)
+ 17 HLDF minimum in the West hand = 28 pts.
One last comment to conclude this segment : how many times
have we read, in Bridge books, references made to “strong” 17 pts
or “weak” 15 pts such as : « Do not open 1 NT, 15/17, the following
two hands » :
ª Qxx ª AKx
© KQx © QJx
¨ QJx ¨ Kx
§ KQxx § Axxxx
« The first one is « a very poor 15 points » (no Ace, lots of Queens and
Jacks), the second is « a very strong 17 points » (lots of Aces and Kings)”.
Unfortunately, such approximations are not accurately quantified
and only serve to perpetuate the perception that only experts are
capable of properly assessing the true value of their hands – when
the reality is that “expert judgment” is not needed to assess strong
or weak hands : an accurate point count will do just that !
The first hand is worth 12 ½ pts (minus 1 point for being 4 3 3 3,
minus 1 point for no Ace, and 1 ½ point for the isolated ª Q), and
the second one is worth 19 pts (9 pts for 2 Aces + 1 for the 5 clubs).
Opening either hand 1 NT, 15/17, won’t even cross your mind…
Actually, even the greatest champions seem to have problems
judging their hands properly – as illustrated by the following
hand, from the final of a 2012 national open, teams, opened 1 NT,
12/14, by the Italian pair Nunes and Fantoni, multiple world
champions :
ª Axx
© xxx
¨ Ax
§ KQJxx

12
 CORRECTIONS TO GOREN’S POINT COUNT : EXAMPLE DEALS

This hand should be counted for 16 HL pts, not 14 H pts.

Wouldn’t you pass on such a weak 1NT opening with the following
hand ? Instead of playing 3 NT :
ª Kxx
© xxx
¨ K 10 x x
§ Axx
These few examples show that to obtain a point count a little more
precise, and more accurate, it would be helpful to start by applying
to the 4 3 2 1 point count the very basic corrections identified as far
back as 1966 and accept to count in half-points – which, unfortu-
nately, neither J-R. Vernes himself nor the majority of experts have
seen fit to adopt and advocate just to keep the point count
as “simple” as possible.
Perhaps as a compassionate gesture toward the average or
intermediate players ?! The same players to whom experts have
tried to teach a multitude of complex conventions intended to
compensate for the appalling lack of precision of most bidding
systems played and advocated.
Ironic – or rather a very regrettable error of judgment which has
only resulted in delaying by several decades the precise evaluation
of Bridge hands. How unfortunate.

13
14
 FIT POINTS FOR 9 OR MORE TRUMPS

3. YEAR 1995 : FIT POINTS FOR 9 OR MORE TRUMPS

Beyond his initial statistical analysis and conclusions published in

1966, J-R. Vernes was asked by the French Bridge Federation, in
the early ‘90s, to assess the statistical accuracy of the point count
generally advocated and taught.
In collaboration with Bernard Charles, mathematician and
statistician whose contribution was essential, they jointly
published in 1995 the results of their extensive research in a book
titled « L’évaluation des mains au Bridge » / Evaluation of Bridge Hands
(my translation as the book was never translated into English).
The new conclusions they reached had to do, this time, with
distributional values related to the total number of trumps held by
the same side – and they were of major importance, not only for
their new discoveries but also, as we will see later, for some key
conclusions they did not reach.
Their most important finding was revolutionary :
« When attributing points to short suits (void, singleton, doubleton), the
authors of the 3 2 1 count for opening hands, and 5 3 1 count for hands
with trump support, focused solely on “side” suits other than the trump
suit. But statistical analysis reveals that another factor is just as
important : the total number of trumps held between the two hands of
a partnership – and the points that should be attributed to this factor
must be added to those counted for short suits ».
And they concluded from their analysis that 2 points should be
added for a 9-card trump fit between two hands and 3 points
should be added for a fit of 10 trumps or more.
Author’s note / reminder : This does not change the need to add
1 point for an honor in the trump suit whenever its initial value
does not exceed 3 pts ( J x x, J 10 x, Q x x, Q 10 x, Q J x, K x x ).
Yet, years after these conclusions were reached, they have yet to
be systematically taught or even integrated in hand evaluation
practices as reflected in most articles or books dealing with this
topic. Illustrations :
1999 Cap Gemini world’s top 16 pairs tournament.
ª KQxxx ª AJxx
© Axxx © Kx
¨ A ¨ xxxx
§ xxx § xxx
Only half the field reached the 4 ª game.

15
 FIT POINTS FOR 9 OR MORE TRUMPS

Most other pairs bid : 1 ª 2ª

Pass
We know now that the error comes from East who should count
his hand for 12 ½ HDFit pts (+ 2 for the 9-card trump fit + 2 for the
doubleton with 4 trumps), certainly not 9 HDS pts. He should
therefore bid 3 ª, not 2 ª. West, with his 16 HLD pts, will then
easily conclude in 4 ª.

2004 Olympiads – France / Brazil

ª AKx ª xx
© Kxx © AQxx
¨ xx ¨ Axx
§ Axxxx § KQxx
The French team plays 3 NT after the following auction :
1§ 1©
1 NT (12/14) 3 NT
And misses the 6 § slam. With 15 H pts, and without a Major-suit
Fit, East did not “see” a slam with a partner limited to 14 H pts.
But West has 16 HL pts, not 14, and should have opened 1 NT,
15/17. With 16 H pts (9 pts for 2 Aces), East should then have no
problem considering a slam should a Fit exist between them – and
this could be discovered quickly with a bidding start such as :
1 NT (15/17) 2 ¨ (forcing Stayman)
3 § (5 clubs, no 4-card Mjr)
Once West’s 5 § 3 3 2 distribution is known, East can add to his 16
pts : 2 pts for the 9-card § Fit + 2 pts for the ª doubleton with
4 trumps – that’s 20 HLDFit pts + 15 HL minimum = 35 pts.
Up to the bidding system now to identify all Key cards of the side.

1995 Cap Volmac world’s top 16 pairs tournament.

ª A ª xx
© KQ © A 10 x x x
¨ xxxx ¨ AKQxx
§ AJxxxx § x
Only one pair out of 8 (13 % !) managed to reach 7 ¨ after this start :

16
 FIT POINTS FOR 9 OR MORE TRUMPS

1§ 1¨
2¨
And only 3 of the remaining 7 pairs (43 %) reached 6 ¨. A clear
case of deficient point count.
With the ¨ Fit, West must now add 1 point for the ª singleton in
a hand with 4 trumps and count his hand for 18 HLDFit points.
He must therefore rebid 3¨, not 2¨. East would then know
immediately that his side is in slam zone : he has 16 HL pts (9 pts
for 2 Aces, 1 point for each 5-card suit) to which he must add 2 pts
for the 9-card ¨ Fit = 18 pts + 17 HLDS pts minimum = 35 pts
minimum. A sound bidding system should now be able to get to
7 ¨ after inquiring about partner’s Key cards.

One last comment on the subject of precision :

while it is undeniable that counting in half points provides
significantly greater precision and accuracy, a legitimate question
would be : how easy is it to learn and to master ?
Actually, it is a lot easier than one might initially think. Indeed,
while complex, multiple conventions can significantly tax the
memory, particularly when the frequency of their use is limited to
a few, specific cases, the point count is continuously practiced, on
every single hand held – regardless of its strength or distribution
– and quickly becomes reasonably easy to master, second nature
once sufficiently practiced.
You will have plenty of opportunities, reading on, to practice it
yourself, if you wish, and be your own judge of it.

17
18
 OPTIMAL HAND EVALUATION

CHAPTER 2

2015 : New Findings and Further Corrections

1. POINTS FOR SUIT LENGTH AND FOR SHORT SUITS

Already in 1966, J-R. Vernes identified in Modern competitive
bidding some important corrections to make to the Distribution
point count. Specifically, he wrote in that book : « 6-card suits are
underrated by at least one point : a 6 3 2 2 hand is worth 3 D pts, not 2,
a 6 4 2 1 hand is worth 4 D pts, not 3, and a 7 2 2 2 hand is worth 5 D
pts. Also, 5 3 3 2 and 6 3 2 2 hands have a distributional value, at a No
Trump contract, of, respectively, 1 point and 2 pts – the long suit often
allowing extra tricks to be developed at NT ».
« Another point : both 5 4 2 2 and 5 4 3 1 hands are traditionally given
the same distributional value of 2 D pts. Yet, the 2 pts given to 5 4 3 1
hands is a theoretical average value which doesn’t apply to any
specific case ! In actual fact, statistics show that 5 4 3 1 hands are worth
3 pts at a suit contract and 1 point only at No Trump ».
Unfortunately, J-R. Vernes did not draw from these observations
the appropriate conclusions on the specific point count which
corresponds precisely to these statistical findings. They should
have been as follows, based on a very simple principle : points
attributed for suit Length must be differentiated from points given
for short suits, as follows :
– Points for suit Length should be attributed as follows : 1 point
fo a 5-card suit – if headed by at least 3 H pts, that is at least K or
Q J, 2 points for a 6-card suit – also if headed by at least K or Q J,
plus 2 pts for each card beyond the 6th card.

– Distribution points for short suits should be attributed as

follows : 2 D points for a singleton, but no point for one doubleton
only. It takes two doubletons for 1 D point.

– As for a void, it must be counted 4 points, not 3 : in their 1995

book, Evaluation of Bridge hands, the authors showed statistical
analysis revealing that, in all cases, a void is worth either 4 or 4 ½
D pts – not less (except when in a trump support hand of less than 4
trumps). Rather than the 3 D pts universally attributed to it since
the beginning of time !
19
 2015 : NEW FINDINGS AND FURTHER CORRECTIONS

The conclusion to this analysis on distribution points for short

suits – as it applies to openers’ hands, not to responders’ hands –
is fundamental : the traditional 3 2 1 distribution point count, for,
respectively, void, singleton and doubleton, is totally erroneous
and must be replaced by the 4 2 0 1 count for : void, singleton, one
doubleton, two doubletons.
And these distribution points for short suits must be added to the
points counted for suit Length.
This point count now totally eliminates the traditional point count
problem which wrongly attributes the same value of 1 D point to
4 4 3 2 and 5 3 3 2 hands, 2 D pts to 5 4 2 2 and 5 4 3 1 hands, 3 D
pts to 6 3 2 2 and 6 3 3 1 hands, etc.
The table below shows the appropriate points to attribute to each
distribution, whether L pts (for Length) or D pts (for Distribution).
The numbers are in bold when they differ from Goren’s HD count.

Balanced L pts D pts L+D pts Single-suit L pts D pts L+D pts
distribution distribution
4333 0 -1 -1
4432 0 0 0
5332 1 0 1
5422 1 1 2
6322 2 1 3
7222 4 1 5
Uneven (3 doubletons = 1 pt)
distribution
4441 0 2 2
5431 1 2 3
5521 2 2 4
6421 2 2 4 6331 2 2 4
5440 1 4 5
5530 2 4 6
6430 2 4 6 7321 4 2 6
7330 4 4 8

Note : The points attributed to 5 and 6-card suits apply to suits

headed by at least Q J or K (at least 3 H pts).
Adding L and D points to assess the true total value of a hand is a
major and fundamental departure from the traditional point count
and, therefore, warrants some analysis, thereafter.

20
 2015 : NEW FINDINGS AND FURTHER CORRECTIONS

From the beginning of Contract Bridge, the practice has been to

count initial distribution points either for long suits or for short
suits, but not for both – and this has been a very serious mistake
which has prevented hand evaluation from being accurate.
1. Counting points for long suits has consisted of counting 1 point
for each card beyond the fourth card of any suit – but this method
does not differentiate between 4 3 3 3, 4 4 3 2 and 4 4 4 1 hands, or
between 5 3 3 2, 5 4 2 2 and 5 4 3 1 hands, etc. A seriously flawed
proposition.
2. Counting points for short suits, an alternative distribution point
count adopted and popularized by Charles Goren since 1948, has
consisted of counting 1 point for each doubleton, 2 pts for each
singleton and 3 pts for each void – but this method does not
differentiate between 4 4 3 2 and 5 3 3 2 hands, or between 5 4 2 2
and 5 4 3 1 hands, or between 6 3 2 2 and 6 3 3 1 hands, etc.
Clearly a proposition just as flawed as the long suit count.
3. To address these flaws, Alan Truscott proposed, in the seventies,
a point count combining counting points for long suits and points
for short suits – called “Assets” – valuing a suit of 5 cards or more
as 1 Asset, a singleton as 1 Asset and a void as 2 Assets. But this
method does not differentiate between 4 3 3 3 and 4 4 3 2 hands, or
between 5 3 3 2, 5 4 2 2, 6 3 2 2 and 7 2 2 2 hands, or between
5 4 3 1, 6 3 3 1 and 7 3 2 1 hands, etc. Yet again, a proposition just
as flawed as the other two.
4. More recently, in 2003, the Canadian statistician Zar Petkov
presented an evaluation point count method counting, in one
single count, both long suits and short suits and applying this
count to all hands, starting with opening hands.
But this count is also inappropriate – for a detailed analysis of why,
see Note # 1, in the Index at the very end of this book.
It seems appropriate to conclude this section by testing whether
our corrected optimal count would account for the “right” points
for the imaginary hand of 13 cards in the same suit which no hand
evaluation method seems to have been able to properly quantify :
ª A K Q J 10 x x x x x x x x © --- ¨ --- § ---
Let’s see : 10 ½ H pts + 16 L pts for length + 12 D pts for 3 voids =
38 ½ HLD pts – The points for a Grand slam !!
Finally, an evaluation point count for Bridge hands which does
quantify properly this famous hand distribution !

21
 2015 : NEW FINDINGS AND FURTHER CORRECTIONS

2. COMBINATION OF HONORS AND THE TRUE VALUE OF TENS.

2.1 COMBINATION OF HONORS
J-R. Vernes states, in Modern competitive bidding, and confirms, 30
years later, in Evaluation of Bridge hands, that « contrary to popular
belief, the combination of honors in the same suit of one hand does not
represent a significant “plus” compared to the same honors in different
suits, and this is true for combinations of 2 or 3 honors ».
But while this is true for combinations of two or three honors in
suits of 3 or 4 cards, this does not hold true for combinations of
3 honors in long suits of 5 + cards, particularly for suits of 6 + cards
which do give this combination a significantly greater value.
Here is why : a 6-card suit will find a Fit of 8 cards or more with
partner 75 % of the time – a suit that will often become the trump
suit – which will enable it to generate, most often, 5 tricks in that
suit, at NT or in a suit contract. And this is, of course, even more
so in the case of three honors in a 7-card suit which will find a Fit
of 8 cards or more with partner… 93 % of the time !
Indeed, the combination of three honors in a long suit is signi-
ficantly under-valued by the traditional point count – and that is one
of the reasons which have motivated many experts and champions
to evaluate such hands in number of tricks rather than in points.

Conclusion : 2 additional points must imperatively be counted for

a combination of 3 honors in a suit of 6 cards or more. Illustration :
K Q J x x x is counted, in traditional point count, for 6 H pts
+ 1 point for the 6 cards = 7 pts.
– Opposite a doubleton in that suit in partner’s hand, this
combination will, most often, generate 5 tricks, and almost always
no less than 4 tricks (with the 5 missing cards distributed 4 – 1).
But 4 tricks are worth considerably more than just 7 pts – they are
worth over 11 pts (27 pts for 10 tricks in a suit or 26 pts for 9 tricks
at NT = average of 2.8 pts per trick x 4 = 11.2 pts).
And the corrected optimal count will credit this hand with 10 pts :
6 H pts + 2 pts for 6 cards + 2 pts for 3 honors in a 6-card suit.
– Opposite a tripleton in partner’s hand, this combination will
generate, almost all the time, 5 tricks, worth 13.5 pts. The corrected
optimal count is : 6 H pts + 2 pts for 6 cards + 2 point for 3 honors
in a 6-card suit + 2 pts for a 9-card Fit = 12 pts.

22
 2015 : NEW FINDINGS AND FURTHER CORRECTIONS

And if this combination finds a singleton in that suit in partner’s

hand (19 % probability, less than the 27 % probability of finding a
tripleton), it will still generate, most often, 4 tricks. Furthermore, if
the contract is played in a trump other than that suit, a single ruff
will, most often, be enough for that suit to generate 5 tricks.
The same calculation applies to A K J x x x, to A K Q x x x and to
A Q J x x x and adding 2 points for 3 honors in a suit of 6 cards or
more is applicable to all combinations of 3 honors.
For similar reasons, a combination of 3 honors in a 5-card suit calls
for adding 1 point for it – a 5-card suit will find a Fit of 8 cards or
more with partner 55 % of the time.
Interestingly, very few references can be found in Bridge literature
about precisely quantifying the specific combination of 3 honors
in suits of 5 cards or more. M. Bergen and D. Kleinman are among
the very few who do factor it in. M. Bergen, in particular, refers to
3 honors in a 6-card suit as a “great suit” and adds 2 pts for them.

2.2 THE TRUE VALUE OF TENs

In 1966, J-R. Vernes wrote that two 10s in the same hand were
worth ½ point. 30 years later, to avoid counting in half-points,
J-R. Vernes and B. Charles, considering that a hand includes, on
average, one 10, advocate instead to add one point for three 10s in
the same hand.
This is a clear case where thinking strictly in terms of statistical
average can lead to a misguided analysis and an even worse
recommendation – with major consequences, as several examples
will illustrate in this book’s section.
In reality, the value of 10s varies considerably depending upon
whether a 10 is totally isolated or combined with an Ace (value close
to zero point), combined with a King (value of ½ point), or combined
with a Queen or a Jack (value of one full point). Consequently,
attributing a value to 10s only when 3 or more are in the same hand,
is just as erroneous as giving 10s a constant value, be it ½ point each
or ½ point for two 10s.
An error, moreover, considerably amplified by the 6 4 3 2 1 and
5 4 3 2 1 point counts which give each and every 10 a constant value
of one full point.
As it happens – and this may surprise many – attributing to 10s
their appropriate value is of critical importance and we will
therefore, hereafter, analyse this particular subject in great detail.

23
 THE TRUE VALUE OF TENs

A) An isolated 10 is, in a large majority of cases, of very little value.

For example, 10 x x opposite x, or x x, or x x x is of no value
whatsoever. And this does not improve (other than having very
limited defensive value) opposite J, or J x, or J x x or Q, Q x, or
Q x x or K, K x, or K x x or A, A x, or A x x or A Q x or A K x.
It only has some value, mostly defensive, opposite Q J, Q J x, K J,
K J x, A J, A J x and K Q x.
In general, an isolated 10 can be estimated to have a value much
closer to zero than to ½ point.

B) A 10 which accompanies an Ace does not add much value. The

following combinations illustrate this :
A 10 x xxx The 10 adds little to no value.
A 10 x Jxx If they avoid leading that suit, the opponents
might make 2 tricks. The 10 adds some value.
A 10 x Qxx If they avoid leading that suit, the opponents
might make 2 tricks. The 10 adds some value
when it generates a 2nd trick (low to the Q
and then low to the 10).
A 10 x Kxx The 10 will usually not generate a third trick.
A 10 x QJx The 10 here allows finessing the K. Succeeding
half the time, it is worth half a trick = 1.4 pts.
A 10 x KJx The 10 allows here finessing the Q from either
hand. It can be attributed a value of 2 pts.
A 10 x KQx The 10 will not generate an additional trick.
A 10 x x KQx The 10 allows here finessing the J or getting
an extra trick when catching the J doubleton.
It is worth more than half a trick = about 2 pts.
A 10 x x KQJ The 10 will not generate an additional trick.
Without going through all possible combinations, it can be seen
that, in most cases, a 10 combined with an Ace does not add much
value, if any. Its value, like when it is isolated, can be estimated to
be closer to zero than to ½ point.
C) A 10 which accompanies a King brings some added value that
can be evaluated, on average, half a point.
K 10 x xxx The 10 adds little value (it will only produce a
trick with Q J x under the K and the A over K)
K 10 x Jxx The opponents will most often make 2 tricks.
The 10 adds little value.
K 10 x Qxx The 10 allows here finessing the J. Succeeding
half the time, it is worth half a trick = 1.4 pts.

24
 THE TRUE VALUE OF TENs

K 10 x Axx The 10 will usually not generate a third trick.

K 10 x QJx The 10 will not generate an additional trick.
K 10 x AJx The 10 allows here finessing the Q from either
hand. It can be attributed a value of 2 pts.
K 10 x AQx The 10 will generate no additional trick.
K 10 x AQJ The 10 will generate no additional trick.
K 10 x x Q J x The 10 will generate one additional trick. But
so would K x x x on a 3 – 3 break (36 % of the
time). The 10 is worth 2/3rd of a trick = 2 pts.
K 10 x x A J x The 10 allows here finessing the Q from either
hand. It can be attributed a value of 2 pts.
K 10 x x A Q x Same as A 10 x x K Q x seen above. The 10
is worth about 2 pts.
K 10 x x A Q J The 10 will generate no additional trick.
For additional examples, deducting a card or adding a 4th card to
either side will rarely alter the above.
When it accompanies a King, a 10 provides some value, even
though rather marginal – It can be attributed, on average, a value
of half a point, at most.
And this confirms the well known principle of “separation” : « The
greater the separation between an honor and a lesser card, the less
effective the lesser card is ».
And this is further confirmed by the fact that the value of a 10
becomes significant when it accompanies a Queen or a Jack. For a
detailed analysis of this, see Note # 2, in the Index at the end of this book.
E) A 10 which accompanies a Jack provides even more significant
value than it brings to the Queen. It should be evaluated, on
average, one full point – and, furthermore, its combination with
a Jack adds value to the Jack which is then worth one full point,
whether or not the Jack is accompanied by a top honor – the J 10
combination is therefore worth 2 points, in all cases.
Here again, just like for Q J x, we find the “synergy” principle at
play : « Touching honors are worth more than separated honors ».
And the J 10 combination is particularly effective when combined
with a Queen. Illustration :
The following West and East hands – all in Aces and Kings – will
only produce 7 to 8 tricks despite their 25 pts between them (we
will see later why these two hands should be counted for less than 25 pts)

25
 THE TRUE VALUE OF TENs

ª Axxx ª Kx
© Axx © Kxx
¨ Kxx ¨ Axxx
§ Axx § xxxx
But if we replace two of their Aces by two combinations of Q J 10
and one J 10 combination – still only 7 pts in traditional point
count, in lieu of the 8 pts attributed to 2 Aces – the two hands now
produce 9 tricks :
ª Q J 10 x ª Kx
© Axx © Kxx
¨ Kxx ¨ Q J 10 x
§ Axx § J 10 x x
Indeed, the Q J 10 combination generates more tricks than a King
or an Ace and that is why it should be valued at 4 pts, not just 3.
One point more than a King, yet not as many points as an Ace
(4 ½ pts) as an Ace provides a “control” feature, offensive and
defensive, that neither the King nor the Q J 10 combination have.

In focusing solely on the relative value of the top 4 honors, the

4 3 2 1 and 3 2 1 ½ point counts totally missed this key specificity !
Just as did those advocating to count half a point for one 10, or for
two 10s. A small revolution, indeed !
And counting only 4 pts for an Ace, 3 pts only for Q J 10 and
1 point only for J 10 in the hand shown above will not reach the 25
pts needed for 3 NT, particularly if 1 point is deducted for West’s
4 3 3 3 distribution, or with the following ones :
ª A J 10 x ª xx
© Axx © Kxx
¨ K 10 x ¨ QJxx
§ Qxx § K J 10 x
When, in both cases above, a 15/17 1NT opening by West (9 pts
for two Aces, minus 1 point for 4 3 3 3 + 2 pts for J 10) would allow
East to bid 3 NT, with 12 pts for the first East, 11 pts for the second
East – assuming that the two West hands above are counted for
15 pts, not 14 !

26
 THE TRUE VALUE OF TENs

And how about the two hands below, without a single Ace :
ª K Q 10 x ª xxx
© Kx © Q J 10 x
¨ Kxx ¨ Q J 10 x
§ Q J 10 x § Kx
Unless 10s are counted for their full value of 4 pts in the above
deal, you won’t get to the 25 pts needed to bid 3 NT, particularly
if the opening hand correctly deducts 1 point for an Aceless hand.
But the appropriate count should be : 15 pts in West (- 1 for no Ace)
+ 11 pts in East for a total of 26 pts, which will allow the side to bid
3 NT, with or without a 2 § stayman response.
Let’s now look at a few example hands bid in tournaments by
world-class players, illustrating the appropriate value of 10s :
Rosenblum Cup (USA), 2006.
ª Jx ª Q 10 x
© xxxx © Ax
¨ AKQx ¨ xxx
§ Axx § K J 10 x x
Only one pair found the 3 NT game, on a 14/16 H 1 NT opening.
On 13/15 H 1NT openings, East players often invited by 2 NT,
invariably passed by openers, and on 12/14 H 1NT openings, East
players invariably passed. The majority of pairs opened 1¨ and
ended up in 1 NT or 3 §.
But East has 13 HL pts, not 10 H pts, and he should bid 3 NT
directly over any 1NT opening! But that requires counting its
points “right” : 1 L point for the 5 clubs and 1 point for each 10.
That is not to say, of course, that you need to have Jacks and 10s
for 3 NT games. Thus, had the two hands above been, for example :
ª xx ª Ax
© xxxx © Jxx
¨ AKQx ¨ xxx
§ Axx § KQxxx
The bidding should be :
1 NT (12/14 H) 2 NT (11/12 HL)
3 NT (14 H, not 13)

27
 THE TRUE VALUE OF TENs

Regional tournament, 2007.

ª Q 10 x x ª Kxx
© Qx © K J 10 x
¨ Axx ¨ Q J 10 x
§ KQJx § xx
On a 12/14 1NT opening by West, over half of East players passed,
counting their hand for 10 pts only, made up mostly of Jacks and
10s – factors generally considered discouraging by most players.
When, in fact, East has 12 pts (2 pts for each J 10) and should invite
West to game. West, having 14 pts (1 point only for the © Q dou-
bleton but 2 ½ pts for Q 10 of ª) should bid 3 NT.

European championships, 2008.

ª Ax ª Kxx
© AKxxx © Qx
¨ Axxx ¨ J 10 x
§ xx § K J 10 x x
About half the field stopped in 2 © after the following auction :
1© 1 NT
2¨ 2©
East’s “timidity” is not so surprising considering that many
players do not count 1 point for a 5-card suit and require 11 pts,
not just 10, to bid a new suit at the level of two. But East has 12 HL
pts, counting 2 pts for each J 10, + 1 point for 5 clubs. He should
bid 2 §, not 1 NT, and then 2 NT, over which West will bid 3 NT.

Final of European championships, teams, 2008.

ª AKx ª Q 10 x x
© K J 10 x © xxx
¨ xxx ¨ J 10 x x
§ AQx § Kx
Only two teams out of 18 (11 %) found the (winning) 3 NT contract.
East players, counting their hand for 6 pts, passed most often on
West’s 15/17 1NT opening. A case of cumulative errors…

28
 THE TRUE VALUE OF TENs

As West has 18 pts, not 17 : 9 pts for two Aces + 5 pts in © (2 pts
for J 10) minus 1 for 4 3 3 3 = 18. He should therefore have opened
1 § instead of 1 NT. After East’s initial response (be it 1 ¨ or 1 ª),
West’s jump-rebid of 2 NT will indicate 18/19 pts on which East,
with 7 ½ pts (2 pts for J 10, 2 ½ pts for Q 10) can now bid 3 NT.
Thus, the total value of several 10s can vary from zero to 3 or 4 pts !
depending upon which honor they accompany. A small revolu-
tion, indeed. Time to come out of the dark ages on this subject!

These examples clearly illustrate how wrong the traditional point

count is concerning the proper valuation of 10s, whether it advo-
cates to count half a point for one 10 or for two 10s or 1 point for
three or four 10s in the same hand – an absurd recommendation
which counts no point for any of the 10s in the five examples above
where they are worth either 2 pts or 3 pts !
On the two topics we just studied – dealing, first, with the
combination of three honors in a suit of 6 cards or more and, then,
with the number of points to be attributed to 10s – the conclusions
and recommendations advocated by J-R. Vernes and B. Charles
were clearly misguided and inappropriate.
Actually, one can also wonder how they failed to assess the proper
value of the specific A J 10 x combination which will generate two
tricks 75 % of the time (double finesse, with adequate entries) –
thus better than A Q x x which will generate two tricks only 50 %
of the time. Yet, the Queen is valued 2 pts but J 10 only 1 point ?
And this under-evaluation is even more pronounced when A J 10
is found in a suit of 5 cards or more, a combination particularly
“massacred” by the traditional point count which : A) counts the
Ace 4 pts, instead of 4 ½, B) does not count 1 point for a 5-card suit,
and C) counts 1 point only for J 10 !
Yet, with just a doubleton in partner’s hand (30 % of the time), this
combination will produce 3 to 4 tricks 75 % of the time (double
finesse) and will produce, most often, no less than 4 tricks when
partner has one of the two missing honors – or better yet, both, or
three cards or more in the suit (55 % of the time). Illustration :
A J 10 x x opposite x x = 3 + tricks = 8 + pts. 7 ½ optimal pts
– certainly not 5 pts only !
A J 10 x x opposite Q x = 4 + tricks = 10 ½ pts. 8 ½ optimal pts
– certainly not 7 pts only !
A J 10 x x opposite K x = 4 + tricks = 10 ½ pts. 10 ½ optimal pts
– certainly not 8 pts only !

29
 THE TRUE VALUE OF TENs

Curiously, little mention is made in Bridge literature of the under-

valuation of this specific combination. M. Bergen is one of the few
writers to attribute one additional point for a suit of 4 cards or
more with 3 of the top 5 cards, including the 10 – which he refers
to as a “quality suit”.
And a 9 which accompanies a 10 in that configuration : A J 10 9 x
should also be counted for 1 point as it is worth just as much as the
Jack or the 10 here and increases the hand’s trick-taking potential.
One other example will illustrate the importance of quantifying
precisely and correctly several elements of the corrected point
count outlined so far, including the “true” value of 10s :
An expert gave the two opening hands below as an example to
illustrate the need to exercise “careful judgment” in assessing the
pluses and minuses of a hand :
ª KQx ª KQx
© KQx © J 10 x
¨ Qxxx ¨ A J 10 x x
§ Kxx § Kx
With the following comments : « The first hand was opened 1 NT
15/17 H and the second 1NT 12/14 H. With the following hand of
10 H pts :
ª Jxx
© Axx
¨ Kxx
§ Qxxx
Partner will bid 3 NT on the first 1 NT opening of 15/17 H pts and will
Pass on the second 1 NT opening of 12/14 H pts. Yet, these bids should
be reversed to obtain the right outcome. But the responder is not at fault :
as the first hand has no Ace, no 5-card suit and its ¨ Q is isolated : the
hand is barely worth 14 pts, while the second hand has a 5-card suit and
some productive intermediate cards (10s) : it is worth close to 16 pts ».
Unfortunately, such judgments are too approximate to be helpful,
particularly since a precise quantification reveals that the
commentator’s point count is actually quite inaccurate : as the first
hand has 12 ½ pts, not “barely 14 pts” (- 1 point for no Ace and - 1
point for 4 3 3 3 distribution), and the second has 17 ½ pts, not
“close to 16 pts” (1 point for the 5-card suit and 2 pts for each J 10).

30
 THE TRUE VALUE OF TENs

And responder’s hand has 8 ½ pts, not 10 (- 1 for 4 3 3 3 distribution

and 2 pts only in total for the isolated Queen and Jack). He should
therefore only invite to game with 2 NT on the 15/17 1NT opening
by the second hand. The opener will then conclude to 3 NT as he
has 17 ½ pts, not “close to 16 pts”.
Isn’t it time to use a rigourous and precise quantification instead
of approximate judgment ?

And counting 1 point for each 10 which accompanies a Queen or

a Jack applies to every hand, regardless of its distribution or
contract bid, whether NT or in a suit. As, in cases of uneven
distribution, a J 10 or Q 10 opposite a singleton in that suit in
partner’s hand does not make for more “wasted honor pts” than
Q J x opposite a singleton, while, when they are not opposite a
singleton, J 10 or Q 10 are just as valuable in a suit contract as they
are in a NT contract. A simple example will illustrate this :
Opener Responder A Responder B Responder C Responder D
ª Axxx KQxx KQxx Q J 10 x Kxx
© Qxx xxx J 10 x J 10 x J 10 x
¨ KQxxx Axx Axx Axx Axx
§ x J 10 x xxx Kxx Q J 10 x
1¨ 1ª 1ª 1ª 2 NT
2ª 2 NT ? 2 NT ? 2 NT ?
3§ 3 ª - Stop 4ª 4ª
Responders A and B have 10 ½ pts (2 pts for J 10 but minus 1 point
for 4 3 3 3) and should inquire about a possible singleton in
opener’s hand via a 2 NT bid.
J 10 of clubs opposite the § singleton being “wasted honor pts” for
responder A, he stops in 3 ª, while responder B has no “wasted
honor pts” in clubs and he can therefore bid to game in 4 ª.
Responder C has 12 ½ pts plus 2 pts for the ª and ¨ Fits. Opposite
a § singleton, he must avoid playing NT and can bid 4 ª instead.
Responder D also has 12 ½ pts and, without a 4-card major, bids
2 NT. The opener, minimum and with a singleton, passes.

As can be seen, J 10, just like Q J 10, have the same value in a suit
contract as they do in NT and numerous examples illustrate this.
One of them is the third deal shown in this book’s foreword. Let’s
re-visit it. The hands were as follows :
31
 THE TRUE VALUE OF TENs

ª A J 10 x x ª Kxx
© KQx © Jx
¨ Qxxx ¨ A J 10 x
§ x § xxxx
In the tournament this deal comes from, the bidding was :
1ª 2ª
Pass
West counts his hand as 15 HLD pts (at best!), not enough for a
game opposite a partner’s hand limited to 10 HS pts.
But East’s hand is worth 11 ½ HDS pts (- 1 for no Queen but 1 point
for the ¨ 10, + 1 pt for the ª K + 1 D point for the © doubleton),
not 9 or 10 pts. The bidding should have been :
1ª 2¨
4 § (¨ Fit, § singleton) 4ª
On East’s 2 ¨ bid, West has 18 HLDS pts with one more point for
the § singleton in a hand with 4 diamonds as trump + 1 point for
the ¨ Q. More than enough pts for a “splinter” bid of 4 §.
These two hands have a combined total of 29 ½ pts, not 25 HLS pts.
Now then, we can distinguish between true and false statements
among the many comments often found in Bridge literature
dealing with hand evaluation, such as :
« Aces are worth 4 ½ pts, rather than 4 pts, when playing a suit
contract » – False! Aces are worth 4 ½ pts in all cases, whether
playing a suit contract or NT.
Or still : « Queens and Jacks are worth, respectively, 1 ½ point and ½
point, rather than 2 pts and 1 point, when a suit contract is played » –
False! Queens and Jacks are worth, respectively, 1 ½ point and ½
point, when they are isolated, regardless of the contract played.
Or still : « 10s have some value, particularly when playing a NT
contract » – False! 10s are worth 1 full point when they accompany
a Queen or a Jack, whether playing NT or a suit contract.
It is high time to come out of the dark ages which have prevailed for too
long on these subjects !
How unfortunate that experts and champions have not focused
more of their attention, over the years, on hand evaluation when
inaccurate point count is the primary source of bidding errors and
reaching the wrong contracts !

32
 THE IMPORTANCE OF KINGS AND QUEENS : POINT CORRECTIONS

3. THE IMPORTANCE OF QUEENS AND KINGS : POINT CORRECTIONS

3.1 THE IMPORTANCE OF QUEENS

The examples just shown clearly illustrate the importance of

Queens as they can generate additional tricks which Aces and
Kings alone do not produce. But while this positive attribute is
translated by the 2 pts given to Queens when in combination with
other honors – instead of its 1.75 pt average statistical value – what
is not translated in points is the negative impact that not having
any Queen has on a hand ! A fact never before identified.
And the solution to quantifying this fact is not, contrary to what is
advocated by some, to deduct points from hands rich in Queens
and Jacks – Queens and Jacks are not a weakness, they bring
additional value! Neither is it, contrary to what is advocated by
the 6 4 3 2 1 and 5 4 3 2 1 counts, to increase the value of Queens
relative to that of Aces and Kings, which is statistically erroneous;
rather, it is to deduct 1 point from a hand which has no Queen.
Thus, this one point deduction (similar to deducting one point for
an Aceless hand) reflects the importance of Queens – not having
any Queen is a drawback – without altering in any way the relative
values attributed to the top 3 honors. Another small revolution !
To illustrate this, let’s go back to the first example previously used
(page 26), in which the hands were as follows :
ª Axxx ª Kx
© Axx © Kxx
¨ Kxx ¨ Axxx
§ Axx § xxxx
West has 15 ½ H pts (4 ½ pts for each Ace minus 1 point for the
4 3 3 3 distribution), minus 1 point for no Queen : that’s a total of
14 ½ pts, not 15 ½ pts. And East has 9 ½ H pts : minus 1 point for
no Queen. Thus, on a 12/14 1NT opening, East should pass.
Here are a few more illustrations of this corrected optimal point
count, with hands played in tournaments :
European championships, Quarter finals, Teams, 2008.
ª Kxxx ª xxx
© Axx © Kxxx
¨ Axx ¨ KJxx
§ AKx § xx

33
 CORRECTIONS FOR NUMBER OF KINGS AND QUEENS

3 NT was played by 20 of 24 teams! – successfully only once.

Another case of hands made of Aces and Kings – without Queens.
West players count their hand for 18 H pts, or more, and jump-
rebid 2 NT (18/19). On which, East players, counting their hand
for 7 pts, naturally bid 3 NT.
But West has 17 ½ pts, not 18 : 13 ½ pts for 3 Aces - 1 for 4 3 3 3 and
minus 1 point for no Queen = 17 ½ . He should therefore open
1 NT, 15/17. On which East, with 6 pts, not 7 (minus 1 point for no
Queen), can now pass.

NEC Cup, Japan, 2009.

ª Axx ª Qxx
© Axx © Qxxx
¨ Kxxx ¨ Qx
§ Axx § K J 10 x
3 NT was played at 38 of 46 tables. On a 15/17 1NT opening by
West, East players counted their hand for 10 pts and bid 3 NT.
Here again, the result of cumulative errors in point count…
Indeed, West has 14 ½ pts, not 15 or 16 : minus 1 for 4 3 3 3 and
minus 1 for no Queen. And East has 9 pts, not 10 : 5 pts in clubs
with 2 pts for J 10, 1 point only for the ¨ Q doubleton + 3 pts only
for the two isolated Queens in ª and ©. On a 12/14 1NT opening,
East should therefore pass. Optimal precision !

3.2 THE IMPORTANCE OF THE NUMBER OF KINGS : 3 Ks OR NO K.

And the same correction applies to Kings as the absence of any King
in a hand is just as negative as the absence of any Ace or Queen is.
Furthermore, another factor has never before been identified and
quantified : and that is that the abundance of Kings (3 or 4) in the
same hand warrants one additional point. Here is why :
First, the average statistical value of a King is 3.15 pts, which means
that half a point should be added for 3 Kings, and second, the true
value of 3 or more Kings in the same hand is actually greater than
half a point as these multiple Kings are sure to add value to any
and all Queens in partner’s hand, whether the Queens are isolated
or not.

34
 CORRECTIONS FOR NUMBER OF KINGS AND QUEENS

Therefore, it is in fact the absence or the abundance of Kings (3 or 4)

which determines, in part, the value of Queens (and of Jacks and,
to a degree, of Aces, as well) in the two hands of the same side.
Thus, the following corrections must be made to the number of
points attributed to Kings :
Deduct 1 point for a hand without any King.
Add 1 point for a hand with 3 + Kings.
And these two corrections apply to all hands – whether opening
hands or responders’ hands, playing NT or a suit contract.
This very important factor constitutes one of the key precision
elements of the corrected optimal point count. Here are a few
examples that will illustrate it :
Technique (« Pas à pas ») – Le Bridgeur, 1978.
« What do you bid, in East, with the following hand, on your partner’s
15/17 H 1NT opening ? :
ª KQxxxx
© Kx
¨ xx
§ Kxx
Comment from the author : « Bid 4 ª – not 3 ª which would indicate
a stronger hand with slam potential ».
Really ?! And miss 6 ª or 6 NT opposite an opener such as :
ª Axx
© Qxx
¨ Ax
§ AQxxx
And that is because East’s hand is worth 15 HLD pts : 2 pts for
6 spades + 1 point for 3 Kings + 1 point for the two doubletons.
And an 8-card ª Fit is guaranteed by the 1 NT opening. The side
will therefore be in slam zone should West have 3 or 4 spades or
3 spades and 5 clubs (double Fit).
In fact, these two hands have 35 HLDFit pts between them (+ 2 pts
for the 9-card ª Fit + 1 D point in diamonds). Contrary to what is
suggested above, the “right” bid is 3 ª, not 4 ª.
The proper evaluation of one additional point for the 3 Kings is the
first step towards contemplating the slam zone. Up to the bidding
system to find West’s 3 spades and 5 clubs.

35
 CORRECTIONS FOR NUMBER OF KINGS AND QUEENS

Regional tournament, 2007.

ª J 10 x ª AKQ
© Axx © Jxx
¨ Axx ¨ Qxxx
§ Axxx § xxx
The bidding was most often :
1§ 1¨
1 NT (12/14) 2 NT
3 NT
Down 2. And quite a few pairs bid 3 NT directly over 1 NT.
This is another example of the wrong point count leading to
misguided bidding and to an awful contract. But then, the right
point count has yet to be taught, so the world’s best pairs would
likely bid these two hands no differently today.
Note : Cumulative point deductions must be limited to a
maximum of 2 pts. Therefore, for hands having no King, nor any
Queen and distributed 4 3 3 3, deduct no more than 2 pts, not 3.
West has 13 ½ pts (13 ½ for 3 Aces + 2 pts for J 10, - 2 pts for no
Queen, no King and 4 3 3 3) and East has 10 ½ pts (2 pts for isolated
Queen and Jack, - 1 for 4 3 3 3), not 12 ! East must pass on 1 NT.

Bidding contest – Bridge World, 2009.

ª xx ª Qxx
© AKxxxx © Qxx
¨ xx ¨ Axxx
§ K 10 x § QJx
The European champions’ team participating in the contest ended
up in 3 NT. And the auction advocated in “Bridge World Standard”
was :
1© 1 NT (forcing)
2 © (6 hearts) 4©
The contest organizer commenting, with humour : « That’s what
The Book says to do… ».
Both contracts were evidently too high. In both cases the problem
starts with East counting his hand for 11 H pts and, later, upon the
discovery of a 9-card © Fit (+ 2 pts), bids up to game (13 opposite
13 = Game, right ?).

36
 CORRECTIONS FOR NUMBER OF KINGS AND QUEENS

But East only has 11 ½ fit pts, not 13 : 8 ½ H pts (only 3 pts for the
two isolated Queens, minus 1 for no King and minus 1 for 4 3 3 3)
+ 3 pts for the 9-card © Fit with the Queen = 11 ½. It is therefore
3 ©, invitational, and not 4 © or 3 NT, which East should have bid.
On 3 ©, West, minimum, would obviously pass.
It is the absence of any King in East’s hand, not the abundance of
Queens and Jacks, that is the key evaluation element of this hand.

European Championships, 2006.

ª AKx ª Q J 10 x
© A 10 x © xxx
¨ Kxx ¨ Qxx
§ A 10 x x § Qxx
3 NT was bid by 43 pairs out of 70 (succeeded 4 times). West
having invariably announced a hand of 18/19 H pts and East
counting his hand for 7 pts, 3 NT is reached more often than not.
But West only has 17 ½ pts (- 1 for 4 3 3 3 and minus 1 for no Queen)
and East only has 5 pts : 3 pts only for the two isolated Queens, - 1
with no King, - 1 for 4 3 3 3. The bidding should have been :
1 NT Pass

Final of the Yeh Bros Cup, Australia, 2009.

ª xxx ª AQJx
© Axx © 10 x x x
¨ Axxx ¨ QJ
§ AKx § xxx
On West’s opening of 1 NT 15/17, East counts 10 pts for his hand
and goes through a 2 § Stayman bid before concluding with 3 NT.
A BBO (Bridge Base Online – Internet) expert commented :
« The 3 NT contract is perfectly normal with 25 H pts, but unfortunately
there were only 7 tricks for the taking ».
Well, these two hands have nowhere near 25 pts : West’s hand has
14 ½ pts (minus 1 for no Queen, minus 1 for 4 3 3 3) while East only
has 8 ½ pts (2 pts only for Q J ¨ doubleton and minus 1 point for
no King) – A total of 23 pts, not 25.

37
 CORRECTIONS FOR NUMBER OF KINGS AND QUEENS

Regional tournament, 2014.

ª A 10 x ª KQJ
© 10 x x © KQx
¨ AQxx ¨ 10 x x
§ A 10 x § xxxx
Several Wests opened their hand 1 NT 15/17 – probably having
read somewhere that three Aces and three 10s were worth at least
1 additional point – while deducting 1 point for a 4 3 3 3 distri-
bution isn’t a must… East, counting his hand for 11 H pts bid 3 NT.
Down two.
Note that the side has all four 10s but not a single one of them
brings much value, and the optimal count does not give credit to
any of them (two 10s are isolated, and two 10s accompany an Ace).
West actually has 13 ½ pts (minus 1 for 4 3 3 3 and minus 1 for no
King) and East only has 10 pts, not 11 (4 3 3 3).
Best not to bid beyond the level of 1 NT… Such as :
1 NT (12/14) Pass
or : 1¨ 1 NT (10 HL maximum)
Pass
One final note : When a hand has all 4 Kings, it is 2 additional
points that must be counted, not just one, as partner obviously has
no King and will deduct 1 point from his own hand. This would
cancel the upgrading which must be credited to the 4 Kings as
partner will only count any isolated Queen in his hand 1 ½ point
when the value of any Queen will be optimized by the King in
partner’s hand.
Note : The addition of 1 point for 3 Kings does not apply to
3 Aces, whose 4 ½ pts already reflect the Ace’s average statistical
value. Nor does it apply to 3 Queens or 3 Jacks as their variable
values are already reflected by whether or not they accompany
other honors.

38
 OPTIMAL HAND EVALUATION

4. 9 + CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

I indicated previously that there is as much to learn from
J-R. Vernes and B. Charles’ statistical findings as there is from
what they did not discover relative to Fit points. Let’s now revisit
and explain this comment.
– Indeed, surprisingly, they failed to realize that the 2 additional
pts applying to a 9-card trump fit apply, in fact, to any 9-card fit,
not just to the trump suit !
– And, furthermore, this applies just as well to NT contracts !
And this is quite logical and simple to understand : a fit which
allows the development of several tricks in a suit allows that same
development in No Trump contracts, as well ! Illustration :
« How to find 6 NT ? » – From a column in Le Bridgeur, 1980.
ª Ax ª xx
© xxx © AKQx
¨ Kxxxx ¨ AQxx
§ AKx § Qxx
In the tournament this deal comes from, not a single pair found
the 6 NT contract. Most opened the West hand 1 ¨, rebid 1 NT
(12/14 H) and the side ended up in 3 NT. The few who opened
1 NT, 15/17, heard East bid a 2 § Stayman followed by a jump to
4 NT “quantitative” on which, minimum, they passed.
And the commentator asks « How to find 6 NT when the two hands
only have 31 H pts ? ».
But we know now that the reality is quite different : first, West has
15 HL pts and East has 17 ½ H pts for a total of 32 ½ HL pts, to
which 3 pts should be added for the 9-card ¨ fit with the King –
a Fit just as pertinent at NT as it is for a ¨ contract ! And the 6 NT
contract could be bid as follows :
1 NT (15/17 HL pts) 2 ¨ (forcing Stayman)
3 ¨ ( 5 diamonds, no 4-card Mjr)
East knows that his side is in slam zone right upon West’s bid of
3 ¨ : 32 ½ HL pts + 2 Fit pts with 9 diamonds = 34 ½ pts minimum.
Note that if you remove a small diamond from each hand and
transfer each to spades, you no longer have a good slam and the
two hands then total only 29 ½ pts (4 3 3 3 and no Fit).

39
 9 + CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

And the argument, made by some, that the benefits of a 5 or 6-card

suit are compensated, when playing a NT contract, by the draw-
back of having a short suit (doubleton) in that hand, making it
vulnerable to the run of a long suit held by the opposing team,
does not, in general, correspond to the reality “at the table”.
In fact, it is generally preferable to play 3 NT rather than 4 © or
4 ª, with a 5 3 3 2 hand facing a 4 3 3 3 distribution in partner’s
hand. That is a well known and documented reality, which the
following example illustrates :
Final of Bridge Olympiad, 1980.
ª AQxxx ª Kxxx
© Axx © xxx
¨ Ax ¨ QJx
§ xxx § Axx
Both teams play 4 ª, down one. 3 NT would have been a better
contract as East’s ª support offers no ruffing value.
Now, let’s see : East has 9 ½ pts (- 1 for 4 3 3 3) + 3 pts for the
9-card ª fit with the King. 3 Fit pts which count just as much at NT
as they do for a spade contract. And West has 15 ½ HL pts, not 14.
And 3 NT should not be hard to bid, after : 1 ª 2 NT

« Evaluation of Bridge hands » – J-R. Vernes and B. Charles, 1995.

The following example comes from the above-mentioned book in
which the authors wonder which point count could justify playing
3 NT with the two hands below which total only 23 HL pts in
traditional point count (21 H pts + 2 pts for 6 clubs) :
ª Ax ª xxx
© xxx © Axx
¨ xx ¨ Axxx
§ A Q 10 x x x § Kxx
To resolve this, for the very specific cases of NT contracts for a side
having an 8 + card fit with the top 3 honors in a 6-card minor
suit, they propose a bizarre “mixed point” count (which consists
of applying the 4 3 2 1 H count to the 6-card minor suit while
applying the 3 2 1 ½ H count to the other suits).
Interesting how it would, all of a sudden, become appropriate to
count in half-points and abandon the “simplicity” of the 4 3 2 1
count to achieve… accuracy ?!

40
 9 + CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

Had they only realized that a 9-card Fit, with a Queen or King in
the support hand, is worth the same 3 fit pts for a NT contract that
it is for a suit contract, they would have spared themselves this
flabbergasting “mixed point” count !
Indeed, once that has been discovered, these two hands are no
longer a challenge : West has 13 HL pts (9 pts for the 2 Aces + 2 pts
for the 6 clubs but – 1 for no King) and East has 10 pts (4 3 3 3 and
no Queen) + 3 pts for the 9-card § Fit with the King = 26 HLFit pts
for 3 NT. And East will discover the 9-card § fit immediately upon
opener’s rebid :
1§ 1¨
2 § (6 + clubs)
All East has to do now is to verify whether the ª suit is controlled
before concluding in 3 NT.

« How good is your Bridge hand ? » - R. Klinger and A. Kambites.

The following deal comes from the above-mentioned book :
ª xxx ª Axx
© xx © Axx
¨ Ax ¨ KQJx
§ AKQxxx § xxx
The bidding was : 1§ 1¨
2§ 3 NT
West counts his hand for only 15 HL pts and East, without
distributional value, does not “see” a slam likely and concludes
with 3 NT.
But West has 18 HL/19 HLD pts, not 15 HL (9 pts for 2 Aces + 4 pts
for the 6 clubs with 3 honors). He should have rebid 3 §, not 2 §.
East can now add to his own 14 pts (9 pts for 2 Aces, - 1 for 4 3 3 3)
+ 2 pts for the 9-card § Fit = 16 + 18 HLD minimum = 34 pts :
slam zone. The auction could then be :
1§ 1¨
3 § (6 clubs, 17/19 HLD) 4 § (RKC Ask, § trump)
4 ¨ (3 Keys) 4 © ? (Queen of § ?)
4 NT (yes, § Q) 6 NT
6 NT with, in traditional count, only 29 HL pts !

41
 9 + CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

From « The No Trump Zone » - D. Kleinman, 2004.

The question is : « What is your opening bid with the following hand ? :
ª xx
© Kxx
¨ AQx
§ AKxxx
« If you open it 1NT, 15/17 H, your Partner cannot be expected to
consider a 6 NT contract with 14 pts – since we know that 34 H pts are
needed for 6 NT. But partner had the following hand and 6 NT was on » :
ª Ax
© Axx
¨ KJxx
§ Qxxx
Indeed !... West actually has 18 HL pts (9 pts for 2 Aces + 1 pt for
the 5 clubs), not 16 or 17. Therefore he should not open 1 NT.
But East’s 14 ½ pts still won’t bring the point total to what’s needed
for 6 NT. What will bring it to the right total is the addition of 3 pts
for the 9-card § Fit with the Queen : 18 + 14 ½ + 3 = 35 ½ pts.
A § Fit which should not be difficult to find after a start such as :
1§ 1¨
2 NT (18/19 HL)
Getting to 6 NT should become easy from there.
The appropriate conclusion is therefore : no, it is not 34 H pts that
are needed for 6 NT but rather 34 HLF pts : Honor + Length + Fit.

Regional tournament, 2006.

ª AQxx ª Kxxxx
© x © xxx
¨ KQxxx ¨ Axxx
§ Axx § x
The bidding was : 1¨ 1ª
3 ª (17/19 HDS pts, ª Fit) 4 ª
And 6 ª was missed. Nothing unusual, as West counts his hand
for 19 HLS pts, 1 point shy of the 20 HS pts needed for a 4 ©
“splinter” bid.

42
 9 + CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

East has no reason to consider a slam with his own 12 HDS pts
(7 H pts + 3 for the 9-card ª fit with the King + 2 for the § single-
ton) : 19 pts maximum + 12 = 31 pts. No slam.
But had West properly counted his hand for 19 HLD pts (+ 1 for
5 diamonds and 9 pts for 2 Aces), he could now add 1 point for the
© singleton with 4 trumps = 20 pts, enough for a 4 © “splinter” bid.
Even though the “splinter” bid is wildly uneconomical, it would at
least have the merit to enable East to consider the slam, with its
13 ½ pts (7 ½ HL pts, + 1 point for the ¨ Fit) : 20 HDS pts minimum
in opener’s hand + 13 ½ = 33 ½ minimum.
Actually, these two hands have 34 ½ pts as the ¨ Fit has 9 cards,
not 8 – and there are no “wasted honor points” in hearts.

Bidding contest - Bridge Magazine, 2007.

« 1) Do you open the following hand ? » :
ª KJxx
© xx
¨ AKxxx
§ xx
« 2) If you opened, what do you bid after this auction ? :
1¨ 1©
1ª 2 NT
?
Comment : « The large majority of champions answering the quiz passed
on 2 NT – Nothing unusual with only 11 H pts facing 11/12 H.
But 3 NT was on – thanks to the diamond suit running – as partner’s
hand was » :
ª Qx
© Axxx
¨ Qxxx
§ Axx
Actually, West has 11 ½ HL pts (4 ½ pts for the Ace + 1 point for
the 5 diamonds minus 1 for no Queen) and East has 10 ½ pts (9 pts
for the 2 Aces, 1 ½ pts for the isolated ¨Q, 1 point only for the
ª doubleton Queen, minus 1 point for no King) for a total of 22 HL
pts. Now add 3 pts for the 9-card ¨ Fit with the Queen and the
total becomes 25 HLFit pts.

43
 9 + CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

And to reach the right contract, all West needs to do is to disclose

his 5 diamonds by bidding 3 ¨ over partner’s 2 NT. Now East can
count his hand for 13 ½ HLFit pts and bid 3 NT over 3 ¨.
Again, the key new finding is that the number of points needed for
a NT contract should be the total of : Honor + Length + Fit points
– Not HL points only !

National selection, by teams - 2009.

ª xx ª Axxx
© Ax © Kxx
¨ K Q 10 x x ¨ xxxx
§ Kxxx § Ax
The expert commenting the deal expresses his surprise that « one
team did find the 3 NT contract, with only 23 points between them ».
Actually, every single team should have reached 3 NT with the right
point count – as these two hands total 27 ½ pts, not 23 !
West has, in fact, 14 ½ HL pts (1 point for the 5 diamonds + 1 for
the ¨ 10) and East has 11 H pts (9 pts for the 2 Aces but - 1 for no
Queen) + 2 pts to be added for the 9-card ¨ Fit = 27 ½ pts.
This is a typical example used by some to illustrate the theory that
« 3 NT can often succeed with less than 25 H pts when the points are
evenly divided between the two hands (facilitating two-way commu-
nications between the hands) ».
But the same contract, with the same honors and number of points,
would be on without the points being evenly divided between the
two hands if you transfer the © Ace from West to East or even one
of East’s Aces to West! Certainly, two-way communications
between hands is an important factor, but the first and foremost
reality here is that these two hands total 27 ½ pts, not 23.

44
 8-CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

5. 8-CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS !

THE “MISSING LINK”

Furthermore, another key conclusion was not drawn from

J-R. Vernes and B. Charles’ study – and this one is as fundamental
and even more important than the one concerning 9-card Fits because
considerably more frequent: and that is that 1 point must be added
for 8-card Fits, and this applies to All suits – not just the trump suit
– and to All contracts, including NT !
That they failed to draw this conclusion is actually quite surprising
in view of the very pertinent comments they made on this subject
in their 1995 book Evaluation of Bridge Hands :
« The number of tricks that can be expected from developing a suit with
a given distribution depends essentially on the number and distribution
of the cards held by the opposite side. Thus, a side holding 7 trumps has
a fragile numerical advantage of only one card over the opposite side
which holds 6 cards, most often distributed 4 - 2. An 8th trump becomes
a major factor which changes that equation as it now leaves the opposite
side with only 5 cards, most often distributed 3 - 2, and significantly
increases its numerical advantage from one card to three cards ».
« And this is confirmed by statistical data which indicates that going
from 7 to 8 cards increases the average number of tricks made by appro-
ximately 0.75 tricks, and by 0.55 tricks when going from 8 to 9 ».
And they conclude : « Clearly the importance of the number of
trumps held by the same side has been seriously underestimated and not
quantified by the traditional doctrine, beyond evaluating short suits ».
Yet, they apply these findings to the trump suit only, and only add
points for 9 and 10-card trump Fits, not to 8-card Fits. Why ?
It is the following conclusions that should have been drawn from
this analysis :
– 1 point must be added for an 8-card Fit, a second point must be
added for a 9-card Fit (2 pts in total) and a third point must be
added for a 10-card Fit (3 pts in total).
Beyond 10 cards, no additional point should be counted for an 11th
card as a duplication of honor and length values frequently occurs
with 11 cards or more.
– And these additional points for 8, 9 and 10-card Fits apply to
side suits just as they do to the trump suit – and they apply to All
contracts, whether NT or in a suit !

45
 8-CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

Here is a simple illustration of how these Fit points assess the true
value of suits in two hands of the same side :
The following 4 - 3 suit with the top 4 honors : ª K Q J x Axx
produces 4 tricks = 11 pts. Counted 10 ½ pts in optimal count.
The same honors, but in a suit now distributed 5 – 3, will produce
5 tricks = 13 ½ pts. Counted 13 ½ pts in optimal count, with 2 pts
for 5 spades with 3 honors + 1 Fit point for 8 spades.
A similar suit, now distributed 6 – 3, will produce 6 tricks = 16 pts.
Counted 16 ½ pts in optimal count with 4 pts for 6 spades with
3 honors + 2 pts for the 9-card ª Fit.

The following examples will now provide clear illustrations of the

appropriateness of attributing Fit points for 8-card Fits :

The Roman Club system.

One can read in the book on the Roman Club system : « On a Roman
1 § opening, describing specifically a 12/16 H pts hand distributed
4 3 3 3 or 4 4 3 2, a partner with less than 8 H points knows immediately
that a game is not an option ».
Probably because 16 H maximum + 7 H = 23 pts, short of the 25 H
pts required for 3 NT.
A clear admission of deficient point count, to be sure – just as
prevalent nowadays as it was in the ‘60s ! Illustration :
ª xxx ª xxx
© Ax © xx
¨ KQJx ¨ Axxx
§ AQxx § K 10 x x
In the Roman § system, these hands would be bid as follows :
1 § (forcing) 1 ¨ (weak hand)
1 NT (balanced, 12/16 H) Pass
When 3 NT should be played.
And the result would generally be no different for those opening
a 14/16 or even 15/17 H, 1NT, with bidding like :
1 NT Pass
In both cases, the game will be missed because points are not being
counted properly! and 8-card Fits are neither found nor counted.

46
 8-CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

The point total of these two hands is, in fact, 27 HFit pts – not 23 H
pts! Opener has 17 H pts, not 16 (9 pts for 2 Aces), and East has
7 H pts (minus 1 for no Queen) + 1 point for the 8-card ¨ Fit + 2
pts for the 8-card § Fit with the King, for a total of 27 HFit pts.
And a minor-suit Stayman, on a 1 NT opening, could be one way
to find the double-Fit in the minor suits.

World championships by team, 1969 (USA/Italy).

ª AQx ª KJxx
© Kx © Axxx
¨ AKxxx ¨ Qxx
§ xxx § KQ
Both teams play 3 NT. Without a Major-suit Fit or a 9-card ¨ Fit,
East, with 14 H pts (minus 1 for K Q § doubleton) does not “see”
a slam facing a 16 or 17 H 1NT opening.
But West has 18 HL pts, not 16 or 17. And if East can discover
opener’s 5 diamonds, the right point count will enable him to count
a total of 35 HLFit pts between the two hands : 18 + 14 + 2 pts for
the 8-card ¨ Fit with the Queen + 1 point for the § doubleton.
A slam which should not be difficult to bid after a start such as :
1¨ 1©
2 NT (balanced, 18/19 HL)

Bidding contest – Le Bridgeur, 1978.

ª Kxx ª Axx
© AKxx © Q 10 x x
¨ AQx ¨ KJxxx
§ xxx § x
Comment from the contest organiser : « Some hands seem to defy
“traditional” hand evaluation! Applying the traditional trump support
count to East’s hand only brings the combined total to : 16 H + 14 S =
30 HDS, far below the 33 pts needed for a slam ».
Yes, indeed! But now, with the right point count, these two hands
could be bid as follows :
1 NT (15/17 H) 2 § (Stayman)
2 © (4 hearts)

47
 8-CARD FIT POINTS FOR ALL SUITS AND ALL CONTRACTS

Right upon West’s 2 © rebid, East can count a minimum of 32 pts

with his partner : 15 HL pts minimum in West + his own 12 HL pts
+ 2 pts for the 8-card © Fit with the Queen + 3 pts for the § single-
ton with 4 trumps = 32 HLDFit pts. Up to the bidding system now
to get the side to 6 ©.
These two hands actually total more than 34 pts with 1 more point
for the 8-card ¨ Fit and no “wasted honor points” in clubs.

World championships by team, Final – 1984.

ª Q J 10 x x x ª Kx
© Axx © J 10 x
¨ Ax ¨ xxxx
§ Ax § Kxxx
One team plays 4 ª, down one, while the other team stops at 2 NT.
The right contract is 3 NT.
To reach it, let’s first apply the right point count : West has 18 ½
HL pts (13 ½ H pts for the 3 Aces, 6 pts in spades, minus 1 for no
King) and East has 7 H pts (no Queen) + 2 pts for the 8-card ª Fit
with the King = 27 ½ HLFit pts. And the bidding could be :
1ª 1 NT
3 ª (6 spades, 17/19 HL) 3 NT
East can count on 17 HL pts minimum in West + his own 9 HFit
pts = 26 HLFit pts. Without distributional values, 3 NT would be a
logical preference to 4 ª. With a 6 ª 3 3 1 hand, West can still revert
to 4 ª.

European championships - 2004.

ª xxx ª Ax
© Ax © KQxx
¨ AKxxx ¨ Q 10 x
§ AQx § KJxx

On a 15/17 H 1NT opening, East players, with 15 H pts and not

finding a © Fit, rule out a NT slam and most teams play 3 NT.
A few teams play 6 ¨.

48
 FIT POINTS FOR 8 CARDS FOR ALL SUITS AND ALL CONTRACTS

But the proper point count allows the side to be much more
ambitious : West has 19 ½ HL pts, not 17 H ! (3 Aces at 4 ½ pts each
and 1 point for the 5 diamonds), ruling out a 1 NT opening – simply
a horrible opening bid – and if East can locate his partner’s
5 diamonds, he will be able to add to his 16 H pts : 2 pts for the
8-card ¨ Fit with the Queen for a total of 36 HLFit pts minimum
between both hands : grand slam zone ! The side actually has 37 ½
total pts. And the slam could be bid as follows :
1¨ 1©
2 NT (balanced, 18/19 HL) 4 § ? (RKC Ask, ¨ trump)
4 © (4 Key cards) 5 § ? (§ Ask for K & Q)
5 © (§ Queen) 7 NT

Regional tournament, 2005 (U.K.).

ª xxxx ª xxx
© Axx © xx
¨ AQxx ¨ K J 10 x
§ Ax § KQJx
The bidding was most often (Acol) :
1 NT (12/14 H) Pass
And the 3 NT game was missed.
But East has 11 H pts (1 point for the ¨ 10), not 10, and should
invite to game, not Pass. West, with 14 ½ H pts (no Queen) will bid
3 NT. These hands actually total 26 ½ HFit pts with the ¨ Fit.

World championships by team - 2008.

ª AKxxx ª Qxx
© KQxxx © Axxx
¨ --- ¨ xxxx
§ Qxx § Kx
Not a single team found the slam! The bidding having been, most
often, as follows :
1ª 2ª
3© 4©

49
 FIT POINTS FOR 8 CARDS FOR ALL SUITS AND ALL CONTRACTS

West counts his hand for 18 HD pts and his 3 © rebid indicates
17 to 19 HLD pts. East describes up to 10 HDS pts in spades and 10
to 12 HDS pts in hearts. This gives West little motivation to bid
further as his own 20 pts (+ 2 for the 9-card © Fit) + 12 pts
maximum in East = 32 pts maximum.
But the right point count is : 20 HLD pts in West : 2 pts for the two
5-card suits and 4 pts for the void, not 3. And East has 12 HDFit
pts in spades, not 10. His first mistake is therefore to bid 2 ª which
limits his hand to 10 HDS pts.
But then, opening West’s very strong hand by a non-forcing 1 ª is
hardly a good beginning! Playing a strong 1 § system (and relays)
would be more helpful in finding East’s exact distribution and
specific honors. But that’s another subject, for another book…

Cavendish tournament, 2011.

ª AQxx ª Kxxx
© A 10 x © KQxx
¨ Kx ¨ xx
§ Axxx § xxx
4 out of 18 pairs (22 %) did not find the 4 ª contract and 10 of the
14 pairs who did find it bid as follows :
1 NT (15/17 H) 2 § (Stayman)
2ª 3ª
4ª
Unfortunately, this auction only confirms that an accurate point
count has yet to be taught !
West has 18 ½ H pts, not 17, and when he indicates that he has
4 spades, East must count his own hand for 8 H pts + 2 pts for the
8-card ª Fit with the King + 2 pts for the ¨ doubleton with
4 trumps = 12 HLFit pts. He should have bid 4 ª directly rather
than just bid an invitational 3 ª. An invitation on which a
minimum opener would pass – For example, an opener with this
hand :
ª QJxx
© AJx
¨ xxx
§ AKx

50
 FIT POINTS FOR 8 CARDS FOR ALL SUITS AND ALL CONTRACTS

A minimum opening hand of 15 pts (- 1 for 4 3 3 3) which actually

makes 4 ª a safer contract than the actual hand in the tournament !
(while 3 NT is not in the cards).

Example used in this book’s foreword.

We can now go back to the second example used in this book’s
foreword – two semi-regular hands totaling only 27 H pts in
traditional count, even though 6 NT is on :
ª Axx ª xx
© xx © Axx
¨ Kxx ¨ AQJxx
§ AQxxx § Kxx
This deal comes from a regional tournament held in 2012, where
most West openers described their hand as a NT hand of 12/14 HL
pts on which East responders, counting their hand for 15 HL pts,
did not “see” a NT slam and invariably concluded in 3 NT.
Contemplating a Fit in a minor suit and a possible suit slam does not
seem to be the thing to do nowadays…
Well, let’s see now : West has 15 HL pts (9 pts for 2 Aces), not 13 H
or 14 HL. On a 15/17 HL 1NT opening, East has 17 HL pts (+ 1
point for 3 honors in a 5-card suit) and can readily envision a
possible slam. The bidding could then be as easy as this :
1 NT (15/17 HL) 2 ¨ ? (forcing Stayman)
3§ (5 §, no 4 card Mjr) 3 ª ? (guard spades ?)
3 NT (Yes, spades guarded) 4 § ? (RKC Ask, § trump)
4 NT (2 Keys, with § Q) 5 ¨ ? (¨ K & Q Ask)
5 NT (¨ K) 6 NT
Once West’s 5 clubs are known, East can add 2 pts for the 8-card
§ Fit with the King and can count a minimum total of 34 HLF pts
for his side = slam zone. All that is left for East to do now is to
inquire about West’s Key Cards before concluding in 6 NT.
(West cannot have another King for 7 NT as this would then give
him 18 HL pts and he would not have opened 1 NT).
The side actually has a total of 36 HLF pts – for two hands counted
29 HL pts in traditional point count.
Conclusion : Clearly, Fit points count just as much in NT contracts
as in suit contracts and not having discovered this until now has
undoubtedly been the greatest deficiency of all point counts.

51
 CORRECTION FOR MIRROR HANDS

Before closing this chapter, a comment needs to be made with

regards to 8-card Fits with three small cards opposite a 5-card suit.
We sometimes read in bridge literature that three small cards is
« the worst possible holding one can have opposite a 5-card suit in
partner’s hand » – presumably because, unless partner’s 5-card suit
is headed by A K Q, one of the three small cards is likely to pro-
duce a loser. But such an assertion happens to be totally erroneous
and is in no way statistically validated. For more details on why
this is not so, see Note # 3, in the Index at the end of this book.

« MIRROR » HANDS : WARNING / DANGER !

The misfit represented by “mirror” hands – meaning two hands of
the same side having exactly the same distribution, including the
same long suit – is well known, such hands allowing no opportunity
to ruff or to promote small cards of a long suit. Any sound
evaluation method should deduct 1 point in such cases.
And now the addition of Fit points for 8-card Fits or more makes
it imperative to deduct 2 points, not just one, for such perfectly
“mirror” hands. Illustration :
Regional tournament, 2004.
ª A 10 x ª Kxx
© A 10 x x © KQxx
¨ AKx ¨ Jxx
§ Kxx § AQx
Counting West’s hand for 18 H pts and East’s hand for 15 H pts,
one will not be surprised that the large majority of pairs ended up
bidding 6 © or 6 NT – unfortunately, these two perfectly “mirror”
hands can only deliver 11 tricks.
But West only has 17 ½ pts, not 18 (- 1 for no Queen and - 1 for
4 3 3 3) and the bidding could / should be instead :
1 NT (15/17 HL) 2 ¨ ? (forcing Stayman)
2 © (4 ©) 2 ª ? (relay – continue)
2 NT (4 © 3 3 3) 3 NT
East stops in 3 NT as he has 14 pts (- 1 for 4 3 3 3) + 1 point for the
8-card © Fit minus 2 pts for two perfectly “mirror” hands = 13 pts.
Facing an opener of 17 pts maximum, the side will have no more
than 30 pts : no slam.

52
 CORRECTION FOR MIRROR HANDS

Another example, this time from a tournament deal published in

the magazine « Le Bridgeur » in 2015. The two hands shown were
these :
ª Axx ª K 10 x
© Axx © K 10 x
¨ Kxxx ¨ AQxx
§ A 10 x § KQx
Betting on their side having between 33 and 34 H pts, many East
players bid 6 NT directly on their partner’s 1 NT opening – but,
here again, these two perfectly “mirror” hands only yield 11 tricks.
The first mistake comes from West who only has 14 ½ pts, not 15
(minus 1 for 4 3 3 3, minus 1 for no Queen), but East commits a
bigger mistake yet by jumping directly to slam, “burning” in the
process, no less than… 24 bidding steps – without inquiring first
about West’s distribution. Instead, the bidding could have been as
follows, starting with a weak 12/14 1NT opening :
1 NT (12/14 HL) 2 ¨ ? (forcing Stayman)
2 NT * 3 § ? (relay – continue)
3 ¨ (4 ¨ 3 3 3) 3 NT
* 2 NT : No 4-card Major, no 5-card minor
East stops in 3 NT as he has 18 ½ pts (-1 for 4 3 3 3, + 1 for 3 Kings)
+ 1 point for the 8-card ¨ Fit, minus 2 pts for two perfectly “mirror”
hands = 17 ½ pts, which, facing an opener of 14 pts maximum will
give his side no more than 31 ½ pts : no slam.
As well, but to a lesser degree, the drawback of no possible ruff or
promotion of small cards from a long suit also occurs when two
hands of the same side have two “mirror” suits, versus two
“mirror” hands. In such cases, 1 point, instead of 2 pts, should be
deducted. Example :
ª Jxx ª Ax
© KQxx © AJxxx
¨ A 10 x ¨ Kxx
§ KQx § A 10 x
Two hands which would often lead to a 6 © contract in “standard”
bidding after a 15/17 H 1NT opening facing 17 HL pts + 2 pts for
the 9-card © Fit for a total of 34 pts minimum.

53
 CORRECTION FOR MIRROR HANDS

But West only has 14 pts, not 15 (4 3 3 3) and the bidding could /
should be instead :
1 NT (12/14 HL) 2 ¨ ? (forcing Stayman)
2 © (4 hearts) 2 ª ? (relay – continue)
2 NT (4 © 3 3 3) 3 NT or 4 ©
East stops in game as 14 H + 17 ½ HL (no Queen) + 2 pts for the
9-card © Fit – 1 point for two “mirror” suits (¨ and §) = 31 ½ pts,
not 34 : no slam.

We will end this section with an updated summary of the optimal

point count, which appears on the next page.

54
 UPDATED SUMMARY OF THE OPTIMAL POINT COUNT

HONOR POINTS

Ace : 4 ½ pts K : 3 pts Q accompanied : 2 pts Q isolated : 1 ½ pt

J accompanied : 1 pt J isolated : ½ pt
10 isolated or with an Ace : 0 pt 10 with a K : ½ pt
10 with a Q or a J : 1 pt J 10 : 2 pts

No Ace : - 1 pt (opening hands only)

No Q : - 1 pt
No K : - 1 pt 3 Ks : + 1 pt 4 Ks : + 2 pts

Singleton Honor : - 1 pt
2 Honors doubleton : - 1 pt Q or J doubleton : - ½ pt
(A K, A Q, K Q, Q J) i.e. : Q x : 1 pt J x : 0 pt
3 Honors (or 4) in a 5-card suit : + 1 pt in a 6-card suit : + 2 pt

LENGTH POINTS

5-card suit (with Q J or K minimum) : 1 pt 6-card suit : 2 pts

6-card suit without Q J or K : 1 pt
7-card suit (or longer) : 2 pts for each card from the 7 th ont

DISTRIBUTION POINTS AND DISTRIBUTION-FIT POINTS

Void : 4 pts Singleton : 2 pts One doubleton : 0 pt 2 doubletons : 1 pt

4 3 3 3 Distribution : - 1 pt A singleton at a NT contract : - 1 pt
FIT POINTS : Count for ALL suits and ALL contracts (suit or NT)
8-card Fit : + 1 pt 9-card Fit : + 2 pts 10 + card Fit : + 3 pts
+ 1 for Honor(s) in suit(s) with a Fit when the Honor(s) are < 4 pts
2 perfectly “mirror” hands : - 2 pts 2 “mirror” suits : - 1 pt
Distribution-Fit points with a trump Fit :
number of trumps : 4 3 2
Distribution-Fit points Void : 4 pts 3 pts 2 pts
Singleton : 3 pts 2 pts 1 pt
Doubleton : 2 pts 1 pt 0 pt

55
56
 OPTIMAL HAND EVALUATION – ILLUSTRATIONS

Now, let’s briefly review the key features that make the optimal
point count a very accurate count and illustrate by a few examples
how effective these features are at precisely differentiating the
value of opening hands that would be considered worth the same in
traditional count – but are not.
These key, differentiating features are the following six :
– Counting Aces 4 ½ pts each, not 4 pts.
– Giving Queens and Jacks variable values depending upon whether
they are isolated or not.
– Counting 10s for their appropriate value – in particular 1 full
point when they accompany a Jack or a Queen.
– Deducting 1 point for a hand with no Ace (opening hand only) or
no King or no Queen.
– Deducting 1 point for a 4 3 3 3 distribution.
– Adding the appropriate points for suit length to the appropriate
distribution points (4 pts for a void, 1 point for 2 doubletons).
It can be said with certainty that without such corrections, no hand
evaluation method can accurately assess the trick-generating value
of an opening hand.
Certainly, these necessary corrections make the optimal point
count less simple than the “basic” 4 3 2 1 count but that is because
the “basic” 4 3 2 1 count is more simplistic than it is simple. By being
grossly inaccurate, the 4 3 2 1 count bears the initial responsability
for the multitude of poor contracts reached every day by experts
and champions alike, as illustrated by so many examples.
Here are a few examples of the optimal point count’s ability just
to differentiate opening hands.
Will it not be more effective to be able to immediately distinguish
the difference in values of the following opening hands – counted
in traditional count as having the same 18 H/19 HL pts :
ª Ax ª Qx
© KQx © KQx
¨ Axxxx ¨ AQxxx
§ KQx § KQx
Two similar opening hands but with the ª Ace replaced, in the
second hand, by two Queens, in spades and diamonds.
The first hand has 20 HL pts, not 18 H pts, while the second has
18 ½ HL pts. Not a significant difference, one would think…

57
 OPTIMAL HAND EVALUATION – ILLUSTRATIONS

Yet, facing a partner with the following hand of 12 HL pts :

ª xx
© Axx
¨ Kxxxx
§ Axx
There is 6 NT to play with the first opening hand, but you’d better
find the 5 ¨ contract with the second !
Wouldn’t you feel silly if your side bid as follows in both cases :
1¨ 2 NT or 2 ¨ (inverted minor raise)
3 NT Pass
A 6 NT contract which won’t even be suspected in 4 3 2 1 count
(31 HL pts). While in optimal count, with the first opening hand
the side has : 20 HL pts + 12 HL pts + 4 pts for the 10-card ¨ Fit
with the King - 2 point for two “mirror” hands = 34 HLFit pts.
While with the second opening hand, the side only has 32 ½ pts
because the opener only has 18 ½ HL pts, not 20.
East should describe his hand, particularly his 5 diamonds, and
West will know to play 5 ¨ rather than 3 NT with the second hand,
while reaching 6 NT with the first hand.
I guess differentiating 20 pts from 18 ½ would be worth it, after all…

Or, this time, to differentiate the following opening hands, each

counted 11 H / 12 HL pts in 4 3 2 1 count :
ª J 10 x ª J 10 x ª J 10 x
© KJx © KJx © KQx
¨ Qxxxx ¨ AQxxx ¨ A J 10 x x
§ Ax § xx § xx
12 H pts 13 ½ HL pts 14 ½ HL pts

Or, here again, to differentiate the following opening hands, each

counted 14 H pts in 4 3 2 1 count :
ª KJx ª KJxx ª K J 10 x
© Axx © Axx © Axx
¨ Qxxx ¨ AQxx ¨ A Q 10 x
§ Axx § xx § xx
13 ½ H pts 15 H pts 17 H pts
(4 3 3 3)
58
 OPTIMAL HAND EVALUATION – ILLUSTRATIONS

These few examples illustrate the considerable additional precision

provided by the optimal count without much added effort :
9 pts for 2 Aces, only count 1 L point for a 5-card suit headed by
Q J or more, deduct 1 point for a 4 3 3 3 distribution, count 1 point
for a 10 which accompanies a Queen or a Jack – and that does it !
A count which can identify a difference in value from 13 to 17 pts
for evenly-distributed hands having the same distribution (except
for one) and the same top 4 honors, surely is worth the effort !
And what about these two opening hands :
ª Qxxxxx ª K Q J 10 x x
© A 10 x © Axx
¨ Jx ¨ xx
§ AK § Ax
13 ½ HL/14 ½ HLD 20 HL/21 HLD
Two hands with the same honors but with the 10, the Jack and the
King transferred to the long suit, resulting in a difference of
6 ½ pts ! A count which captures perfectly the significant difference
between the two hands and the second hand’s 7 winning tricks.
Or these two opening hands, each counted 13 H pts in 4 3 2 1 count:
ª Kxxx ª A Q 10 x
© Kxx © xxx
¨ Kxx ¨ AKxx
§ A 10 x § xx
12 ½ H pts 15 H pts
Where the first hand only has 1 sure trick while the second has
3 sure tricks. How could a sound evaluation method give these two
hands the same count ?!... Not the optimal point count.
Hereunder is an interesting example where the optimal count
leads, this time, to evaluate a Queen… 4 ½ pts !
Opener’s hand Responder A Responder B
ª AQxxx ª KJx ª KJx
© xxx © Ax © Ax
¨ Kx ¨ Axx ¨ Axx
§ AKx § Jxxxx § QJxxx
18 HL pts 12 ½ H pts (no Q) 17 HL pts

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 OPTIMAL HAND EVALUATION – ILLUSTRATIONS

On a strong 1 § opening, responder A has 12 ½ H pts (- 1 without

Queen, no L point for the 5 clubs not headed by at least Q J) and
the side has a game but no slam.
While responder B, with the addition of the § Q, has this time
17 HL pts – 4 ½ pts more, not just 2, because : no deduction for no
Queen, the § J, with the § Q, counts 1 full point + 1 point for the
5 clubs headed by Q J. And this side now has 7 NT to play with a
total of 38 HLF pts with 1 point for the 8-card ª Fit and 2 pts for
the 8-card § Fit with Q J ( 31 H/33 HL pts in traditional count).

Already, the findings of the optimal point count revealed so far in

this book point to a major conclusion :
4 key factors – not just honor pts – are of critical importance in
assessing the appropriate value of two hands to be played in a NT
contract : honors, of course, but also : suit length (5 3 3 2, 6 3 2 2),
hand’s distribution (- 1 point for a 4 3 3 3 hand) and Fit pts !
This clearly debunks the fallacy which has prevailed for so long,
that « for NT contracts played with evenly-distributed hands, honor
cards have to do most of the work and thus only HL pts need to be
counted to assess the hands’ potential »…
The above has two major implications :
1. Invitational responses of 2 NT or 4 NT on partner’s 1 NT opening
– meant to invite the opener to Pass or bid on based solely on his
honor point count – are pure heresy, to be stricken from Bridge
bidding as far as their current meaning stands for.

2. The traditional conversion table linking “expected” NT tricks to

H or HL pts only – i.e. 25 H or HL pts for 3 NT – is a total fallacy
without any validity, statistical or otherwise. Particularly since
honor points are not counted properly in the first place…

And this explains why so many odds-on 3 NT contracts can and

should be bid with as little as 22 “traditionally counted” H pts or
even 20 H pts! In a later chapter, we will cover comprehensively
what the appropriate conversion table should be.

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 OPTIMAL HAND EVALUATION

CHAPTER 3

« Wasted Honor points » count

1. « WASTED HONOR POINTS » – OPPOSITE SHORT SUITS

In 1995, J-R. Vernes and B. Charles quantified another important

finding, this one concerning the evaluation of “wasted honor pts”,
relative to the favorable or unfavorable location of honors opposite a
short suit in partner’s hand. They wrote this on the subject :
« A factor likely to significantly modify the value of a hand, particularly
when it comes to suit contracts, is the effective value of honors
depending upon whether they are opposite a long suit or a short suit in
partner’s hand. Statistical data reveals that honors lose an important part
of their value when they are opposite a singleton or a void in partner’s
hand : in these cases, on average, 2 points should be deducted as
“wasted” opposite a singleton, while 3 points should be deducted as
“wasted” opposite a void. The only exception being the Ace with no other
honor, which should not be downgraded when opposite a singleton.
Conversely, a suit without a single honor opposite a singleton or a void
has no “wasted honor points” and the hand must be upgraded: by
2 points when opposite a singleton, by 3 points when opposite a void.
That is because, in these cases, on one hand, the honors in the hand
opposite the short suit complement fully those held by the hand with
the short suit and, on the other hand, the short suit will often allow the
elimination, through ruffing, of several small losing cards held by the
hand opposite the short suit ».
One remains quite surprised to find very few references in Bridge
literature to this very precise and very important quantification –
even though the large majority of champions and experts are very
aware of this factor’s significance and do take it into account.
Yet, just “taking it into account” won’t be enough to reach the right
contract : to do so requires, first, to integrate all other corrections to
make to the traditional point count and, second, to play a bidding
system that specifically identifies the precise location of a short suit.
Illustrations :

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 WASTED HONORS POINT COUNT – ILLUSTRATIONS

« Prepared deals » tournament, 2002.

ª QJxx ª AKxx
© x © xxx
¨ AKxxx ¨ Qxx
§ Axx § Kxx
The bidding was : 1¨ 1ª
3ª 4ª
And the 6 ª slam was missed, West counting his hand,
in traditional count, for 16 HD pts – instead of 18 HLD – makes a
rebid indicating 17 to 19 HDS pts which does not allow East, with
his 12 pts, to “see” more than 31 total pts and consider a slam –
particularly without knowledge of West’s © singleton.
But the reality is quite different : on his partner’s 1 ª, West should
count his hand for 18 HLD pts + 2 pts for the 8-card ª Fit with Q J
+ 1 for the © singleton with 4 trumps = 21 pts. That’s enough points
for a “splinter” bid in hearts. East could then add to his 11 H pts
(- 1 point for 4 3 3 3) 2 pts for “no wasted honor pts” in hearts for a
minimum of : 20 + 13 = 33 total pts = slam zone. Actually, we can
see that the total number of points between the two hands adds up
to 36 pts, with 2 additional pts for the 8-card ¨ Fit with the Queen.

Regional tournament, 2006.

ª xxx ª Axx
© AKQxxx © xx
¨ Axx ¨ KQJxx
§ x § xxx
The bidding was often :
1© 1 NT
2© 2 NT or 3 ¨
3© 4©
And the 6 © slam was missed. Here again, nothing unusual in
traditional count. But West’s hand has 20 HLD pts, not 15 HL (4 pts
for the 6 hearts with 3 honors and 9 pts for 2 Aces). However, this
more accurate point count will still not be sufficient to point to a
slam unless West describes a much stronger hand than described
by his 2 © and 3 © rebids – truly horrible rebids – and his § singleton.

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 WASTED HONORS POINT COUNT – ILLUSTRATIONS

A strong § opening could lead, instead, to the following auction :

1§ 1¨
2 © (6 + ©, 18/20 HLD) 2 NT (relay)
3 § (§ singleton)
East could then count, in his own hand : 12 ½ HL pts + 2 pts for
“no wasted honor pts” in clubs + 1 point for the 8-card © Fit = 15 ½
pts + 18 minimum = slam zone. All East has to do now is to check
on Key cards before concluding in 6 ©. With 28 HLD Goren pts…

Regional tournament, 2009.

ª KQxxx ª Axx
© AJx © xx
¨ xxxx ¨ A J 10 x
§ x § xxxx
The bidding was often : 1ª 2ª
Pass
Here again, nothing unusual… But East has 11 HDFit pts (9 pts for
2 Aces, 2 pts for ¨ J 10, minus 2 pts for no King and no Queen but
+ 1 for the 8-card ª Fit + 1 D point for the © doubleton). Thus,
East’s 2 ª raise (limited to 10 pts) is not the right bid.
But the key is to detect West’s § singleton opposite East’s four
small clubs. Not very difficult, if East “relays”, such as :
1ª 1 NT (forcing)
2 ¨ (5 ª 4 ¨, 13/15 HLD) 2 © (3rd suit forcing)
3 © (3 hearts = § singleton) 4ª
Upon learning of West’s § singleton, East can add to his 11 HDFit
pts, 2 pts for “no wasted honor pts” in § + 1 point for the 8-card ¨
Fit = 14 HLDF pts + 13 HLD pts minimum in West = 27 pts = 4 ª.

Rubber game.
ª Axxx ª Kxxxx
© x © xxxx
¨ KQxx ¨ xx
§ Axxx § Kx
« How do you bid after the following start :

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 WASTED HONORS POINT COUNT – ILLUSTRATIONS

1¨ 1ª
2 ª (ª Fit, 14/16 HDS) ?
?
Even if East bids 3 ª, it will give West no useful information to
help him determine whether to pass or to go on to game, not
knowing whether his © singleton will be favorable or not.
The key is for East to know of West’s © singleton to count + 2 pts
for “no wasted honor pts” in that suit. And the opener, who started
with 16 HD pts now has 18 pts with 1 point for the 8-card ª Fit +
1 point for the © singleton with 4 trumps. West’s 2 ª rebid is
simply awful! The following bidding would be more useful :
1¨ 1ª
3 © (“mini-Splinter”) 4ª
The 3 © “mini-Splinter” indicates a 4 card ª Fit, 17/19 HLDF pts
and a © singleton. East can now add to West’s 17 pts minimum his
own 6 HL pts (- 1 for no Queen) + 2 pts for “no wasted honor pts” in
hearts + 2 pts for the 9-card ª Fit with the King (West only counted
pts for an 8-card Fit) + 1 point for the two doubletons = 28 pts
minimum, 30 pts maximum = no slam, but game = 4 ª.

Example used in this book’s foreword.

Now we can revisit the first example used in this book’s foreword,
from G. Versini’s 1968 Bridge Dictionary, where he commented :
« in a case like this one, where the hands are valued at 23 HDS pts – far
below the 27 HDS pts needed for a game – the Goren point count does not
adequately translate this perfect Fit ». The hands were :
ª KQxx ª AJxx
© x © xxxx
¨ xxx ¨ xx
§ AQxxx § Kxx
50 years after this very pertinent observation, we should have
found a more accurate hand evaluation method that counts 29
total pts between these hands, not 23! Now, finally, we have :
West has 14 ½ HLD pts, not 11 H/12 HL! And East has 7 ½ H pts
(no Q) + 1 point for the 8-card ª Fit + 2 pts for the 8-card § Fit with
the King + 2 D pts for the ¨ doubleton with 4 trumps + 2 pts for
“no wasted honor pts” in © = 29 HLDFit pts.

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 WASTED HONORS POINT COUNT – ILLUSTRATIONS

And bidding the 4 ª contract could very easily be done, in a

manner very similar to the auction in the previous deal :
1§ 1ª
3 © (“mini-Splinter”) 4ª
Nowadays, such hands should no longer pose a problem – thanks
to an optimal hand evaluation in super accuracy !
Assuming that the bidding system played provides pertinent information
such as describing a singleton and its precise location.

THE GOREN « GAME EQUATION » FALLACY.

The quantification of “wasted honor points” totally debunks Goren’s
well known “game equation” which is : « An opening (13 HD pts)
opposite an opening (13 HDS pts) equals Game (26 total pts) ».
It is now clear that this equation has no validity. First, because
points are not counted correctly in the first place, but second,
because if one of the two hands has a singleton opposite “wasted
honor pts”, then the point total of the combined hands may only be
24 pts (- 2 pts). Illustration :
Here are two hands which, according to Goren’s “equation”, call
for a 4 © game :
ª KJxxx ª xx
© A 10 x x © Kxxx
¨ Kxx ¨ QJxx
§ x § KQx
Goren would have these hands bid as follows :
1ª 1 NT (forcing) or 2 ¨
2© 4 © (13 + 13 = 26)
Goren adds one point for the © King and 1 point for the ª
doubleton. Well, clearly 13 + 13 does not add up to game.

Let’s conclude this chapter with one final point on “wasted honor
points” by using an example deal found in the great American
champion, A. Morehead’s 1990 update of « Morehead on Bidding ».
In that book, the following hands of the same side are used to
illustrate that the valuation of honor cards can be all wrong :

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 WASTED HONORS POINT COUNT – ILLUSTRATIONS

ª KJxxx ª AQxxxx
© AKQJxx © xx
¨ x ¨ Kxx
§ x § Kx
With the following comment from the authors : « Despite their
combined honors, East/West cards cannot produce a slam because they
are missing two Aces. Here, it would be absurd to insist that two Kings
are worth 6 pts but one Ace is only worth 4 pts, when replacing the two
minor-suit Kings by one Ace would produce a slam, but with 2 pts less ».
Well, that is actually not the right way to account for the value of
honors. The right way is to accurately evaluate the combined value
of two hands. That is done by deducting 2 pts for each of the
“wasted honor pts” because the Kings are opposite singletons.
This reduces the value of the two Kings to 2 pts, from their initial
6 pts. The proper point count is therefore : 6 pts for the two Kings,
minus 4 “wasted honor pts” + 4 D pts for the two singletons = 6 pts.
While an Ace without any other honor, opposite a singleton,
retains its full value of 4 ½ pts. That’s : 4 ½ pts + 4 D pts for the two
singletons = 8 ½ pts. That’s 2 ½ points more, not less… And 2 more
points should be added for no “wasted honor pts” opposite the
second singleton. That’s 4 ½ points more, not 2 points less…
The key principle to retain from this is : the right hand evaluation
method and the right bidding system must provide the ability to
accurately re-evaluate the correct combined value of two hands.

2. MISFITS OPPOSITE LONG SUITS.

The precise quantification of “downgrading” and “upgrading” into
“wasted honor points” that we just reviewed refers to corrections
made by the hand that is opposite one with a short suit.
Surprisingly, J-R. Vernes and B. Charles say nothing about any
correction to be made by a player being short in partner’s long suit
– yet, that situation is just as important.
Indeed, one of the most critical challenges in Bridge is to be able to
detect quickly a misfit between two hands – before reaching too
high a level. And to avoid exceeding the bidding safety level, an
effective hand evaluation method should accurately, and quickly,
quantify misfits. Yet, the traditional point count totally ignores the
precise quantification of misfits – and it is indeed on the key issues
of Fits and Misfits that its deficiencies are the most significant.
It is these two key issues that we will now address.

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 MISFIT POINTS

A. « MISFIT » POINTS IN A SUIT.

A misfit in a suit occurs when one player is short in his partner’s
long suit – which is always a weakness when the long suit is of
5 cards or more. And being short in partner’s long suit must be
defined not just as a singleton or a void but also as a doubleton
without honor. And the negative value of such misfits can and must
be quantified, similar to that applying to the “wasted honor points”
count :
– When a player has a void in partner’s long suit (5 + cards), he
must downgrade his hand by 3 points ; when he has a singleton in
partner’s long suit, he must downgrade his hand by 2 points
(whether or not the singleton is an honor) and when he has a
doubleton without any honor in partner’s long suit, he must
downgrade his hand by 1 point. So, in summary :
– Deduct 1 point for a doubleton without any honor in partner’s long
suit ( 5 + cards).
– Deduct 2 points for a singleton in partner’s long suit – even if the
singleton is an honor.
– Deduct 3 points for a void in partner’s long suit.
Here are some examples illustrating this misfit point count :
Final of National selection, 1981.
ª xx ª Qxx
© A J 10 x x © Q
¨ Kxxx ¨ QJxx
§ Kx § AQxxx
3 teams out of 4 played 3 NT, down two. The auction was often :
1© 2§
2¨ 3¨
3© 3 NT
Comment by the reporter : « Can East, with his 13 points, be expected
to stop short of game ? That is beyond anyone’s capability… ».
13 points ? What 13 points ?!... East only has 9 ½ pts, not 13. Let’s
count them together : 7 ½ HL pts in clubs, 3 pts in diamonds, ½ pt
for the singleton © Q minus 2 misfit pts for the © singleton in
West’s 5-card suit, + 1 ½ pts for the isolated ª Q, minus 1 for no
King = 9 ½ pts. Therefore, East’s 2 § bid is inappropriate and the
bidding should have been :

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 MISFIT POINTS

1© 1 NT
2¨ 3¨
Pass
West does not have the points for a game – in NT or in diamonds.

From M. Bergen’s 2002 booklet on « Hand Evaluation ».

In this booklet, M. Bergen refers to the following hand, opened
1 NT by one of his readers :
ª A 10 x © A 10 x x x ¨ Axx § Ax
And comments on it as follows : « This hand is much too strong to be
opened 1 NT. Counting 4 ½ pts per Ace + ½ point for two 10s = 18 ½
pts, plus upgrading for the 5-card © suit. Open this hand 1 © ».
Well, such a point count will lead straight to 3 NT opposite this hand :
ª KQx © xx ¨ Kxxx § xxxx
A contract guaranteed to fail with opponents taking at least 3 clubs
and 2 heart tricks before Declarer can score 9 tricks.
But the right point count will avoid bidding this doomed contract :
First, the opener’s hand is, in fact, worth 17 HL pts, certainly not
18 ½ +, once 2 pts have been deducted for no King and no Queen
– and zero pts for two worthless 10s.
But it should be opened 1 © anyway (rather than 1NT with a
5-card Major) on which partner should count his hand 7 H pts
only, not 8, as 1 misfit point should be deducted for a doubleton
without honor in opener’s 5-card suit. The bidding will then be :
1© 1 NT (7/9 HL pts)
2 NT (invites game) Pass (minimum)

European championships, 2006 (France/Israël).

ª Axxxx ª ---
© Jx © Kxxx
¨ Axx ¨ KQJxxx
§ Kxx § 10 x x
The French team plays 3 NT, down two. That is because the misfit
was not precisely quantified.

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 MISFIT POINTS

On West’s 1 ª opening, East must count his hand for 10 pts : 13 HL

pts (10 pts in diamonds for 6 ¨ with 3 honors + 3 for the © K)
minus 3 for the ª void in opener’s 5-card suit = 10 pts. Therefore,
a 2 ¨ response by East is inappropriate. Instead, a 1 NT response
would avoid over-reaching to 3 NT and the side should be able to
sign off in 2 or 3 ¨.

Bidding contest – Bridge World, 2009.

ª KQJxx ª x
© Qx © AKxx
¨ Axxx ¨ xxxx
§ Jx § Axxx
Both pairs participating in the contest end up in 3 NT, after the
following auction :
1ª 1 NT (forcing)
2¨ 2 NT or 3 ¨
3 NT
A very poor contract, the result of East’s seriously deficient point
count. On West’s 1 ª opening, East only has 9 pts (no Queen) as
2 pts must be deducted for his ª singleton in partner’s 5-card suit.
The auction should be :
1ª 1 NT (10 H maximum)
2¨ 3¨
Pass
3 ¨ being an invitation to game to an opener having 16 H pts, for
3 NT or 5 ¨ – West, minimum, must pass.

Exercise : « Whose fault is it ? » – Le Bridgeur, 2012.

ª x ª KJxxx
© Kxx © Axx
¨ AKxxxx ¨ Qx
§ Kxx § J 10 x
The bidding was : 1 ¨ 1ª
2¨ 2 © (3rd suit forcing)
3¨ Pass

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 MISFIT POINTS

Comments from the East player who submitted the deal : « Result :
made 4 ¨ – but 3 NT was missed… After my strong 2 © bid, I had
nothing more to add ! But shouldn’t West, with his strong 13 pts, have
bid 3 NT ? ».
It’s always the partner’s fault, isn’t it ?!... But having 3 NT bid by the
player having a singleton is not the solution. And a poor point
count has its part of blame too. East has himself 12 ½ HL pts (2 pts
for J 10 of §) which justifies a perfectly descriptive 2 NT bid on his
part and bids NT from the right hand, too – instead of 2 © which
describes… nothing. Now West, with his 15 ½ HL pts (2 pts for 6
diamonds + 1 point for 3 Kings but minus 1 point for no Queen)
can deduct 2 pts for his singleton in partner’s long suit but add
1 point for a guaranteed 8-card ¨ Fit : that’s 14 ½ pts + 11 HL
minimum = 25 ½ pts. West can now bid 3 NT.

« Practice your defence » – Le Bridgeur, 2011.

ª Qxxx ª AJ
© Q © Axxxx
¨ Qxxxx ¨ 10 x x
§ AQx § Kxx
The bidding was : 1¨ (!) 1©
1ª 2§
2 NT 3 NT
Down 2. Impressive opening by West !… who must have “seen”
11 or 12 H pts in his hand ?!... when he only has 9 !
With no point to be added for the 5 diamonds not headed by at
least Q J or K, - 1 for no King and only ½ point for the © singleton
Q, West only has 9 H pts and 11 HLD and should not open.
The bidding should have been :
Pass 1©
1ª 1 NT (12/14 HL)
Pass
Now, on East’s 1 © opening, West has even fewer than 9 H pts !
He only has a grand total of 7 pts once 2 pts are deducted for the
© singleton “misfit” in opener’s 5-card suit. West must pass on
opener’s 1 NT.

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 MISFIT POINTS

B. « MISFIT » POINTS IN NO TRUMP.

“Misfits” also exist at NT : when there is a singleton (or, worse, a
void) in one hand or the other. These cases were identified by
J-R. Vernes and B. Charles who wrote this on the subject, in 1995 :
« A singleton is always a weakness at NT. But statistical analysis reveals
that the importance of a singleton, in a NT contract, varies considerably
depending upon whether it is in dummy’s hand or in declarer’s hand.
This is quite naturally explained by the fact that a singleton in declarer’s
hand exposes dummy to a lead “through” its guard(s) in the suit, which
is not the case when the singleton is in dummy’s hand.
It results from this that one must avoid, as much as possible, to
announce NT himself when his hand has a singleton ».
Wow ! Just think about how often we see champions bid NT with
a singleton in their hand! Some even open 1 NT with a singleton !
Unfortunately, J-R. Vernes and B. Charles did not quantify their
very pertinent conclusion and the clear distinction that should be
made between the two situations.
This can and should be precisely quantified, as follows :
when a NT contract is proposed by one player or the other, 2 points
should be deducted for a singleton in the hand of the player bidding
NT first, while 1 point should be deducted for a singleton in the
hand of the player who supports a NT bid by his partner.
If the short suit is a void (playing a NT contract with a void is not
recommended!), deduct 3 points in the first case, 2 points in the
second case. Illustration :
ª KJxx ª Axx
© xxx © Axxx
¨ KQxx ¨ x
§ A 10 § KJxxx
If the bidding is : 1¨ 1©
1ª 2§
2 © (3 hearts) ?
To bid NT now, East will have to deduct from his 13 HL pts 2 pts
for his singleton in the hand bidding NT first.
It is therefore 2 NT, invitational, which he should bid, not 3 NT, as
West has not guaranteed more than 12 HL pts.

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 MISFIT POINTS

While if the bidding is :

1 NT (12/14 H) 2 § (Stayman )
2 ª (4 spades) 3 § (5 + clubs, no ¨ guard)
3 NT (¨ guarded) Pass
NT being bid, this time, by the hand without singleton, East only
has to deduct one point for his singleton, giving him 12 HL pts.
Over his partner’s 3 § bid, West, with 13 ½ H pts and guarding the
diamond suit, can now bid game in NT.
And one can see that 3 NT has much better chances of succeeding
if declared by West, the diamonds being protected, first, against a
diamond lead, and later, against a diamond return should a spade
finesse fail, putting North back on hand. While if NT is declared
by East, a diamond lead through West’s honors, followed by a
diamond return, will seriously endanger the 3 NT contract.

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 OPTIMAL HAND EVALUATION

CHAPTER 4

Point count in response to partner’s opening

Although J-R. Vernes and B. Charles did state that « honors located
in partner’s long suits have greater value than when located in partner’s
short suits », they did not quantify their difference in value.
The traditional point count partially quantifies this statement when
it advocates to add 1 point for an honor in the trump suit –
whenever the initial count of this honor (or honors) did not exceed
3 points. And this additional point, as previously identified in this
book, should be added for an honor in any and all suits having a fit
of at least 8 cards – not just in the trump suit – and this applies to
NT contracts, as well.
But there is another case where 1 point must be added for an honor
(other than the Ace) in partner’s long suit : that is when there is a
“semi-fit” in partner’s long suit (5 cards or more).
1. « SEMI-FIT » POINTS
There is a “semi-Fit” when a hand has an honor doubleton (other
than an Ace) in partner’s long suit (5 + cards). This applies to both
hands – whether in response to an opening bid or in rebid position.
While an Ace doubleton does not need to be upgraded, as its value
remains the same regardless of its location and its 4 ½ pts value
already factors this in, any of the other three honors must be upgraded.
– In such cases, 1 semi-fit point must be added. Thus, a
Jack doubleton in partner’s long suit counts for 1 point (instead of
zero), a Queen doubleton counts for 2 points (instead of 1) and a
King doubleton counts for 4 pts (instead of 3).
This upgrading of a doubleton King, Queen or Jack in partner’s
long suit is justified as follows :
– For a Jack or Queen doubleton : their value being counted for
one point less than the traditional count (zero and one point,
respectively, instead of 1 and 2 pts) should be upgraded when
opposite a partner’s long suit where they could now considerably
facilitate the development of partner’s long suit – opposite, for
example : K x x x x or K J x x x, or K Q x x x or A J x x x or
A Q x x x or A K x x x.
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 SEMI-FIT POINTS

– As for the King doubleton, its value will most often become quite
superior in partner’s long suit to what it would be in another suit.
The examples below illustrate this quite clearly :

The combinations below : are better than these :

ª Axxxx Kx ª Axxxx xx
© xxx xxx © xxx Kxx

ª Axxxx Kx ª Axxxx xx
© Axx xxx © Axx Kxx
ª Axxxx Kx ª Axxxx xx
© Qxx xxx © Qxx Kxx
ª QJxxx Kx ª QJxxx xx
© xxx xxx © xxx Kxx
ª QJxxx Kx ª QJxxx xx
© Axx xxx © Axx Kxx
ª QJxxx Kx ª QJxxx xx
© Qxx xxx © Qxx Kxx
Here are a couple of examples of semi-Fit point calculations :
European Championships, open by teams, 2008.
ª Ax ª Kxx
© AKxxx © Qx
¨ Axxx ¨ J 10 x
§ xx § K J 10 x x
40 % of the teams stopped in 2 © after the following auction :
1© 1 NT (limited to 10 H pts)
2¨ 2 © (simple © preference)
Pass
40 % of the best world-class players, in 2008, do not count East’s
hand worth more than 10 pts ?!
Well, that does correspond to what is still being universally taught
today : no point counted for the two 10s accompanying Jacks and
no additional point counted for the 5 clubs, even with K J 10 !...

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 SEMI-FIT POINTS

And since the correct point count is not taught, judgment must
replace it. Does that mean that 40 % of the world’s best players do
not have the judgment needed for proper hand evaluation ?
If so, what about the rest of us ?...
What should be taught is this : East has 13 HL pts : 6 pts in clubs,
2 in diamonds, 3 in spades, 1 point for the © Q doubleton + 1 semi-
fit point for the Q doubleton in opener’s 5-card suit = 13 HL pts.
East players should therefore have bid 2 §, certainly not 1 NT.
Followed by 2 NT, or 3 NT, on West’s 2 ¨ rebid.
West has, himself, 16 ½ HL pts (- 1 for no Queen), minus 1 for a
doubleton without honor in partner’s 5 club-suit = 15 ½ HL pts
opposite 12/13 HL pts = 27 ½ pts minimum. He will bid 3 NT on
East’s 2 NT.

International tournament, 2010.

ª AKxxx ª Qx
© QJxxx © K 10 x
¨ x ¨ xxxxx
§ Ax § xxx
Most teams stopped in 2 ©, missing the 4 © game. And the expert
reporting the deal comments : « stopping in 2 © is understandable;
after all, the side only has 19 H pts and 23 HDS pts ».
A comment similar to the one made by G. Versini with regards to
the first deal presented in this book’s foreword. But his comment
was made in 1968 – while this one was made in 2010 !
Let’s bring some modern era counting to all this : West has 19 HLD
pts (1 point for each 5-card suit, 9 pts for 2 Aces) and his rebid
should be 3 ©, not 2 ©, which will describe a strong 5 ª 5 ©.
On 3 ©, East can count his own hand as : 4 ½ H pts ( ½ point for
the 10 of © + 2 Fit pts for 8 © with the King + 1 D point in spades
+ 1 semi-fit point for the ª doubleton Q + 2 pts for “no wasted honor
pts” in a minor suit, as opener’s bid promises a singleton in a minor
suit with 5 – 5 in the Majors = 10 ½ HDFit pts. Opposite a minimum
of 18 HLD pts, East will have no problem concluding in 4 ©.
Had East’s honors been in his minor suits, instead, he would only
have 1 Fit point in hearts, not 2, and would deduct 2 pts for “wasted
honor pts” in his minor suits, instead of adding 2 pts. With only
5 pts, he would pass om 3 ©. Optimal hand evaluation !

75
 POINT COUNT IN RESPONSE TO A NO TRUMP BID

2. POINT COUNT IN RESPONSE TO A NT BID

In response to a NT opening, 1 NT or 2 NT, almost all Bridge books
refer to points being counted as honor pts, HL pts for some, and
specifically that the 2 § Stayman response requires a minimum of
8 H pts on a 15/17 1NT opening, 10 H pts on a 13/15 1NT opening
or 11 H pts on a 12/14 weak 1 NT opening.
The justification being that : 1) if a 4 - 4 fit in a major is not found,
the side must have at least 23 H pts to play at the level of 2 NT, and
2) there is no reason to look for a NT game if the side has less than
25 H pts.
But we know now, don’t we, that this is a serious point-count error
– as it doesn’t take into account Fit points which apply just as well
to NT contracts. As a result, a side where a hand has been opened
1 NT can be upgraded by as many as 4 or 5 points just by finding an
8-card Fit. The following examples will illustrate this :
ª Axx ª x
© AQxx © Kxxx
¨ Kxx ¨ QJxx
§ Axx § xxxx
One can see that 4 © should be played with these hands, even
though East only has 6 H pts. The “correct” count should be : West
has 17 ½ pts (- 1 for 4 3 3 3) and, once the © Fit is found, East has
6 H pts + 2 pts for the 8-card © Fit with the King + 3 pts for the ª
singleton with 4 trumps = 11 HDF pts for a total of 28 ½ pts = 4 ©.
For East to pass on a 15/17 1NT opening for having less than the
“prescribed” 8 H pts would therefore be a serious mistake.
Where is the problem? The problem is that East must count his
2 Distribution pts for the ª singleton, which gives him 8 HLD pts.
And, once modified by removing East’s singleton and the © Fit,
these hands, with the same honors, could still have a point count
justifying, this time, playing 3 NT, such as :
ª Axx ª xx
© AQx © Kxxx
¨ Kxxx ¨ QJxxx
§ Axx § xx

76
 POINT COUNT IN RESPONSE TO A NO TRUMP BID

Now, West still has the same 17 ½ pts, and East has 7 HL pts/
8 HLD pts (1 D pt for the 2 doubletons) + 3 pts for the 9-card ¨ Fit
with Q J for a total of 27 ½ HLFit pts = 3 NT.
Here again, passing on a 15/17 1NT opening, counting East for only
6 H pts or 7 HL pts, would be a mistake.
It is, in fact, 8 HLD points that are needed for a 2 § Stayman on a
15/17 1NT opening – not 8 H pts !
Conclusion : in response to a NT opening – or to a NT rebid by the
opener – the responder must count his HLD points, not his H
points only! And the principle of this rule is quite logically this :
one of the two hands of the same side must always count its
distribution points. Therefore, on a 1 NT opening – which does not
count distribution pts – the responder must count his D pts.
When the responder does not find an 8-card Fit in a Major while
the opener is minimum, in most cases it will be better to play a suit
contract, even at the level of 3, rather than 2 NT.
Therefore, making it a pre-requisite for the side to have no less than
23 H pts (to be able to play 2 NT) to make a 2 § Stayman inquiry
over a 1 NT opening is a serious error.
Here are some examples illustrating the proper point count after a
1 NT opening :

Regional tournament, 2005 (U.K.).

ª xxxx ª Axx
© Axx © xx
¨ AQxx ¨ K 10 x x
§ Kx § Q J 10 x
The Acol bidding was :
1 NT (12/14 H) Pass
And the 3 NT game was missed. But East has 12 H pts (1 ½ pts for
the two 10s), not 10, and should bid on. And on an invitational bid
by East, the opener should bid 3 NT as he has 14 H pts (9 pts for
2 Aces), not 13.
Actually, these two hands add up to 28 HF pts with 2 pts for the
8-card ¨ Fit with the King.

77
 POINT COUNT IN RESPONSE TO A NO TRUMP BID

Here is another example of the upgrading of a side – by 6 pts, this

time !
Regional tournament, 2010.
ª Kxxx ª AQxx
© Ax © xx
¨ KQxx ¨ Axxxx
§ Axx § xx
The most frequent auction was :
1 NT (15/17) 2 § (Stayman)
2ª 4ª
With 12/13 HDS pts in spades, East players did not “see” a slam
opposite a maximum of 17 pts and concluded in 4 ª.
But the correct point count tells a different story : upon West’s 2 ª
bid, East can count : his own 12 HLD pts (minus 1 for no King but
1 D point for two doubletons) + 1 point for the 8-card ª Fit + 2 D
pts for a doubleton in a hand with 4 trumps for a total of 15 HLD
pts. Should West be maximum and/or have a ¨ Fit, the side would
be in slam zone. And this could be found out simply and quickly
with the following auction, instead of the one above :
1 NT (15/17) 2 ¨ (game forcing Stayman)
2ª 2 NT (relay – continue)
3 ¨ (4 diamonds)
Upon West’s 3 ¨ bid, East can now add 2 pts for the 9-card ¨ Fit
and is now sure to be in slam zone, without needing West to be
maximum. Now Key card slam Asking bids by East will get him
to 6 ª. The side actually has 35 HLDF pts.

DN 1 (National Division) by team of four, 2015.

One last example now, illustrating several elements of the optimal
point count identified earlier in this book :
ª Axx ª x
© Kxxx © A J 10 x
¨ AKxx ¨ J 10 x x x
§ Kx § xxx

78
 POINT COUNT IN RESPONSE TO A NO TRUMP BID

In the semi-finals of DN 1, in 2015, 3 teams out of 4 bid as follows :

1 NT (15/17) Pass
East, not having the “prescribed” 8 H pts, passes…
Cumulative point count errors which lead to missing the 4 © game.
East must count his HLD points on a NT opening! and, of course,
his 10s. Which gives him 8 ½ HLD pts (- 1 for no King, - 1 for no
Queen), not 6 H pts. He should make a 2 § Stayman inquiry. With
1 point for the © Fit + 1 more point for the ª singleton with
4 trumps, the side may now be able to find the © game. Maybe...
But the first point count mistake comes from the opener who has
18 H pts, not 17 (9 pts for two Aces + 1 point for 3 Kings - 1 point
for no Queen). A 1 NT opening was therefore not the right bid.
A 1¨ (or a strong §) opening would enable West to reach the
proper point count immediately upon East’s 1 © reply as he could
add 2 pts for the 8-card © Fit with the King + 2 D pts for the
§ doubleton with 4 trumps = 22 HDF pts + 5 H pts minimum in
East = 27 HDF pts. Actually, the side has 3 additional pts for the
9-card ¨ Fit with J 10.
Better yet, a strong 1 § opening would allow to bid the game from
the “right” hand, West, thus protecting the § K against a club lead :
1 § (16 H +) 1¨
1 NT (18/20 HL) 2 § (Stayman)
2© 4©
10 ½ HDF pts in East + 18 HL pts in West = 28 ½ pts = 4 ©
Surely, either auction above would be better than :
1 NT Pass

And there is more yet to properly evaluate the “true” worth of two
hands than counting the hand responding to a NT opening in HLD
points. The NT opening hand itself can be re-evaluated by many
points when finding the “perfect” Fit in a long suit held by the
responder – would you believe by as many as 8 or 9 pts ?!
Interestingly, in his excellent book « The No Trump Zone », Danny
Kleinman alludes to it – yet misses the mark by many points.
In that book, he refers to the « 20 point No Trump », illustrated by
the two hands below :

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 POINT COUNT IN RESPONSE TO A NO TRUMP BID

ª A J 10 ª xx
© Kx © Axx
¨ AQxx ¨ KJ
§ K 10 x x § QJxxxx
About these two hands, Danny Kleinman makes the following
comments : « The auction at nearly every table was short and sweet :
1 NT (15/17) 3 NT
But 6 § was odds-on! What went wrong? A transfer bid for clubs by the
responder would have more appropriately described a club suit deserving
mention (strong 5 or decent 6) and, in support of partner’s clubs, opener’s
hand is now worth a good 20 pts, and West should be able to envision a
slam and proceed to get to 6 § ».
« Worth a good 20 pts » means that D. Kleinman adds 3 pts to West’s
initial 17 H point count – but we know now, thanks to the optimal
point count, that this is well short of what the re-evaluation should
be, that is : 5 or 6 pts with East having 5 + clubs: 3 or 4 Fit pts (+ 1
point for the King) + 2 D pts for a doubleton in a hand with
4 trumps. So, after a transfer bid for clubs by the responder, West
can count his own hand for 22 HDF pts minimum and, assured of
at least 11 HL pts in responder’s hand, he can envision a possible
slam and proceed to 6 §.
Of course, I am sure that you didn’t fail to note that West’s hand
has 19 ½ H pts, not 17, and that opening it 1NT, 15/17, was really
horrible. East having himself 13 ½ HL pts, and the two hands
having 4 Fit pts for their 10-card § Fit, missing the slam truly
requires an extraordinary effort – if you count your points properly!

And here is now, below, an opener’s hand that is worth 8 more pts
than initially counted, once a 10-card Fit has been found :
ª Kxxx ª AQxxxx
© Ax © Kxx
¨ AKQx ¨ xxx
§ xxx § x
Say the bidding started as :
1 NT (15/17) 2 © (transfer bid for ª)
2 NT (super-accept) 4 § (§ singleton, 6 spades)

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 POINT COUNT IN RESPONSE TO A NO TRUMP BID

West can now count 25 HDF pts for his own hand : 4 Fit pts for the
10-card ª Fit with the King + 2 D pts for the © doubleton with
4 trumps + 2 pts for “no wasted honor pts” in clubs. That’s 8 more
pts than the 17 H pts initially counted !
The bidding could then proceed, for example, as follows :
4 ¨ ? (RKC Ask, ª trump) 4 ª (1 Key Card)
4 NT ? (ª Q ?) 5 © (yes, ª Q + © K)
6ª
And had East indicated a void in clubs, instead of a singleton, then
West would add a 9th point to his own count ! (+ 3 pts for “no
wasted honor pts” opposite a void) and would be headed for the
grand slam in spades. Optimal point count in super accuracy !
It is the “26 point No Trump” that D. Kleinman should have alluded to
in his book « The No Trump Zone », not the 20 point NT !

81
82
 OPTIMAL HAND EVALUATION

CHAPTER 5

Point count : Summary chart and Conversion Table

Converting Points into Corresponding Tricks

conclusion : The analysis just completed on hand evaluation

leads to a clear conclusion : the traditional M. Work/C. Goren point
count, as universally taught and subscribed to still today, is
seriously flawed and certainly inadequate to offer a sufficient or
satisfactory level of precision. Nevertheless, it does constitute the
soundest foundation from which a very precise optimal point
count hand evaluation can be designed – with the appropriate,
necessary corrections to make to it.
And achieving this can essentially be done by, first :
– Applying the most basic and fundamental corrections identified
over 50 years ago now, and that is : deduct 1 point for a 4 3 3 3
distribution, for an Aceless opening hand, for an honor singleton,
or for two honors doubleton.
And, second, by integrating to the traditional count four key
elements identified and illustrated in this book :
– Counting 4 ½ pts per Ace, deducting half a point for an isolated
Queen or Jack, counting 1 point for each 10 which accompanies a
Queen or a Jack, and deducting 1 point for a hand without a King or
a Queen. And this applies to all suits and all contracts.

– Counting appropriately Length points – which remain the same

for a suit contract or for NT.

– Counting appropriately Distribution pts and “wasted honor pts”.

– Counting appropriately Fit and misfit points which apply to all

suits and all contracts.
This book has hopefully demonstrated that these necessary
corrections are well worth the accuracy they bring to hand
evaluation. And mastering them will quickly become routine as
constantly practiced on every hand.

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 OPTIMAL HAND EVALUATION

THE OPTIMAL POINT COUNT VS « TRICK » COUNTS

Now, let’s see whether the optimal point count can also evaluate
with great precision unevenly-distributed hands with one long
suit, typically the very hands for which some experts advocate to
use a “trick” count : the LTC count (Losing Trick Count) for some,
the PTC count (Playing Trick Count) for others, or the TTC count
(Total Tricks Count) for some others.
To do this, let’s take long single-suit hands with a singleton or a
void, with enough honor points to justify a normal opening :
1) 8-card single-suit hand :
ª AKQxxxxx
© Ax
¨ xx
§ x
This hand can be expected to deliver, by itself, 9 tricks in spades as
trump, the equivalent of 24 ½ HLD points (27 HLD total points for
10 tricks = 2.7 points per trick x 9 = 24.3 pts).
The traditional count gives this hand only 21 HLD pts (2 pts for 6
spades + 1 point for each card after the 6th, 2 pts for the singleton
and 1 point for each doubleton). That’s only 8 tricks, not 9.
While the optimal count gives this hand 25 HLD points (2 pts for
each card after the 6th + 2 pts for 3 honors in a long suit, 9 pts for
2 Aces, only one point for two doubletons).
2) 9-card single-suit hand :
ª KQJxxxxxx
© Ax
¨ xx
§ ---
Here again, this hand can deliver, by itself, 9 tricks in spades as
trump, the equivalent of 24 ½ HLD points.
The traditional count gives this hand only 20 HLD points (3 pts for
the void and 1 point for each doubleton). Not quite enough for 8
tricks.
While the optimal count gives this hand 25 ½ HLD points (4 pts for
the void, 1 point for two doubletons).

84
 THE OPTIMAL COUNT VS « TRICK » COUNTS

Extending this demonstration to 10 + cards single-suit hands will

produce the same result – the optimal count eliminates the need
to consider using another method to accurately evaluate unevenly
distributed hands.
And what about long suits headed by Q J 10 instead of Ace and
King? A legitimate question from anyone wondering whether
these weaker honors would value such hands appropriately,
compared to a trick count. Well, let’s see :
8-card single-suit hand headed by Q J 10
ª Q J 10 x x x x x
© AKx
¨ x
§ x
This hand will deliver, by itself, 8 tricks in spades as trump, the
equivalent of 21 ½ HLD points.
The traditional count gives this hand only 18 HLD points. That’s
barely 7 tricks, not 8.
While the optimal count gives this hand exactly 21 ½ HLD points.
And the count will be the same with a void instead of 2 singletons.
And what about 5 – 5 hands ? Let’s see :
ª Q J 10 x x
© xxx
¨ AKxxx
§ ---
This hand can be expected to deliver between 6 and 7 tricks in
spades or diamonds as trump, the equivalent of 17 ½ HLD points.
The traditional count gives this hand only 15 HLD points, the
equivalent of 5 ½ tricks, not 6 ½.
While the optimal count gives this hand exactly 17 ½ HLD points.
Conclusion : unevenly-distributed hands are very precisely assessed
by the optimal count, which totally eliminates the need to consider
using another method to evaluate unevenly-distributed hands.
A few examples will now illustrate that there is no need to resort
to such trick counts. Let’s compare, for example, the optimal point
count to the Losing Trick Count.

85
 THE OPTIMAL COUNT VS THE “LOSING TRICK” COUNT

First, let’s review some fundamentals about the LT Count :

It was conceived, in the 1930s, because the traditional point count
was considered, justifiably, inadequate to accurately evaluate
irregular hands and the full value of trump Fits.
It applies only to situations where a trump Fit (8 + cards) has been
identified between partners.
– It consists of counting losers for only the first three cards of a suit
i.e. A x x x = 2 losers A Q x x = 1 loser / 1 ½ losers for some.
– Kings or Queens are considered winners even when not accom-
panied by another honor (the Queen was later revised to ½ winner).
– It estimates the number of tricks a side can make by adding the
losers from each hand and then deducting the total from 24 (24 being
the most losers two hands can have, 12 per hand).
A typical example would be :
ª Axxxx ª KQxx
© xxx © Ax
¨ AKxx ¨ Qxx
§ x § xxxx
The West hand has 7 losers and is opened 1 ª. Responder has
7 losers and bids 3 ª (equivalent to 10/12 H pts in traditional point
count). Opener counts 7 + 7 losers = 14 deducted from 24 = 10
tricks are likely to make. Opener will therefore bid 4 ª.
Question : « Would you find that 4 ª game using the “traditional” point
count which gives these two hands only 22 H pts and 25 HDS pts ?... ».
Well, today the answer to that question is : yes, certainly !
As the LTCount was conceived long before we learned from
J-R. Vernes that 2 additional points should be counted for a 9-card
trump Fit, that a doubleton in a hand with 4 trumps is worth 2 pts
and of course about “wasted honor points”.
East’s hand is not at all worth 12 HD pts but 15 HDF pts ! (+ 2 pts
for the 9-card ª Fit + 2 D pts for the © doubleton with 4 spades).
The bidding could therefore be : 1 ª 4ª
However, neither auction is satisfactory as neither one identifies
West’s singleton and therefore whether there are, opposite the
singleton, “wasted honor points”. With the ¨ Queen being in clubs,
instead, the LTC count will be the same but 4 ª is no longer a good
contract as its success now depends on the opponents’ diamonds
being divided 3 - 3 (only a 35 % probability).

86
 THE OPTIMAL COUNT VS THE “LOSING TRICK” COUNT

A better auction would be :

1ª 2 NT (Jacoby raise)
3 § (§ singleton) 4ª
With a ¨ singleton, instead of clubs, East would sign off in 3 ª.
Another typical example used by LTC advocates would be :
ª AQxxx ª KJxx
© A Q 10 x x © Kx
¨ x ¨ xxxx
§ xx § xxx
The argument being : « Upon a 2 ª reply from partner on your 1 ª
opening, would you bid again with only 12 H pts or 14 HL pts in
“traditional” point count ? The LTC count would be more helpful to find
the 4 ª game as opener can count 5 losers in his own hand + 9 losers in
partner’s hand = 14, deducted from 24 = 10 tricks are likely to make ».
Well, that may be a problem for those counting only points either
for Length or for short suits but it won’t be a problem with the
optimal point count which counts points for both Length and
Distribution as the above opener has 17 HLD pts (no King), not
12 H or 14 HL pts! Therefore, West will obviously bid again : 3 ¨,
short-suit trial bid, invitational, or 3 ©, describing a strong 5 – 5 in
the Majors. East can now add 2 pts for “no wasted honor pts” in
either minor suit + 2 pts for the 9-card ª Fit + 2 D pts for his ©
doubleton with 4 trumps = 12 HDF pts + West’s 16 pts minimum
= 28 pts = 4 ª. No need for an LTC count.
Here is another example, taken from « How good is your Bridge
hand », 2000, by R. Klinger and A. Kambites :
ª xx ª xxxx
© AQxx © J 10 x x x
¨ xx ¨ AK
§ A Q 10 x x § Jx
Comments from the authors : « West has 6 losers and East has 8 losers.
The bidding should go :
1§ 1©
3© 4©
« The Losing Trick Count has just enabled you to reach an excellent game
with only 21 H pts ».
87
 THE OPTIMAL COUNT VS THE “LOSING TRICK” COUNT

21 H pts – really ?... Let’s see : East can count, initially, 7 ½ H pts in
his hand (2 pts only for his ¨ K doubleton, no L pts for his 5 hearts
not headed by at least 3 H pts and - 1 point for no Queen). On his
1 © response, West can count 17 HLDFit pts and raises hearts to
3 ©. East can now add to his hand 2 pts for a 9-card © Fit with J 10
(West is unaware of East’s 5 hearts) + 1 point for his two
doubletons + 1 point for J x in opener’s long suit = 10 ½ HFit pts
/11 ½ HDF pts. Opposite a minimum of 17 HLDF pts, East can now
conclude in 4 ©. Optimal point count accuracy !
Note that the two hands have a combined total of 28 ½ HLDFit pts
– who cares about how many H pts the two hands have !
There is no need for the LTC count.

One last example, from the same book :

ª KQxxxx ª J 10 x x
© AJxx © xx
¨ Axx ¨ K 10 x
§ --- § Qxxx
Authors’ comments : « West has 5 losers and East 9 ½ losers. But
because the West hand has more Aces than Queens, you should upgrade
it to 4 losers only ». The bidding should go :
1ª 2 ª (9 losers)
4ª Pass
« You have reached an excellent game despite having only 20 H pts ».
But had East’s King been in clubs instead of diamonds, 4 ª would
go down – while with Q 10 in hearts, instead of clubs, it is a 6 ª
slam that should be bid! Such “blind” bidding is pure nonsense !
And so is bidding these two hands without even identifying West’s
short suit in clubs – this defies Bridge bidding most elementary
common sense…
And referring to H points… with West’s 6 4 3 0 hand is absurd !
Now, let’s do some serious Bridge counting and bidding here :
West has 21 HLD pts (2 pts for the 6 spades, 9 pts for 2 Aces and
4 pts for the void), and on a 1 ª opening, East has 12 HDFit pts
(2 pts for J 10 of spades, 3 pts for a 9-card ª Fit with J 10, ½ point
for the 10 of diamonds with the King and 2 pts for the © doubleton
in a hand with 4 trumps).
So, a : 1 ª – 2 ª bidding start is not even conceivable !

88
 THE OPTIMAL COUNT VS THE “LOSING TRICK” COUNT

Much better bidding dictates to open West’s hand with a forcing

bid and, playing a strong Club system, bidding would look like :
1 § (16 + HL pts) 1 ª (4 + spades, 7 + HL pts)
3 § (“splinter”, ª Fit) 4ª
On East’s 1 ª bid, promising 7 + HL pts, West can now safely, and
economically, “mini-splinter” to indicate his ª Fit and his short §
suit. On which East, despite his “wasted” § Queen, concludes in
4 ª as he has 5 ½ H pts (no pts for his § Queen) + 1 point for J 10
in the trump suit (which partner is unaware of) + 2 pts for a
doubleton in a hand with 4 trumps = 8 ½ HDFit pts and he is
assured of at least 19 + HLDFit pts in West (16 + HL pts + 1 Fit
point minimum in spades + 2 pts for his § singleton).
While, with his King in clubs, instead of diamonds, East would
only bid 3 ª. And with Q 10 in hearts, instead of clubs, he would
invite slam (no “wasted honor pts” in clubs) by bidding his
¨ control : 3 ¨.
Accurate point count should and will lead to better results –
assuming that your bidding makes sense in the first place !

Conclusion : the optimal point count is all that is needed to

correctly evaluate the combined trick potential of two hands,
whether their distribution is even or uneven. There is no need to
have to resort to any kind of “Trick Count” whatsoever.
____________________________________________________________

This concludes the analysis of the various corrections to make to

the traditional point count to obtain an optimal hand evaluation.
You will find hereafter in this book :
1) A complete and final summary chart, which includes all the
elements quantifed earlier – followed by a conversion table
converting total points into corresponding tricks.
2) Exercise hands illustrating the optimal point count.

89
COMPLETE AND FINAL SUMMARY OF THE OPTIMAL POINT COUNT

HONOR POINTS

Ace : 4 ½ pts K : 3 pts Q accompanied : 2 pts Q isolated : 1 ½ pt

J accompanied : 1 pt J isolated : ½ pt
10 isolated or with an Ace : 0 pt 10 with a K : ½ pt
10 with a Q or a J : 1 pt J 10 x : 2 pts

No Ace : - 1 pt (opening hands only)

No Q : - 1 pt
No K : - 1 pt 3 Ks : + 1 pt 4 Ks : + 2 pts

Singleton Honor : - 1 pt
2 Honors doubleton : - 1 pt Q or J doubleton : - ½ pt
(A Q, A K, K Q, Q J) i.e. : Q x : 1 pt J x : 0 pt
No deduction for a J doubleton
with an A or a K as A J is better than A x and K J better than K x.
3 + honors in a 6-card suit : + 2 pts in a 5-card suit : + 1 pt

LENGTH POINTS

5-card suit (with Q J or K minimum) : 1 pt 6-card suit : 2 pts

6-card suit without Q J or K : 1 pt
7-card suit (or longer) : 2 pts for each card from the 7th on

DISTRIBUTION POINTS

Void : 4 pts Singleton : 2 pts One doubleton : 0 pt 2 doubletons : 1 pt

4 3 3 3 Distribution : - 1 pt A singleton at a NT contract : - 1 pt

MISFIT POINTS

Opposite a long suit (5 + cards) in Partner’s hand :

- 3 pts for a void - 2 pts for a singleton
- 1 pt for a doubleton without honor
2 perfectly “Mirror” Hands : - 2 pts 2 “Mirror” suits : - 1 pt

90
 COMPLETE AND FINAL SUMMARY OF THE OPTIMAL POINT COUNT

SEMI-FIT POINTS

+ 1 for K, Q, J 10 or J doubleton in partner’s long suit (5 + cards).

FIT POINTS

8-card Fit : + 1 pt 9-card Fit : + 2 pts 10 + card Fit : + 3 pts

+ 1 for honor(s) in suit(s) with a Fit when the honor(s) are < 4 pts
These Fit points count for All suits and All contracts (suit or NT)

DISTRIBUTION-FIT POINTS WITH A TRUMP FIT

The difference between the number of trumps and the number of

cards in the support hand’s shortest suit.

number of trumps : 4 3 2
Distribution-Fit points Void : 4 pts 3 pts 2 pts
Singleton : 3 pts 2 pts 1 pt
Doubleton : 2 pts 1 pt 0 pt

WASTED HONOR POINTS

Honors (except Aces) opposite a singleton : - 2 pts

opposite a void : - 3 pts
Not a single honor : opposite a singleton : + 2 pts
opposite a void : + 3 pts
An Ace without any other honor : opposite a singleton : + 1 point

This summary of the many corrections needed to obtain an

accurate point count clearly exposes the historical fallacy that the
“basic” 4 3 2 1 point count is simple and reasonably accurate.
The reality is that there is no such thing as a simple and reasonably
accurate “basic” 4 3 2 1 point count and it is high time to stop
making that false claim and teaching such count as this does not
advance Bridge Hand Evaluation progress.

91
92
 CONVERSION TABLE : CONVERTING POINTS TO CORRESPONDING TRICKS

To be of value, a point count must translate into corresponding

tricks, at each specific playing level – partial, game or slam.
Interestingly, the optimal point count conversion table happens
to be identical to the traditional conversion table – but this is purely
coincidental as the traditional conversion table is pure fiction, is
totally flawed and defies statistical findings.
But this coincidence is very practical and sure makes things easy
for everyone !
CONVERSION TABLE

At the level of : 2 Tricks 3 Tricks 4 Tricks 5 Tricks 6 Tricks 7 Tricks

________ _______ ________ ________ ________ _______
Total points needed
At No Trump : 22 23 24 25 26 27 28 29 30 31 32 33 33 34 35 36 37 +

At a suit : 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 +

The number in bold, in the middle of each series of three numbers,

represents the optimal number of total points between two hands
needed to bid to the level of tricks indicated :
For part-score and game contracts, the first number in each series
corresponds to a probability of success around 40 to 45 % and
constitutes the minimum number of points required to play at that
level, the middle number corresponds to a 50 to 55 % success
probability, the third number corresponds to a 60 + % probability.
For small slams, the first number corresponds to a success
probability of 50 % (minimum number of points required to play at
that level), the middle number corresponds to a success
probability of 55 to 60 %, while the third number corresponds to a
success probability of 65 % +.
For grand slams, the first number corresponds to a 70 % success
probability (minimum number of points required to play at that
level), the middle number corresponds to a success probability of
70 to 75 %, while the third number corresponds to a probability
over 75 % +.
The key reference numbers to remember – those allowing an
optimal chance of success – are therefore the following :

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 CONVERTING POINTS INTO CORRESPONDING TRICKS

26 points (HLF pts) to play 3 NT, 27 pts (HLDF pts) to play a game
contract in a Major suit, 30 pts (HLDF pts) for a game contract in a
minor suit, 33 pts (HLDF pts) for a small slam in a suit and 34 pts
(HLF pts) for a small slam in NT.
The major and significant difference between the above
conversion table and the traditional table is that the total number
of points shown here includes All points : Honor pts, Length pts,
Distribution pts, Fit and Misfit pts, and “wasted honor pts” !
The optimal point conversion table above is based on empirical
statistics (see the methodology used in N.B. below).
While the traditional table is based on silly principles such as :
33 pts minimum for a small slam because this number guarantees
that no more than one Ace will be missing ! or : « a total of 40 honor
pts in the deck, divided by 13 tricks = 3 pts per trick… », as if only
honor pts counted ! Such flawed principles explain the multitude of
wrong contracts reached every day by world-class players.
Finally, an important precision and reminder : having 27 total pts
between two hands does not mean that a game in a Major can be
“predicted”, just as having 34 total pts does not mean that a 6 NT
contract can be “predicted” – the count only indicates that the two
hands have the points corresponding to a potential game or slam.
The true test of an accurate evaluation method is the reverse, and
that is : when a contract has the right odds to be bid and played,
then the evaluation method should show the corresponding points for
it. And not just 75 or 80 % of the time or 85 or 90 % of the time.
The optimal point count meets that criteria over 95 % of the time !
and you can easily verify this yourself through your own
compilation of empirical statistics.
The corollary of the above statement is just as important : when
the optimal point count does not add up to a minimum of 25 total
pts for 3 NT or 32 total pts for a suit small slam, you can be 95 %
sure that these contracts are not odds-on and should not be bid.
And both above statements can certainly not be made by any other
existing hand evaluation method !
The above optimal point conversion table was arrived at through
the analysis of some 6875 contracts (observed in over 4000 deals dealt) as
follows : on every deal dealt, any odds-on contract, whether in
North/South, West/East or both, was recorded with the corresponding
probability of success and the number of optimal points the side had.

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 CONVERTING POINTS INTO CORRESPONDING TRICKS

For example, of all 3 NT contracts recorded as odds-on, only 3.4 % had

fewer than 25 total optimal pts, 25 % had 25 pts, 34 % had 26 pts,
28 % had 27 pts and about 10 % had over 27 pts; then 25 optimal pts
was identified as the minimum needed for a 3 NT contract. Then, each
number was compared to the success probability (%) to establish which
point count corresponded to a contract with a 45 % success probability,
or 50 to 55 %, etc.
The historical fallacy of the « 40-point deck » equation.
It is worth looking into the “40-point deck” equation which
resulted from M. Work’s 4 3 2 1 honor point count as it has long
ruled the way we have been taught to convert points into
“expected” tricks, such as : « it takes just over 60 % of the 40-point
deck to bid and expect to make 3 NT about 50 % of the time, with two
evenly-distributed hands ». The 40-point deck equation being that
the sum of the points attributed to an Ace, a King, a Queen and a
Jack adding up to 10 pts, a complete deck of cards with all of these
honors adds up to 40 points.
Before assessing the validity of the « just over 60 % of 40 points for
3 NT » rule, let’s assess a more easily verifiable equation : would
50 % of all 40 points (20 pts) be expected to produce half the tricks
(6 ½) available ? Let’s do this with an example of two evenly-
distributed hands, each having all top 4 honors, such as :
ª Axxx ª Jxx
© Kxx © Qxx
¨ Qxx ¨ Kxx
§ Jxx § Axxx
But these two hands together will only produce 4 to 5 tricks, not
6 ½ ! And the optimal point count gives them only 16 pts, not 20
(- 2 pts for the two 4 3 3 3 distributions, 4 pts for the isolated
Queens and Jacks, not 6, minus 1 point for two “mirror” suits).
And adding K Q to either West’s spades or East’s clubs will only
produce 7 tricks ! (21 pts, not 25). How come ?
Well, that’s because the 40-point deck equation is a complete fallacy !
Here is why :
1) Queens and Jacks have variable values and therefore an Ace + a
King + a Queen + a Jack do not add up to 10 pts ! They add up to
between 9 ½ and 10 ½ pts. The total deck therefore adds up to
between 38 and 42 pts.

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 CONVERTING POINTS INTO CORRESPONDING TRICKS

2) The appropriate value of tens is not accounted for in this

equation – but we know now, don’t we, that a ten which
accompanies a Jack or a Queen is worth one full point.
Therefore, when counting 10s, a hand with one each of the top 5
cards adds up to between 9 ½ and 11 ½ pts and the full deck now
adds up to between 38 and 46 pts.
3) The negative value of 4 3 3 3 distributions is not accounted for.
As a result, the two hands below are counted the same 10 H pts in
M. Work’s 4 3 2 1 point count :
ª Axxx ª A J 10 x
© Kxx © xxx
¨ Qxx ¨ KQxx
§ Jxx § xx
The same 10 pts? Nonsense! In optimal count, the first hand is
counted 8 ½ pts (4 3 3 3), the second 11 ½ pts.
So now, a full deck adds up to between 34 and 46 pts.
4) And 1 point must be added for a 5-card suit (headed by at least
3 H pts). So, now, a hand with one each of the top 5 cards adds up
to between 8 ½ pts (- 1 for 4 3 3 3) and 12 ½ pts (+ 1 for 5 3 3 2) and
a full deck therefore adds up to between 34 and 50 pts.
So, which 60 % does it take to bid 3 NT? 60 % of 34 pts or 60 % of
50 pts ?!... Obviously, there is simply no such thing as the “40-point
deck” and that any rule derived from such a fallacy has no validity
whatsoever, including counting 3 pts per trick because “40 pts
divided by 13 tricks = 3 pts per trick”. But then bidding 3 NT should
take 27 H pts, not 25… Or the one stating that “if you count half a
point for each 10, then the total number of points needed to bid 3 NT
becomes 26 pts (just over 60 % of a full deck now adding up to 42 pts)”.
Well, see if you can you find 26 pts in 4 3 2 1 ½ count with the
hands below, which call for a 3 NT contract :
ª AQx ª J 10 x
© xx © Axxx
¨ A J 10 x x ¨ Kxx
§ xxx § Axx
In optimal point-count you can. These two hands add up to :
13 HL pts (no K) + 12 H pts (4 3 3 3 and no Q) + 2 ¨ Fit pts = 27 pts.
Minus 1 point for two “mirror” suits = 26. 3 NT is odds-on with one
of two finesses succeeding (75 % chance) and missing hearts or
clubs (whichever suit the opponents lead) no worse than 4 – 3.
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 OPTIMAL POINT COUNT – EXERCISES

EXERCISES / ILLUSTRATIONS
Now, let’s conclude with a few exercise hands to practice the
optimal point count.
Rather than just showing hands to be counted, let’s, instead, look
at some deals counted in traditional count and commented on by
experts, as these deals will not only help us practice our count,
they will enable us to compare it to the traditional count – an
enlightening exercise as it will vividly illustrate how serious errors
in traditional point count, still universally taught today, can lead to
the wrong bidding and, as a result, to the wrong contracts.
The following examples all come from one single “teaching”
column appearing, in November 2011, in the French Bridge
magazine « Le Bridgeur ». The object of that column was stated as :
« The responder’s second rebid is generally of critical importance and
must be mastered », with the following question being asked :
« What should you rebid with each of the following hands, after the
bidding shown below ? » :
1) ª Qxxx
1¨ 1© © K 10 x x
1ª ? ¨ xx
§ Axx
The “appropriate” rebid advocated by the writer of the column, a
world-class champion, was : « 2 ª : ª Fit, between 6 and 10 S points.
3 ª would require 11 to 12 S points ».
Wow! What a huge point miscount! As this hand has 13 ½ HDF
pts : 9 ½ H pts + 2 pts for the 8-card ª Fit with the Queen + 2 pts
for the ¨ doubleton with 4 ª trumps. And 4 ª should be played
with an opener such as :
ª KJxx
© QJx
¨ AKxx
§ xx
An opener who, with this hand, would have no reason to bid
further over his partner’s 2 ª rebid limited to a maximum of 10 S
pts. (An appropriate rebid would be 2 §, if this 4th suit bid is only
considered forcing one round, not game forcing, to learn about a
possible © singleton in opener’s hand, which would be in misfit).

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 OPTIMAL POINT COUNT – EXERCISES

2) ª KQJxxx
1§ 1ª © Axx
1 NT ? ¨ Kx
§ xx
Rebid advocated by the same champion : « 4 ª : conclude directly to
game as a slam can be excluded opposite an evenly-distributed 12/ 14 H
points opener’s hand ».
Another huge point miscount! As this hand has 18 ½ HLD pts
(opposite a NT bid, it is HLD pts that should be counted) and the
side could be in slam zone opposite a 14 HL point opener with
3 spades – for a 9-card ª Fit worth 2 additional points, such as :
ª xxx
© Kx
¨ AQx
§ Axxxx
An opener who, with this hand, would have no reason to bid
further over his partner’s 4 ª conclusion. (the initial response of
1 ª was incorrect. Responder should have jumped directly to 2 ª).

3) ª KQxx
1§ 1ª © Axx
2 NT ? ¨ Kxx
§ Kxx
Rebid advocated by the same champion : « 6 NT : the bid is automatic
opposite an evenly-distributed hand described by the opener as 18/19 H
pts, without 4 ª ».
Another inappropriate bid! When knowing, today, that an 8-card
§ Fit would add 2 Fit pts to the side to bring it to 15 ½ pts (4 3 3 3
but 3 Kings) + 19 HL pts + 2 Fit pts = 36 ½ HLFit pts. That’s grand
slam zone ! But the side isn’t about to discover the § Fit after such
an hallucinating triple jump to 6 NT, “burning” no less than 19
bidding steps! An excellent way to “bury” a 7 NT contract,
opposite an opener such as :
ª AJx
© Kx
¨ Axx
§ AQJxx

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 OPTIMAL POINT COUNT – EXERCISES

A hand worth 22 ½ HL pts but which would be counted for 19 H

pts by most experts and champions... Indeed, these two hands add
up to 40 HLFit pts, not 34 or 35 pts.
(The 1§ opening/2 NT rebid would not apply with the above
opening hand which has well over 19 pts).

4) ª Kxxxx
1§ 1ª © xxx
3ª ? ¨ xx
§ Kxx
Rebid advocated by the same champion : « 4 ª : Do not give in to
the temptation to pass on partner’s non forcing 3 ª invitation. The
9-card ª Fit and the § Fit with the King in partner’s long suit (with less
than 5 clubs he would have opened 1 NT) allow you to be optimistic ».
An “optimistic” 4 ª bid to be preferred to pass ?!... But it is nothing
less than a ª slam that is at stake here – not just a game! As this
hand has 12 HLDFit pts : 6 HL pts (no Queen) + 3 pts for the 9-card
ª Fit with the King + 2 pts for the 8-card § Fit with the King
+ 1 point for the ¨ doubleton. And a 4 ª bid isn’t about to commu-
nicate this to the opener! Actually, it is not very difficult to imagine
a © singleton in opener’s hand – which would add 2 pts for “no
wasted honor pts” in hearts, a hand such as :
ª AQxx
© x
¨ Axx
§ A Q 10 x x
Surely, missing this 6ª slam with such hands can hardly be
acceptable! (The proper rebid is 4 § showing the § Fit and control,
and slam interest – but West’s rebid should have been a “splinter”).

5) ª Kxxx
1© 1ª © Axx
2© ? ¨ AJxx
§ xx
Rebid advocated by the same champion : « 4 © : Conclude directly
to game as you are not in slam zone opposite a 6 © single-suit hand
limited to a maximum of 16 H points ».

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 OPTIMAL POINT COUNT – EXERCISES

An astonishing assessment and conclusion when a 12 to 14 H pts

opener, such as the one below, would be sufficient to produce a
© small slam :
ª Axx or with 12 ½ H : ª A x
© KQxxxx © Kxxxxxx
¨ KQx ¨ KQx
§ x § x
Responder has 15 HLDFit pts (no Queen) with a 9-card © Fit, and
17 to 18 HLD pts in opener’s hand is all that is needed to be in slam
zone. But a 4 © rebid will give the opener no useful information
enabling him to bid further.
In any case, it is the total number of points, H, L, D and Fit pts, that
is relevant, not just honor pts. Particularly once a trump Fit has
been found. (The proper rebid is 3 ¨, 3rd suit forcing, to have the
opener describe the rest of his hand).

6) ª xx
1 NT 2§ © AJxx
2ª ? ¨ AKx
§ KQxx
Rebid advocated by the same champion : « 4 NT, “Quantitative”.
Slam invitation for an opener at the top of his point range : 17 H pts ».
Again, another serious point miscount! Because the right count has
yet to be taught and practiced. Thus, an opener with the following
hand of only 15 H pts will have to pass on 4 NT :
ª Axxx
© KQx
¨ Qx
§ Axxx
When a 68 % 6 NT slam should be played. Responder has, in fact,
18 pts, not 17, and the additional point for the 8-card § Fit gives
the side a total of 34 pts.
But the best way to reach 6 NT surely cannot be to make a 4 NT bid
which “burns” 10 bidding steps without adding any precision.
Pure nonsense! (A game-forcing 2 ¨ Stayman response to 1 NT
would have been more helpful – as the responder could now rebid
2 NT to have the opener describe his hand further and discover the
8-card § Fit).

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 OPTIMAL POINT COUNT – EXERCISES

In conclusion, the above examples illustrate perfectly the major

and often decisive impact the point count has on bidding and,
therefore, on reaching the right contract.
Certainly, the responder’s rebid is of critical importance but it will
only help reaching the right contract if points are counted correctly !
Evidently, this is not the case yet…
And this is obviously even more true if the responder’s first bid is
the wrong one due to an erroneous initial point count !
And examples of such errors are easy to find, as illustrated in
another “teaching column” appearing in Septembre 2015 in the
Bridge magazine « Le Bridgeur », entitled “Bidding technique” and
presented by another world-class champion who asked the
following question :
« What will you rebid with each of the following hands, after the bidding
shown below ? » :

1) ª Q 10 x
1ª 2ª © xx
2 NT ? ¨ AQxxx
§ Jxx
But this hand has 12 ½ HLDF pts ! (no King) and the initial reply of
2 ª, limiting the hand to a maximum of 10 S pts, is simply out of
the question ! Even without knowing that 1 point should be added
for the 8-card ª Fit, how is it possible not to count 1 additional
point for the ª Queen + 1 D point for the © doubleton and 1 point
for the 5 diamonds ?!... No rebid can now fix that initial error !
Just imagine, on such a 2ª reply, what an opener with the
following hand could bid other than Pass :
ª AKxxx
© xxx
¨ KJxx
§ x
Thus missing an easy 4 ª game, when the side has a total of 28 pts
with the additional 2 pts for the 9-card ¨ Fit !
(The appropriate initial response, was, of course, 2 ¨).

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 OPTIMAL POINT COUNT – EXERCISES

2) ª Kxx
1ª 2ª © K 10 x x
3¨ ? ¨ Jxxx
§ Kx
But this hand has 13 HDF pts! (no Queen but 3 Kings) and the
initial reply of 2 ª is, once again, simply out of the question.
Again, even without knowing that 1 point should be added for the
8-card ª Fit, how is it possible not to count one additional point
for the ª K + 1 D point for the § doubleton ?
No rebid can now fix that initial error! On such a 2 ª reply, what
else but pass could we expect from an opener with the following
hand :
ª AQxxx
© QJx
¨ x
§ QJxx
Thus missing an easy 4 ª game, when the side has a total of 27 ½
pts with + 2 pts for no “wasted honor pts” in diamonds.
(An appropriate initial response could be 1 NT forcing, or a Jacoby
2 NT raise to uncover a possible singleton in opener’s hand).

3) ª QJx
1ª 2ª © xx
3§ ? ¨ J 10 x x
§ KQxx
Here again, this hand has 13 HDF pts and the initial reply of 2 ª,
limiting the hand to a maximum of 10 S pts, is once again simply
out of the question. No rebid can now fix that initial error.
On such a 2 ª reply, what else but pass could we expect from an
opener with the following hand :
ª AKxxx
© Axx
¨ x
§ Jxxx
Thus missing an easy 4 ª game, when the side has a total of 27 ½
pts with + 2 pts for the 8-card § Fit with the Jack. (The appropriate
initial response is, of course, 2 § or a 2 NT Jacoby raise) .

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 OPTIMAL POINT COUNT – EXERCISES

4) ª Kxx
1ª 2ª © xx
3¨ ? ¨ Qxxx
§ A 10 x x
Another example of poor point count as this hand has 12 HDF pts
and the initial reply of 2 ª is, once again, the wrong one. And no
rebid can now fix that initial error.
On such a 2 ª reply, what else but pass could we expect from an
opener with the following hand :
ª AQxxx
© xxx
¨ A J 10 x
§ x
thus missing a possible 4 ª game, when the side has a total of
29 pts with 2 pts for the 8-card ¨ Fit with the Queen.
5 ¨ is also a playable contract, for which the side has 30 pts with
2 pts for the © doubleton with 4 trumps.
(An appropriate initial response could be 1 NT forcing for some,
or 2 § for others, or a 2 NT Jacoby raise).

5) ª Axx
1© 2© © Qxx
3¨ ? ¨ Kx
§ xxxxx
Here again, this hand has 12 HDF pts and the initial reply of 2 © is
totally inappropriate. No rebid can now fix that initial error.
And on such a 2 © reply, what else but pass could we expect from
an opener with the following hand :
ª Kxx
© A J 10 x x
¨ Q J 10 x
§ x
Thus missing a 4 © game, when the side has a total of 28 ½ pts –
who said, again, that 10s only have some value at NT contracts ?...
(An appropriate initial response could be 1 NT forcing for some,
or 2 NT Jacoby raise for others).

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 OPTIMAL POINT COUNT – EXERCISES

We just identified five additional examples of highly inaccurate

point count, all in one same magazine column, in 2015, and on the
very first reply to the opening bid! One can expect some difficulty
reaching the right contract after such a poor beginning... Isn’t it high
time to bring the point count up to 21st Century expectations ?!
One can now trust that the accuracy delivered by the optimal
point count ought to significantly contribute to improving Bridge
bidding and to reaching the right contracts more often, whatever
the bidding system used.

The Optimal Hand Evaluation Constitution

« We hold these truths to be self-evident

that “all hand evaluation methods are not created equal”,
that “in Precision We Trust” and pledge to deliver a hand
evaluation point count that accurately assesses the value of
combined hands of the same side, enabling better bidding
leading to much improved results ».

The author

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 OPTIMAL HAND EVALUATION

CHAPTER 6

OTHER POINT COUNTS

OTHER POINT COUNTS : OVERVIEW

This book has shown numerous examples which clearly illustrate
that, contrary to what is generally claimed, the 4 3 2 1 honor point
count falls seriously short of evaluating with precision evenly-
distributed hands played in a No Trump contract – and this
assessment is nothing new, several other point counts having
previously contested the point values attributed to the Ace, King,
Queen and Jack by the 4 3 2 1 count (G. Reith’s 6 4 3 2 1 point count
and the « Four Aces » 3 2 1 ½ count, well over 80 years ago, and
more recently the ZAR point count, R. Cowan’s 5 4 3 2 1 count and
others as well, including the optimal point count described here).
It therefore seems appropriate now to conclude this presentation
by a brief overview of other point counts, the most notable ones.
CULBERTSON’S HONOR TRICK COUNT
E. Culbertson’s honor trick count, THE reference at the beginning
of Contract Bridge through to the late ‘30s, consisted of assessing
the trick-taking potential of honors, including 10s, whether alone
or in combination. Thus, an Ace was counted as one trick, a King by
itself as ½ trick, A K together as 2 tricks, A J 10 x as 1 ½ tricks, etc.
This count was rather complex to master due to its numerous
possible combinations but, more importantly, it was totally
erroneous because it was based on the assumption that all missing
honors were held by the opponents. For example, A Q x was counted
as 1 ½ tricks. But this count has no statistical validity whatsoever
as it does not reflect the cases in which the partner holds some of
the missing honors, such as :
AQx J x x (2 tricks) or A Q x J 10 x (2 ½ tricks)
or A Q x K x x (3 tricks) or A Q x K J x x (4 tricks)
As can be seen, the true value of A Q x is, statistically, considerably
greater than 1 ½ tricks. (Its actual average statistical value is just
under 2 ½ tricks which is exactly the value the optimal count gives
it : 6 ½ pts).

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 OTHER NOTABLE POINT COUNTS

Similarly, the value of an Ace, by itself, is considerably greater

than 1 trick as the Ace has considerable defensive value, in addition
to its trick-generating potential when combined with other honors
in partner’s hand (see page 6/7 for the Ace’s statistical value).
Thus, Culbertson counts 8 winning tricks for one hand holding
A K in every suit, but only counts 6 winning tricks for the same
honors split between the two hands of the same side, a huge
difference of two tricks! As a result, he totally undervalues the trick-
generating potential of the following two hands :
ª Axx ª Kxx
© Kxxx © Axx
¨ Axx ¨ Kxx
§ Kxx § Axxx
Clearly two hands with 8 sure winning tricks, not 6, and which call
for a 3 NT contract as a 9th trick can be expected to be generated
in one of the two suits distributed 4 - 3. Two hands adding up to
26 pts in optimal point count : 13 pts each after deducting, from each
hand, 1 point for the 4 3 3 3 distribution and 1 point for no Queen.
Bottom line, this highly inaccurate honor trick count had no
statistical validity, and was abandoned by the late ’30s.

THE 4 3 2 1 HONOR POINT COUNT

Milton Work’s 4 3 2 1 honor point count was the first to propose
an evaluation method which attributed 4 3 2 1 points to the top
4 honors as an alternative to E. Culbertson’s honor Trick count.
Successfully promoted by the great American champion C. Goren,
initially for hands played in NT and later for all hands or contract
contemplated, the 4 3 2 1 point count became universally adopted
because of its simplicity and the belief that it was well adapted to
evaluate evenly-distributed hands and reasonably accurate overall.
But it has been known for some time now that this count is in fact
far from accurate – and this, regardless of the hands’ distribution or the
contract contemplated. And this book has clearly illustrated why :
– It undervalues the Ace which must be counted 4 ½ pts, not 4.
– It overvalues Queens and Jacks when they are isolated.
– It ignores the value of 10s – worth 1 point when with a J or a Q.
– It does not deduct 1 point for a 4 3 3 3 distribution.
– It does not deduct 1 point for a hand without a King or a Queen.
– And, most importantly, it does not count Fit points for all suits
and all contracts !

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 OTHER NOTABLE POINT COUNTS

As well, C. Goren’s distribution point count, which has tradition-

ally accompanied M. Work’s 4 3 2 1 honor point count, is seriously
flawed (as illustrated earlier in this book).
The results are witnessed every day : missed games and slams as
well as overeaching into unmakable contracts.

THE 6 4 3 2 1 POINT COUNT

In the late ‘20s, another American player, George Reith, proposed
a different honor point count as an alternative to Culbertson’s trick
count and to M. Work’s 4 3 2 1 count : the 6 4 3 2 1 point count.
This count attributed 6 pts to the Ace, 4 pts to the King and on
down through to the ten to which it gave a value of 1 point.
In this count, G. Reith had in fact recognized very early that the
Ace was generally, and for any contract, worth 50 % more than the
King and that the ten had some value which needed to be
quantified. He also attributed points for length: 2 additional points
for a 5-card suit (equivalent to a Jack), and another 2 points for a
6-card suit (a total of 4 pts, equivalent to a King).
Furthermore, G. Reith proposed this count to apply to any and all
hands, regardless of their distribution or the contract played.
However, this count has very serious flaws which doom it, notably :
– It attributes a fixed, constant value to Queens, Jacks and 10s, a
fatal flaw and, furthermore, it significantly overvalues these second-
ary honors relative to the King and Ace.
– It does not deduct 1 point for a 4 3 3 3 distribution.
– And it does not attribute Fit points for 8, 9 or 10-card Fits.
A few example hands, below, will illustrate these flaws, starting
with two 4 3 3 3 hands, without an 8-card Fit :
ª A 10 x ª KQJ
© Kxxx © AQx
¨ A 10 x ¨ Jxxx
§ KJx § AQx
6 NT must not be bid here but the 6 4 3 2 1 count gives these two
hands 53 pts, two more than the 51 pts it needs for 6 NT, as it
overvalues Queens, Jacks and 10s, but most importantly, does not
deduct points for the 4 3 3 3 distributions (and the 4 3 2 1 count does
not fare any better : in the tournament this deal comes from, the
bidding was most often : 1 NT – 6 NT, down 2).

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 OTHER NOTABLE POINT COUNTS

The Optimal count gives these two hands 31 ½ pts (- 2 pts for two
4 3 3 3s, - 1 for no Queen in West’s hand, - 1 for two “mirror” suits).
Another example, with two 4 4 3 2 hands with an 8-card Fit :
ª AJxx ª KQx
© Kx © Ax
¨ xxxx ¨ xxxx
§ KQx § J 10 x x
3 NT should be bid here (68 % chance), but the 6 4 3 2 1 count only
reaches 35 pts, well short of the 39 pts it needs for 3 NT (and the
4 3 2 1 count does not fare any better).
The Optimal point count gives these hands 25 pts (two “mirror”
suits). Note that the 1 point attributed by the optimal count to the
8-card ¨ Fit is quite appropriate, even though the suit is without a
single honor, as the 8 diamonds limit the remaining diamonds in
opponents’ hands to 5 cards, distributed 3 - 2 68 % of the time.
Now one with a 5 3 3 2 hand and a Fit :
ª AQxx ª Kxx
© xx © Ax
¨ xxxx ¨ xxx
§ AKx § QJxxx
3 NT should be bid here (over 80 % chance) but the 6 4 3 2 1 count
only reaches 36 HL pts – too few Jacks and 10s and no Fit pts (and
the 4 3 2 1 count does not fare any better).
The optimal count gives these hands 27 ½ pts with 2 Fit pts.
One may wonder how this count came to be so flawed ? To explain
it, one has to consider that G. Reith’s count was conceived and
introduced, in the late 1920s, as a way to translate E. Culbertson’s
complex honor Trick count into a much simpler point count.
But Culbertson’s Trick count being seriously flawed and inaccurate, it
follows that any translation of his trick count into points carried
with it the very same inaccuracies.
And it’s all downhill from there when G. Reith’s point count
introduces the additional inaccuracy of assigning to the Queen,
Jack and ten fixed, full point values when their values are variable
and should be counted in half points, when applicable.
Furthermore, Jacks and tens representing as much as 18 % of all
points in the 6 4 3 2 1 count, this count can never, ever possibly reach
the points required for a 7 NT contract.

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Clearly, this count did not prove to be better than the 4 3 2 1 count.
In any case, it was not communicated nor promoted properly and,
as a result, was not adopted. No matter, as even though it did give
the Ace its proper relative value to the King, it had as many
deficiencies as the 4 3 2 1 count – some of them just being different :
– It significantly overvalues Queens and Jacks, particularly when
they are isolated.
– It gives the ten a fixed, constant value – a serious mistake as its
value can vary considerably.
– It does not deduct 1 point for a 4 3 3 3 distribution.
– It does not deduct 1 point for a hand without an Ace, a King or a
Queen.
– And, most importantly, it does not count Fit points for all suits
and all contracts.

THE 3 2 1 ½ HONOR POINT COUNT

In the 1930s, the « Four Aces », probably the greatest American
team ever assembled, led by the great American champion
Howard Schenken, used a different point count method : the
3 2 1 ½ honor point count, attributing 3 pts to the Ace, 2 pts to the
King, 1 point to the Queen and ½ point to the Jack.
On one hand, they also recognized very early the true value of the
Ace (50 % more than the King) while, on the other hand, they felt
that both other point counts overvalued the Queen and the Jack.
Furthermore, they would not give the ten a fixed, constant value
of 1 point – when their own count only gave ½ point to the Jack.
They also attributed points for length : 1 additional point for a
5-card suit (equivalent to a Queen), and a 2nd point for a 6-card
suit (a total of 2 pts, equivalent to a King). They also attributed
1 additional point for 3 honors in a 6-card suit.
They established the correlation between points and expected tricks
by dividing by two the total of points between the two hands : i.e.
18 total pts = 9 tricks for 3 NT, 24 total pts = 12 tricks for 6 NT.
They used this point count for any and all hands, regardless of
their distribution or the contract played (even though some members
of the « 4 Aces » team used the 4 3 2 1 count for balanced hands).
While this count was often reasonably accurate, particularly for
suit contracts, it was not adopted as it suffered from counting in
half-points (for the Jack) and for having to convert points into
expected tricks by way of a division. It was also ill-suited for NT
contracts as it very seriously undervalued Queens, Jacks and tens.

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No matter, as well, because even though it did give the Ace its
proper value as compared to the King, it had about as many
deficiencies as either of the other two counts – just that some of
them were different :
– It severely undervalues Queens and Jacks, particularly when they
accompany another honor.
– It ignores the value of tens – a serious mistake when they
accompany a Queen or a Jack.
– It does not deduct 1 point for a 4 3 3 3 distribution.
– It does not deduct 1 point for a hand without an Ace, a King or a
Queen.
– And, most importantly, it does not count Fit points for all suits
and all contracts.

THE 5 4 3 2 1 POINT COUNT

In 1987, the Australian statistician Richard Cowan presented, in
“Applied statistics”, the results of his statistical research concerning
the evaluation of evenly-distributed 4 3 3 3 and 4 4 3 2 Bridge
hands. He observed that, for contracts played in NT with such
hands, tens have a value far from negligible – yet not quantified
by the 4 3 2 1 point count – and that the winning trick potential of
Queens, Jacks and tens seemed under-valued and inadequately
translated by the 4 3 2 1 point count. He concluded that, for these
evenly-distributed hands, a more appropriate count should
integrate tens and attribute to the Ace, K, Q, J and 10 the respective
values of 5 points, 4, 3, 2 and 1.
Author’s note : One might wonder why come up with such a count,
over 55 years after G. Reith presented the 6 4 3 2 1 count which is exactly
the same with the only difference being that G. Reith’s count values the
Ace properly, at 6 pts or 50 % more than the King, while the 5 4 3 2 1
count does not, giving it only 5 pts, just 25 % more than the King…
Furthermore, the 5 4 3 2 1 count’s stated mission is to evaluate
evenly-distributed hands only, intended to be played in NT. This
requires using another method to evaluate unevenly-distributed
hands, not a very practical or attractive proposition…
There isn’t much that needs to be said about this count that hasn’t
already been covered in the overview of G. Reith’s 6 4 3 2 1 count
except that it is simply much worse because of its significant under-
valuation of Aces. As a result, either it reaches the appropriate
points corresponding to an odds-on contract, but so would the
6 4 3 2 1 count, or it does not, while the 6 4 3 2 1 count might,
because of its proper valuation of Aces. Illustrations :

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 OTHER NOTABLE POINT COUNTS

With the hands below, rich in Aces and Kings, 6 NT is 100 % :

ª Axxx ª Kxx
© AQx © KJxx
¨ AKx ¨ Qxx
§ Kxx § AQx
In 5 4 3 2 1 count, these hands add up to 47 pts, short of the 49 pts
needed for 6 NT in that count. But, in 6 4 3 2 1 count, they do add
up to the right total of 51 pts, needed for 6 NT in that count.
Of course, if points were deducted for the two 4 3 3 3 distributions,
as they should, both counts would fall short.
In optimal count they add up to 33 ½ pts (20 ½ H + 14 H), after
deducting 2 pts for 4 3 3 3s and 1 point for two “mirror” suits.
More frequently, though, both counts will fail with hands such as :
ª Ax ª xxx
© xxx © Axx
¨ AQxxx ¨ Kxx
§ Kxx § Axxx

In 5 4 3 2 1 count, these two hands add up to 33 pts, well short of

the 37 pts this count needs for 3 NT, while they add up to 37 pts in
6 4 3 2 1 count, also short of the 39 pts this count needs. And, in both
cases, that’s without deducting 1 point for East’s 4 3 3 3 distribution.
In optimal count they add up to 27 pts : 15 HL + 10 H (4 3 3 3 and
no Queen) and 2 Fit pts.
The bottom line on the 5 4 3 2 1 count is this :
1. Yes, some 10s do count and should be quantified. But assigning
to them a fixed, constant value is totally misguided. From there,
assigning points to the higher honors relative to a fixed value of
1 point given to the 10 dooms the 5 4 3 2 1 count.
2. The Ace is worth, on average, 50 % more than the King and the
King is worth, on average, 70 % more than the Queen – there is
simply no getting around that statistically validated fact.
Valuing the Ace only 25 % more than the King, and the King only
33 % more than the Queen dooms the 5 4 3 2 1 count.
The 6 4 3 2 1 count, at least, has the Ace count right, relative to the
King, and there was no need to create a new wheel, particularly one
which defies statistics and is worse than the initial proposition.

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In summary, the 5 4 3 2 1 count has many more deficiencies than

the other counts :
– It severely undervalues the Ace which it values only 25 % more
than the King. Reason enough to disqualify this point count !
– It significantly overvalues Queens and Jacks, particularly when
they are isolated.
– It gives the ten a fixed, constant value of 1 point – a serious mistake
as its value can vary considerably.
– Together, Jacks and tens represent 20 % of total points! An
absurd overevaluation which prevents this count from reaching the
points total needed to bid even small slams in a majority of cases.
– It does not deduct 1 point for a 4 3 3 3 distribution.
– It does not deduct 1 point for a hand without a King or a Queen.
– And, most importantly, it does not count Fit points for all suits
and all contracts.
For all the above-stated reasons, this count is probably the most
flawed of all counts. While it should not really be given serious
consideration, I have included it in this overview of other point
counts for the sole purpose of bibliography.

THE ZAR POINT COUNT

In 2003, the Canadian statistician Zar Petkov presented a new and
different hand evaluation point count method assigning points to
three key factors – Honors, Distribution and “Super-Fits” –
according to “weights” statistically derived (?). The method
factors in distribution points for every hand, starting with opening
hands (contrary to the practice which has prevailed since the early ‘60s),
allowing aggressive opening bids, “light” on honor points.
This count places major emphasis on distribution and is intended
for suit contracts only. In that context, it can effectively capture the
trick-taking potential of unevenly-distributed hands, often more
accurately than the traditional M. Work/C. Goren point count.
The ZAR point count consists of the following :
1. Honor points : The Ace, K, Q and J are counted, respectively,
6, 4, 2 and 1 pts. That’s the “Four Aces” 3 2 1 ½ point count
multiplied by 2 (which avoids counting in half-points).
While this correctly assesses the value of Aces as 50 % more than
that of Kings, it gives Queens and Jacks a fixed, constant value
whether they are isolated or not, and the value of tens, when
appropriate, is not factored in. This count is, therefore, just as
flawed as the other counts previously described.

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2. Distribution points : the method combines, in one count, the

value of suit Length and that of short suits. Distribution points are
the sum of the length of the two longest suits added to the difference
between the longest suit and the shortest suit i.e. a 5 4 3 1 hand has
13 D pts : 5 + 4 + 5 – 1 = 13. This gives the following scale :

--------- ZAR points ---------- --- Optimal count points --

L pts + D pts = Total pts L pts + D pts = Total pts
4 3 3 3 : 4+ 3 + 4 – 3 = 8 pts 0 + -1 = -1 pt
4 4 3 2 : 4+ 4 + 4 – 2 = 10 pts 0 + 0 = 0 pts
5 3 3 2 : 5+ 3 + 5 – 2 = 11 pts 1 + 0 = 1 pt
5 4 2 2 : 5+ 4 + 5 – 2 = 12 pts 1 + 1 = 2 pts
5 4 3 1 : 5+ 4 + 5 – 1 = 13 pts 1 + 2 = 3 pts
6 3 3 1 : 6+ 3 + 6 – 1 = 14 pts 2 + 2 = 4 pts

The method calls for 26 H + D pts to open and for 52 total points
between two hands of the same side to make 10 tricks, a game in a
Major suit, exactly twice the traditional equivalent : 13 + 13 = 26.
The emphasis put on distribution pts vs honor points can readily
be seen from the above chart. However, the point difference between
the above hands is far too small to accurately reflect their true
distributional values and, as a result, the more unbalanced a hand
is, the less accurate the ZAR distribution count is. Yet, the more
evenly-distributed a hand is, the more likely this method is to
underestimate the hand’s value.
Surprisingly, this can also be true with a short suit in one of the two
hands, such as :
ª xx ª Q 10 x x
© A Q 10 x © Kxxx
¨ xxx ¨ x
§ KQxx § Axxx
ZAR pts : 14 H + 10 D = 24 pts : Pass 12 H + 11 D = 23 : Pass
West’s Queens (and 10) are seriously undervalued as is East’s
4 4 4 1 shape, valued only 1 point more than West’s 4 4 3 2 shape.
Optimal pts : 12 ½ H + 0 D = 12 ½ pts 10 H + 2 D = 12 pts
12 ½ H + 12 HD + 2 pts for the 8-card © Fit with the King + 2 pts
for the ª doubleton with 4 trumps + 1 point for the ¨ singleton
with 4 trumps + 1 point for the 8-card § Fit = 30 pts = 4 ©.

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3. “Super-Fit” points : the method rightly attributes + 1 or + 2 pts

for honors in partner’s trump suit or long suit, and additional
points for short suits with a Fit : with an 8-card Fit : + 1 point for a
singleton, + 2 pts for a void, and for a 9-card Fit : + 1 point for a
doubleton, + 2 pts for a singleton, + 3 pts for a void, and so on for
10 + card Fits. In addition, it adds points for secondary 9 + card Fits :
+ 1 point for a 9-card Fit, + 2 pts for a 10-card Fit.
All this is very good and it has several similarities with the optimal
point count. However, this still does not cut it as it misses the most
important and frequent Fits : the 8-card Fits !
Interestingly, recognizing that the method overvalues distribution
over honors, the ZAR count requires 5 more points to reach game
when the two hands have no 8-card Fit or if they have a 5-card suit
(as the D pts going to the 5-card suit will likely be missing in honor
points) i. e. 57 pts, instead of 52 pts, are then needed to play 3 NT.
Thus, instead of valuing 8-card Fits, it penalizes the side for not
having an 8-card Fit! That’s a huge mistake and a major handicap !
A few examples will illustrate this :
Prepared deals tournament, 2002.
ª QJxx ª AKxx
© x © xxx
¨ AKxxx ¨ Qxx
§ Axx § Kxx
In the tournament, the slam was generally missed after bidding
which was often : 1 ¨ 1ª
3 ª (17/19 HLS) 4ª
ZAR “counters” would not have fared any better.
ZAR pts : 19 H + 13 D = 32 + 2 DF pts 16 H + 8 D = 24 + 1 DF pt
= 59 total pts : Not enough pts for 6 ª.
Optimal pts : 16 HL + 2 D = 18 pts 12 H - 1 D = 11 pts = 29 pts
+ 1 point for the © singleton with 4 trumps + 2 pts for the 8-card ª
Fit with Q J + 2 pts for no “wasted honor pts” in hearts + 2 pts for
the 8-card ¨ Fit with the Queen = 36 HLDF pts = slam.
The bidding would be : 1 ¨ 1ª
3 © (mini-splinter) 4 § (RKC Ask)
4 NT (2 Keys + ª Q) 5 ¨ (¨ Ask)
5 NT (¨ K) 6ª
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 OTHER NOTABLE POINT COUNTS

Upon West’s ”mini-splinter” indicating a 4-card ª Fit, 5 diamonds,

a © singleton and 20 + HLDF pts, East can add to his 11 pts : 2 pts
for the 8-card ¨ Fit with the Queen + 2 pts for no “wasted honor pts”
in hearts = 35 pts minimum = slam zone.
In the above example, what is missing are points for 8-card Fits !
Needless to say, hands rich in Queens, Jacks and tens instead of
Aces and Kings, and without an 8-card Fit, can be a real killer for
the ZAR count for balanced hands – but then, this count was not
designed for such hands.
In summary, the ZAR point count rightfully integrates several key
elements which often make it a fairly accurate evaluation method
for suit contracts but it still has several serious deficiencies :
– It overvalues distribution pts over honor pts while severely
undervaluing the point difference between distributions.
– It severely undervalues Queens and Jacks, and it gives them a
constant value whether isolated or not.
– It ignores the value of tens – a serious oversight when they
accompany a Jack or a Queen.
– It does not factor in “wasted honor points”.
– And, most importantly, it does not count Fit points for 8-card
Fits – for any suit and any contract !

OTHER POINT COUNTS

As can be expected, there are other point counts.
Among the most notable ones, Marty Bergen’s “Adjust 3” count
deserves mention and warrants a short analysis :
It starts with M. Work’s 4 3 2 1 count for the top 4 honors and then
adjusts the count up or down : + 1 point when the number of
“underrated” honors – Aces and tens – exceeds by 3 the number of
“overrated” honors (Queens and Jacks), minus 1 point when it is
the reverse.
An original attempt to avoid counting in half-points which, in the
process, wrongfully links Aces and tens in valuing them, while
ignoring the variable value of tens, Queens and Jacks, depending
upon whether they are isolated or not. This defies statistical findings
as 10s are underrated only when they accompany a Queen or a Jack
and Queens and Jacks are overrated only when isolated.
The following deal, whose opening hand is one used by M. Bergen
himself, clearly illustrates the serious flaws mentioned above.

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In this example, M. Bergen rightfully gives the following hand

only 17 pts (18 pts in 4 3 2 1 count) as he deducts 1 point for the
hand having five Queens and Jacks but only one Ace (5 – 1 = 4) :
ª KJx © KQxx ¨ QJxx § AQ
But his 17 point count (17 ½ in optimal count) is based on the wrong
rationale as the Queens and Jacks in this hand are not overrated in
any way. The right reason to give this hand only 17 ½ pts is because
1 point should be deducted for the two honors doubleton in §.
Now, opposite the above opener’s hand, the following hand
would be counted for 6 pts only in M. Bergen’s Adjust 3 count :
ª Qx © J 10 x ¨ K 10 x x § J 10 x
As a result, this hand would elicit a pass on a 15/17 1NT opening
– and the 3 NT game will be missed.
But this hand has 8 ½ H pts, not 6, because each J 10 combination
is worth 2 pts ! Responder should invite to game, which will be bid
as the opener is maximum.
Note how every single Queen and Jack in opener’s hand warrants
full value as they mesh perfectly with his partner’s honors.
A 3 NT contract very likely to be bid by “basic” 4 3 2 1 counters.
The moral is this : the value of Queens, Jacks and 10s has nothing
to do with their relative numbers in a hand; it has to do solely with
whether or not they accompany an honor and, in the case of 10s,
whether they accompany a Queen or a Jack. And Aces are worth
4 ½ pts in all cases – regardless of the number of tens.
M. Bergen’s Adjust 3 rightfully deducts 1 point for a 4 3 3 3 distri-
bution and adds 1 point for what he calls a “quality suit”, that is a
4 or 5-card suit which has 3 of the top 5 cards, counting tens.
Hereafter is an example which illustrates the reverse of the
previous example :
ª KQJx ª A 10 x
© KQx © A 10 x
¨ Jxx ¨ xxx
§ Axx § Qxxx
M. Bergen’s Adjust 3 count correctly gives the West hand 15 pts,
not 16, as it adds 1 point for the “quality spade suit” and deducts
1 point for the 4 3 3 3 distribution and another point for 4 overrated
honors vs only one underrated honor.

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 OTHER NOTABLE POINT COUNTS

But then, it gives the East hand 10 pts as the 1 point deduction for
the 4 3 3 3 distribution is offset by adding 1 point for 4 underrated
honors vs only one Queen.
This will lead East to drive to 3 NT on a 1 NT opening – a contract
guaranteed to fail as the side will lose no less than 4 diamonds and
the § King. Again, tens are only underrated when they accompany
a Queen or a Jack.
Note that the optimal point count gives the East hand only 8 ½ pts
(minus 1 for 4 3 3 3 and minus 1 for no King) and the side will stop
in 2 NT as West is minimum.
M. Bergen’s Adjust 3 count does deduct 1 point for singleton and
doubleton honors and adds 1 point for length for every card over
4 cards in any suit. While this count does not, initially, count
distribution pts for short suits, once an 8-card Fit has been found,
the trump support hand counts distribution pts for short suits
(singleton and void) and two doubletons, as follows : 3, 2, 1 with
3 trumps, 4, 3, 1 with 4 trumps. Once a trump raise has been made,
the partner then adds points for “extra” length in the trump suit :
1 point for a 6-card suit, 2 pts for 7 cards, and D pts for short suits,
as well : 1 point for two doubletons, 2 pts for a singleton, 4 pts for
a void. Points are not attributed for Fits (8, 9 or 10-card Fits).
All in all, some valid acknowledgments : 1 point deduction for a
4 3 3 3 distribution, point deductions for singleton and doubleton
honors, 4 pts for a void, 1 point for two doubletons, but a count
which ignores some key statistically validated findings such as the
absolute need to add 2 pts for a 9-card trump Fit, and others.
Overall, an original count but which brings only marginal precision
to the 4 3 2 1 point count while introducing some flawed elements
which, overall, makes it a questionable proposition.
Another notable point count is Danny Kleinman’s count which
attempts to reflect statistically validated findings, starting with the
proper relative values between the top 4 honors : Ace 50 % more
than the King and King 70 % more than the Queen. This count also
downgrades isolated Queens and Jacks while upgrading Queens
and Jacks when they accompany a King or an Ace ( the only other
count, to my knowledge, giving a variable value to Queens and Jacks),
counts points for 10s accompanied by honors, deducts points for
4 3 3 3 distributions and for singleton and doubleton honours.
All in all, very much the required adjustments highlighted in the
optimal count.
Unfortunately, this count is handicapped by some very serious
flaws :
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 OTHER NOTABLE POINT COUNTS

– To account for the right relative values between honors,

D. Kleinman opted for a 13 9 5 2 count for the top four honors –
again, to avoid counting in half-points. Then dividing the total by 3
to convert it to the familiar HCP (High Card Point) totals.
An unecessary complexity not conducive to gaining players’
acceptance… Counting in half-points is a lot easier !
But, this division by 3 now wrongly distorts several values, such
as : now Q J x x is only worth 2 pts (5 + 2 - 1 divided by 3), J 10 x x
is only worth .7 point (2 + 1 - 1 divided by 3). Wrong count !
– It attributes to 10s, when accompanied by an honor, the same fixed
value of one 3rd of a point (one Kleinman point divided by 3) = no
variability and a point count too small to be significant.
– Insufficient Length point count for suits of 5 cards or more : thus
Q J x x x is given a total of only 2.7 pts (6 pts for Q J + 2 pts for the
5-card suit = 8 Kleinman pts divided by 3). Wrong count !
– No point deduction for a hand with no Ace or no King or no
Queen.
– And no Fit points for all suits and all contracts.

All in all, a serious attempt to integrate most statistically validated

elements but doomed by a misguided point conversion.
__________________________________________________________

This concludes the overview of other, most notable point counts.

It confirms that the path to optimal accuracy goes much beyond just
trying to assess the proper relative value of honors. It requires :
1) making the proper point corrections for 4 3 3 3 distributions, for
singleton or doubleton honors, and for isolated Queens and Jacks,
2) counting 10s for 1 point when they accompany a Queen or a
Jack, and counting appropriate Fit points for all suits and all
contracts. Without these elements, no point count can achieve optimal
accuracy.
That is not to say that the optimal point count gets it right every
time. Needless to say, that is not the case : the variety of Bridge
hands is too vast… but it does get it right in over 95 % of the time.
Hereafter is an example where the optimal count over-reaches – as
do all other point counts – as duplicate values, such as Jacks and
10s, are “wasted” :

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 THE OPTIMAL POINT COUNT – ACCURACY

3 NT should be avoided with the two hands below :

ª AKx ª QJx
© KJx © AQx
¨ A 10 x x ¨ xxx
§ xxx § A 10 x x
The 6 4 3 2 1 count gives these hands 44 pts, well over the 39 pts it
needs for 3 NT (overvalues Queens, Jacks, 10s and 4 3 3 3 shapes),
the 3 2 1 ½ count gives them 19 pts, good for 3 NT, and the 4 3 2 1
count gives them 28 pts. And the ZAR point count, Bergen’s Adjust
3 count and Kleinman’s count fare no better.
Well, the optimal point count also goes overboard and adds up to 25
pts – despite all the appropriate point deductions : - 2 pts for two
4 3 3 3 distributions, - 2 pts for no Queen in West and no King in
East, and – 1 for two “mirror” suits.
Certainly, some will say that, had West’s © J been in diamonds,
3 NT would be odds-on and all counts would have it right.
Nevertheless, this example is a reminder that having the points for
a given contract does not “predict” that the contract is odds-on.
And going down is O.K. too! As you do want to go down 50 % of
the time for a game, because if you only go down 40 % of the time,
or worse yet only 30 % of the time, that likely means that you are
only bidding games that have a 60 %, or worse yet, a 70 % chance
of success – Well, do the math : at that rate you are not likely to
ever end up in the top 15 of any professional tournament !
Which brings me to one last comment on the subject of comparing
the effectiveness of various point counts: we find in Bridge
literature several references about sophisticated computerized
analyses (double dummy simulations, computerized hand evaluators,
etc.) looking for the hand evaluation method which “predicts” most
accurately how often a contract bid actually succeeds. But this
happens to be totally misguided for the following reasons :
1. They are comparing evaluation methods that are totally flawed in
the first place as they do not attribute variable values to Queens,
Jacks and 10s (with a couple of exceptions), and neither do they
attribute Fit pts for NT contracts !
2. They are evaluating the wrong thing ! Again, no hand evaluation
method can “predict” the number of tricks likely to be made !
It can only indicate that the partnership has the number of points
corresponding to a given number of tricks.

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 THE OPTIMAL POINT COUNT – ACCURACY

It is the reverse that should be evaluated, that is : which hand

evaluation methods do, or do not account for the correct number
of points corresponding to a contract that is odds-on.
And do so at least 95 % of the time! Not 80 % or 85 % of the time.
Bottom line : What is the point of comparing hand evaluation
methods that cannot even find the points for 6 NT with two hands
as basic as the deal shown in this book’s foreword, which was :
ª Axx ª xx
© xx © Axx
¨ Kxx ¨ AQJxx
§ AQxxx § Kxx
Optimal count : 15 HL + 17 HL pts + 4 Fit pts = 36 HLF pts
Woolsey count : 14 ¾ HL + 15 ½ HL = 30 ¼ HL pts = 11 tricks
Bergen Adjust 3 : 14 HL + 16 HL pts = 30 HL pts = 11 tricks
Kleinman’s count : 43 KL pts + 46 KL pts = 29 ½ HL pts (89 KL
pts divided by 3) = 10 ½ tricks
ZAR count : 29 Z pts + 30 Z pts = 29 ½ HL pts (59 Zar pts
divided by 2) = 10 ½ tricks
4 3 2 1 count : 14 HL + 15 HL pts = 29 HL pts = 10 ½ tricks
Kaplan’s 4 Cs : 14 HL + 15 HL pts = 29 HL pts = 10 ½ tricks
3 2 1 ½ count : 10 HL + 10 ½ HL pts = 20 ½ HL pts = 10 tricks
6 4 3 2 1 count : 21 HL + 23 HL pts = 44 HL pts = 10 tricks
5 4 3 2 1 count : 19 HL + 21 HL pts = 40 HL pts = 9 tricks !
And good luck searching for another hand evaluation method that
comes up with the points corresponding to this 6 NT contract…
Unless Fit pts are counted for 8-card Fits in NT contracts, no
method will !
Furthermore, note that giving West a 6-card § suit, instead of
5 clubs, would now produce a 7 NT contract and the optimal point
count would account for the points corresponding to a grand
slam : 37 ½ pts (1 additional point for the 6th card + 1 additional
point for a 9-card Fit) – while every other count would still not add
up to the points needed for a small slam !

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CHAPTER 7

Which Point Count for Opening Hands ?

H or HL or HD points ?

One would think that hand evaluation in Bridge – whatever the

methodology used or its degree of accuracy – would have no
influence on the personal “systemic” choice to count an opening
hand in H or HL or HD points. Why, then, raise the question ?
For two good reasons : 1) because, over the years, these three
different ways to count points for opening hands have been
advocated and, to some degree, still co-exist today. Which one,
then, is the best one or the “right” one ? and 2) because the one
which ended up prevailing – since the mid-‘50s among pioneers
and the mid-‘60s among other experts, champions and players
who “followed the trend” – is, incorrectly, to count opening hands
in honor points only, or in HL points for some. Actually, as we will
see, none of these three practices is appropriate !
Counting opening hands in H or HL or HD points happens to be
an enormous mistake – probably the most serious error ever in Bridge
history! It is therefore mandatory to deal with this essential topic
and that is what this particular chapter is all about.
_________________________________________________________

PURPOSE OF HAND EVALUATION

The very purpose of hand evaluation is : 1) to assess, as accurately
as possible, the number of tricks that a hand can be expected to
generate – including its trick-stopping potential, not just its trick-
taking potential, and 2) to communicate it, as best as possible, to the
partner.
And not distinguishing the major difference between trick-taking
potential and trick-generating potential is at the very heart of a key
issue which has blindsided Bridge experts since the beginning of
Contract Bridge – starting with E. Culbertson’s Honor trick count
which gave the Ace credit for only one trick (when it is worth
considerably more, see page 24/25), A K for only two tricks, etc.

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 COUNTING OPENING HANDS : IN H OR HL OR HD POINTS ?

And counting an opening hand in either H or HL or HD points is

just as serious a mistake! Here is why :
The trick-generating potential of a hand varies considerably depen-
ding upon its distributional structure – and that distributional
structure is precisely what L and D points are meant to translate !
Therefore, counting H or HL points only does not achieve the very
purpose of hand evaluation. Illustration :
ª AKQJxxx © xxx ¨ xx § x
With just a ª singleton in partner’s hand (a 7-card suit will find a
void in partner’s hand only 7 % of the time), the above hand will
generate 7 tricks 96 % of the time. 7 tricks correspond to 19 total
points (27 pts for 10 tricks at 4 ª = 2.7 pts per trick x 7 tricks = 19).
Yet, this hand is given only 13 total HL or HD pts in traditional
count – far, far below the hand’s value. That is because it does not
capture the full trick-generating potential of the hand – in addition
to counting points wrongly.
The optimal point count gives this hand 18 ½ total pts (+ 2 pts
for 3 honors in a 6-card suit, 4 L pts for a 7-card suit).
Now, if a heart, a diamond or a club is moved to the long suit,
spades, the hand becomes, for example :
ª AKQJxxxx © xx ¨ xx § x
and one additional trick gets generated : 8 tricks = 21 ½ pts. Yet, the
traditional HL or HD counts give this hand only 1 additional point,
to 14 pts! This fails to capture the hand’s trick-generating potential
that moving any card to the long suit represents. The optimal point
count goes up by 3 pts, to 21 ½ pts.
Let’s now look at various examples illustrating this trick-generating
potential point count and its applications to bidding.
Counting the following two hands as having the same 14 H points
and opening both hands 1 ª would be absurd as it would in no way
convey to partner their very different trick-generating potential :
Hand A Hand B
ª AKxxx ª AKxxxx
© AKx © AKxx
¨ xxx ¨ xxx
§ xx § ---
While Hand A has 15 HL pts (no Queen), Hand B has 20 HLD pts.

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 COUNTING OPENING HANDS : IN H OR HL OR HD POINTS ?

And a partner with the following hand of 2 ½ pts (no King) would
have no reason to bid anything but pass on a 1 ª opening :
ª Qx
© Qxxx
¨ xxx
§ xxxx
And while with hand A, it would be best to stop in 1 ª, there is
4 © or 4 ª to play with hand B! And the South hand does bring to
hand B 8 ½ HF pts (3 Fit pts in spades, 3 ½ Fit pts in hearts, minus
1 for no King + 3 pts for “no wasted honor pts” in clubs). Therefore,
hand B calls for a forcing opening bid, such as a strong 1 §.
And we can easily find in Bridge literature plenty of examples
illustrating how counting opening hands in H or HL pts only will
often lead to the wrong bidding – and therefore to the wrong
contracts. Illustrations :
In his 1972 book on the Precision Club system, « Precision Bidding
and Precision Play », British champion Terence Reese states that
« with less than 8 H pts, and no trump Fit, there is no reason to bid over
a 1 ª opening (5-card major) limited to 15 H pts, such as with this hand:
ª xx
© Qxxxx
¨ Kxx
§ xxx
Yet, a 4 © contract should be played opposite this opening hand :
ª Axxxx
© AKxx
¨ Axx
§ x
And that’s because the opener has, in fact, 18 ½ HLD pts (minus
1 point for no Queen), not 15 H pts !
And while East starts with 3 ½ pts (4 ½ H pts minus 1 point for a
doubleton without honor in opener’s long suit), the side can add
3 Fit pts for the 9-card © Fit with the Queen + 1 point for the §
singleton with 4 trumps + 2 pts for “no wasted honor pts” in clubs =
28 HLDF pts! Here again, a forcing opening bid, such as a strong
1 §, would prevent a possible pass from partner.

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 COUNTING OPENING HANDS IN HLD POINTS

Here is another example which clearly debunks T. Reese’s above

stated assertion – taken from the 2009 United States championship
in which the following hand was invariably opened 1 © :
ª A K 10 x x
© A Q 10 x x x
¨ x
§ x
Well, the hand only has 13 H “M. Work” pts, right? Therefore, no
harm could come from hearing partner pass should he have less
than 8 H pts without a © Fit… Perhaps, with a hand such as :
ª Qxx
© Jx
¨ xxxx
§ xxxx
Unfortunately, a 4© or 4ª game will be missed, and that’s because,
far from having 13 H pts, the opening hand has 22 ½ HLD pts !
It should have been opened by a forcing bid, such as a strong 1 §.
And these two hands do combine for 28 total pts, good for a game.
South’s hand has : 1 ½ pts in spades + 2 Fit pts in spades + 2 Fit pts
in hearts + 1 D point in hearts (with spades as trump) - 1 point for
no King = 5 ½ HDF pts. Opposite 22 ½ pts, that’s a total of 28 pts !
Note that “luck” has nothing to do with this : had South’s honors
been in the minor suits, instead, he would have 1 H point (1 ½ + ½
and – 1 for no King), and, at best, 1 Fit point + 1 D point : he would
pass on opener’s rebid of 2 ª (after a strong § opening, rebid 1 ©,
forcing, followed by a 2 ª rebid).

A different example can be found in Charles Goren’s 1975 book on

the Precision Club system « Charles Goren presents the Precision
system » where he shows the following hand opposite a 1 ©
opening (5-card major) limited to 15 H pts :
ª Axx
© Kxx
¨ xx
§ Kxxxx

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 COUNTING OPENING HANDS IN HLD POINTS

And makes this comment : « With the © Fit, responder : 1) knows

immediately the trump suit, 2) knows that a slam is excluded, and 3)
knows that a © game can be considered if the opener is at the maximum
of his point range, 14/15 H pts. All he needs to do now is to make an
invitational bid for a © game ».
An astonishing assessment and conclusion when, with an opening
hand such as the one below :
ª x
© A Q 10 x x
¨ Ax
§ A 10 x x x
Instead of just considering and inviting a © game – it is 7 § that
should be played ! Here again, the wrong point-count leads
straight to the wrong conclusion, as the responder has a hand of
13 ½ HLDF pts (no Queen), not 11 or 12, and a simple invitation is
not at all appropriate.
The appropriate response is, of course, 2 §, which will allow the
discovery of the § Fit. Once the double Fit in hearts and clubs is
found, the side can count : 19 ½ HLD pts (no King) + 10 ½ HL pts
(no Queen) + 2 pts for the 8-card © Fit with the King + 4 pts for the
10-card § Fit with the King + 1 point for the ª Ace facing a
singleton = 37 HLDF pts. Grand slam zone !
But the initial error is due, here again, to counting the opening
hand in H pts only. Had West properly counted his hand and had
opened a strong, forcing 1§, East could have described his
5 § 3 © 3 ª 2 ¨ hand and 10/12 HL pts and the side would easily
find the 7 § contract.
__________________________________________________________
Why then has the H or HL point count for opening hands been
advocated and adopted for so long and what could have been its
justification ?
– For some, it came from the idea that, not knowing whether the
final contract played would be in No Trump or in a suit, it would
be prudent not to count distribution pts, initially, for its opening bid.
But this argument does not stand up to analysis as the practice has
also been for the responder not to count his distribution points – and
this means that neither player counts his D points, initially !

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 COUNTING OPENING HANDS IN HLD POINTS

And that happens to be a complete mathematical nonsense in

Bridge bidding where one of the two hands must imperatively
count its distribution points, or a multitude of game and slam
contracts will be missed! And that is, indeed, exactly what we
observe every day in tournaments and world championships.
– For others, the concern comes from the dialogue bidding
methodology used : what if a game forcing 2 over 1 response falls
in opener’s singleton for which the opener has counted 2 D pts ?
But why should that possibility prevail over all the other cases
where it will not be in opener’s singleton ?
– And for others yet, the rationale is the following : in order to
achieve some point-range precision, while staying within the bidding
level safety, it is necessary to describe a hand in zones of 3 points.
And no system having been able to do so in 3 HLD point zones (too
many zones to describe), some advocated, as the next best alternative,
to describe opening hands in zones of 3 H, or HL pts instead.
This does reduce considerably the number of zones to describe :
Zone 1 : 12/14 H pts, zone 2 : 15/17 H pts and zone 3 : 18/20 H pts
– other, higher zones of 21/23 H pts and 24 + H pts being opened
through special, “artificial” opening bids, such as 2 § (and 2 ¨).
But this does not stand up to analysis either, once we have learned
to count points correctly as we just did together with this book –
as it is an “optical illusion” to think that a point zone calculated in
H or HL points respects the bidding level safety. Here is why :
Each zone of 3 H points can represent up to 4 zones of HLD pts !
Illustration : a 12/14 H pts hand can be a 4 4 3 2 hand of 12 H pts
just as it can be a 7 3 3 0 hand of 22 HLD pts ! (14 H pts + 4 L pts +
4 D pts). That is, indeed, 4 HLD zones : 12/14, 15/17, 18/20 and
21/22! And the fact is that it is simply impossible to describe,
precisely and economically, more than two zones. Many experts,
champions and bidding theoreticians have tried… to no avail.
Most importantly, this means that any hand with irregular
distribution opened within a 3 H point range immediately places
the partnership outside the bidding level safety, as bidding safety
requires that bids remain within a 6 HLD point range on an
opening bid at the level of 1 and within a 3 HLD point range on a
rebid at the level of 2 – or on an opening bid at the level of 2.
Illustration : in the following auction :
1© 1ª
2¨

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 COUNTING OPENING HANDS IN HLD POINTS

If the opener’s point zone exceeds 3 HLD pts – such as 13/17 HLD,
for example – a responder of 9 pts is already placed outside the
bidding level safety as, if he bids on, hoping for an opener of 17 pts,
but finds, instead, an opener of 13 pts without an 8-card Fit, his
side will have a total of 22 pts only, without a Fit, short of the
points needed to play at the level of three.
From this analysis, an unequivocal conclusion can be drawn :
The optimal point count makes it clear that opening hands must
imperatively be counted in HLD points, not in H points only !
A small revolution – after so many years in “medieval darkness”…
On that specific issue, Goren, as well as the ZAR point count (even
though their distribution point count is flawed), were in fact well
advised to count distribution pts for opening hands.
And Bridge bidding has never recovered from this historic mistake of
counting opening hands in H or HL pts only – and strong 1 § openings,
as generally practised (16 + H pts), do not escape this historic mistake.

Hereafter are a few more examples of some serious misbidding

resulting from the misguided practice of counting opening hands
in H or HL points only :
Rosenblum Cup, USA, 2010.
ª x ª xxxx
© AJxxx © Kx
¨ --- ¨ Jxxx
§ AKxxxxx § J 10 x
The team considered the World’s best for a 15 to 20 year period
opened West’s hand a non-forcing 2 § ! (natural, 5 + clubs, 11/15
H pts). Rather difficult for East to be motivated to bid over this
opening…
Astonishing opening of a non-forcing 2 § – the hand doesn’t have the
16 H pts prescribed to open 1 strong § ! – a hand which has a 5-card
major and a 7 – 5 two-suiter of 8 to 9 winning tricks ! (23 HLD pts).
But then, the pair may not have to worry too much about their
bidding system in view of their opponents’ bidding which was as
follows, in the other room :
1© 1ª
3 § (non forcing < 16 H pts) Pass

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 COUNTING OPENING HANDS IN HLD POINTS

A good demonstration of how to miss a game contract, or a

possible slam !... and further illustration that counting an opening
hand in H points only can lead to embarrassing situations !
That’s because one point zone counted in H pts can equate to
4 point zones of HLD pts. Here, 12 H pts = 23 HLD pts !
It is mandatory to count opening hands in HLD pts and a forcing
opening bid should be used here, such as a strong 1 §.
Furthermore, beyond the initial lack of precision of the opening
bid, counting opener’s hand in H points only extends the lack of
precision to opener’s rebids, having to jump to make up for a non-
forcing opening.
Here is an example, from M. Bergen’s 2002 book on Hand
Evaluation where he shows the following opening hand :
ª xx © A K J 10 x x x ¨ --- § KJxx
And asks : « What would you rebid after : 1 © 1 NT
?
His solution : « Bid 4 ©. Over a 1 NT response, you must insist on
game with such a terrific hand ».
Wow! Burning 12 bidding steps… only to miss a 6 § slam with a
partner having, for example, the following hand :
ª Axx © xx ¨ xxxx § AQxx
Unaware of a § Fit, partner will have no reason to bid on.
And the non-forcing 1 © opening could be just as bad opposite a
partner having a hand such as :
ª xxx © Qx ¨ xxxx § Q 10 x x
Partner will pass and the 4 © game will be missed !
Had the opening hand been counted fot the 22 ½ HLD pts it is
worth – instead of in H points only – and been opened 1 strong §,
forcing, neither accident would occur, as the opener would not
need to jump rebid to show his hand’s strength and would have
ample bidding space to identify, first, his © suit, then his § suit
and then rebid his © suit.
Lesson : Counting this opening hand in H points only is total
heresy! Elementary logic tells you that if it is worth a double-jump
rebid on a response promising just 6 H pts, it warrants a forcing
opening !
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 COUNTING OPENING HANDS IN HLD POINTS

The « Rule of 20 ».
The misguided phobia of counting only H or HL points for opening
hands has, naturally, led to different opinions among experts –
and legitimate question marks for many players – about when to
open, or not, “borderline” irregular hands. Thus was born the Rule
of 20 (the rule of 19 for some…) to help resolve such dilemma, this
rule being : « Open when the total of your two longest suits + your
honor points adds up to 20 or more – but still make the needed
adjustments guided by sound judgment… ».
But how will this rule be helpful when honor pts are not counted
properly in the first place and if it still requires sound judgment
which, if we had it, would not require this rule to “guide us” ?!...
To illustrate the judgment needed to use this Rule of 20, the
American champion M. Bergen, uses the following examples in his
2002 booklet on « Hand Evaluation » :
ª xx
© A Q 10 x x
¨ A 10 x x
§ xx
His comment on this opening hand is : « 5 © + 4 ¨ + 10 H pts =
19. But with all your honors in your long suits and two Aces + two
10s, you should upgrade your hand to at least 20 + and open 1 © ».
But isn’t that saying that two Aces are worth more than 8 pts and
that 10s have some value not accounted for ? – in other words, that
your point count was flawed in the first place. Why then use this
Rule if its very premise is flawed? Furthermore, the distribution of
the residual 4 cards is not accounted for…
Wouldn’t it be simpler to use the optimal count which precisely
quantifies all pertinent factors including distribution? It counts :
12 HL pts (9 pts for two Aces, 1 point for the © 10, 1 point for the
5 hearts, minus 1 for no King) + 1 D point for 2 doubletons = 13
HLD pts. So, yes, open 1 ©. No need for an extra Rule or for expert
judgment.
Here is his second example : ª AQxxxx
© xxx
¨ ---
§ K 10 x x

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 COUNTING OPENING HANDS IN HLD POINTS

His comment on this hand is : « 6 ª + 4 § + 9 H pts = 19. But

upgrade your hand and open 1 ª because the 3 – 0 distribution of
your residual cards, which is not factored in in the Rule of 20, has
greater potential than a 2 – 1 residual, just as a 5 4 4 0 distribution
is better than 5 4 3 1, which, itself, is better than 5 4 2 2 ».
But then why use such a Rule while acknowledging that it doesn’t
factor in a key element, distribution, which you should factor in ?!
Wouldn’t it be simpler to use the optimal count which precisely
quantifies all pertinent factors including distribution and counts :
12 HL pts (4 ½ pts for the Ace, ½ point for the § ten with the King,
2 pts for the 6 ª) + 4 D pts for the ¨ void = 16 HLD pts – Of course,
open 1 ª! No need for an extra Rule or for expert judgment.
And, actually, a 2 – 1 residual, instead of 3 – 0, shouldn’t alter the
opening bid : the hand will be worth 14 HLD pts and you should
still open 1 ª !
Here is his third example : ª KQ
© Jxxxx
¨ QJxx
§ QJ
His comment on this hand is : « 5 © + 4 ¨ + 12 H pts = 21. But you
should seriously downgrade your hand and Pass. It has too many
minuses : no Ace, honors in doubletons and your longest suit is
terrible ; this hand is a pile of garbage ! ».
No doubt! But again, the initial point count is wrong and
downgrading an Aceless hand as well as two honors doubleton
has nothing to do with judgment! It has to do with knowledge – the
knowledge that 1 point should be deducted for an Aceless opening
hand and for two honors doubleton as statistically established by
J-R. Vernes, as far back as… 1966 !
Actually, there is simply nothing borderline about this hand.
Unfortunately, a doubt is introduced by the very Rule of 20 whose
count adds up to… 21 !
The optimal count will give you the answer : 4 H pts in spades,
not 5, ½ point in hearts, - 1 point for no Ace + 1 D point for the two
doubletons = 10 ½ HD pts. Opening this hand won’t even cross
your mind ! No need for an extra Rule or for expert guidance...
Surely, creating new “gadget rules” to compensate for poor point
counts cannot be the answer – there must be a better way !

130
 OPTIMAL HAND EVALUATION

CHAPTER 8

SUMMARY AND CONCLUSION

A. SUMMARY
We have learned the following from the optimal point count :
– M. Work’s 4 3 2 1 honor point count is almost entirely erroneous :
Aces are worth 4 ½ pts, not 4, Queens and Jacks have variable
values, not fixed values of 2 and 1 point respectively, and the count
does not factor in the value of tens, when appropriate. The only
correct count is therefore 3 pts for the King.
Nor is 1 point deducted from a hand with no Queen or no King.
– C. Goren’s distribution point count is also almost entirely
erroneous: distribution points for an opening hand should be
4 2 0 1 pts for, respectively, a void, a singleton, one doubleton and
two doubletons, not 3 2 1 points.
Furthermore, 1 point must be deducted for a 4 3 3 3 distribution.
As for distribution points with a trump Fit, it is also erroneous to
count 5 3 1 pts with a Fit of 4 trumps : distribution points are the
difference between the number of trumps held by the support
hand and the number of cards it has in its shortest suit. Thus, 2 pts,
not 1, should be given to a doubleton in a hand with 4 trumps.
– As for the count of Length points, it should be 1 point for 5 cards,
2 pts for 6 cards plus 2 points for each card beyond the 6th, not
1 point per card beyond 4 cards. And this count should not be an
alternative to the count of Distribution points for short suits, it
should be in addition to them.
– Furthermore, Fit pts of 1, 2, 3 pts for, respectively, 8, 9 and 10 +
card Fits must be counted for all suits and all contracts and,
similarly, Misfit pts of 1, 2, 3 pts must be deducted for, respectively,
a doubleton without honor, a singleton, a void opposite a 5 + card
suit in partner’s hand, and at least 1 Misfit point must be deducted
for a singleton when playing a NT contract.
– As well, “wasted honor points”, or lack of, must be accounted for
(- 2 or 3 pts or + 2 or 3 pts) opposite a short suit.

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 THE OPTIMAL POINT COUNT – CONCLUSION

– Last, but certainly not least, the optimal point count exposes the
fallacy perpetrated by the traditional conversion table converting
25 H or HL pts into 9 “expected” tricks for a 3 NT contract or
26 HDS pts into 10 “expected” tricks for a 4 ª or 4 © game, etc.
These numbers have been pure fiction and without any rational
foundation or statistical validity as they do not count appropriate
points for honors and for tens, or for 4 3 3 3 distributions, or for
long or short suits, or for Fits or Misfits or for “wasted honor pts”.
Furthermore, points cannot “predict” the number of tricks that can
be made – they only correspond to a given number of tricks.
All of the above does not leave much that is valid or accurate in the
traditional hand evaluation point count still universally used and
taught today – which explains the multitude of wrong contracts
reached every day by all players, including world-class players.
From that standpoint alone, calling the findings of the optimal
point count revolutionary would not be an extravagant statement.

B. CONCLUSION
Before moving on to this book’s Epilogue, let’s briefly assess what
the optimal point count brings to precise hand evaluation which
can be summarized as follows :
Quantified accuracy, statistically validated : it is high time to
replace “expert suggestions”, often too vague when not inaccurate,
by a precise quantification, statistically validated, which does not
continuously call on ”enlightened judgment” which only multiple
world champions would seem to have.
Indeed, we could well summarize what we have read in most
Bridge books written since the beginning of Contract Bridge on the
subjects of hand evaluation and its application to Bridge bidding
as follows :
« The C. Goren/M. Work point count is simple and, generally,
reasonably accurate. It just needs some “adjustments” based on sound
judgment which experienced experts will provide through their
numerous books.
Carefully appraise each and every hand and make the appropriate
“upgrading” and “downgrading” adjustments based on judgment
that will be helped by the following guidelines :

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 THE OPTIMAL POINT COUNT – CONCLUSION

Upgrade your hand with : Downgrade your hand with :

____________________________ ______________________________
Combination of honor cards Isolated Queens and Jacks, parti-
in the same suit cularly when in short suits
Honor cards in Long suits Honor cards in short suits
Several Aces and Kings No Ace, few Kings (vs Qs & Js)
5-card suit(s) or longer No 5-card suit, 4 3 3 3 hands
Intermediate cards (Jacks, 10s) Lack of intermediate cards
Honors in partner’s suit(s) No Honor(s) in partner’s suit(s)
Hands with Long suit(s) and The same short suit as partner’s
Short suit(s), such as : 5 4 3 1,
6 4 2 1, 5 5 2 1, 6 4 3 0, 7 3 3 0
A Fit (8 + cards) with Partner Misfit with partner
9 trumps are better than 8 Duplication of honors & length
As to by how many points should you upgrade or downgrade :
1 point or 2 pts or 2 ½ pts? – you figure it out, use your judgment...
If still not sure, add into the mix factors such as Quick tricks, Losing
tricks, Playing tricks, the Rule of 20 or the Rule of 19, etc. and keep
reading as many books as possible from experts and world
champions which will guide and train your judgment –
Eventually, through trial and error, you may get it right…
– Every single one of these factors is factored into the optimal
point count ! One count does it. Period.
No need to torture yourself on every hand, to refer to guidelines,
to Rules of whatever, or to exercise the “appropriate judgment” to
appraise your hand correctly !...

But then, what is the point of trying to evaluate as precisely as

possible the value of a hand and its trick-generating potential if the
bidding then cannot or does not communicate it to partner ?!..
And this observation seems to reflect the following comment,
often made by many players :
« Beyond the traditional honor point count, I evaluate my hand rather
“intuitively” as no “methodical” hand evaluation seems to be precise
enough to be very helpful. Besides, I have yet to find a bidding system
that communicates such information very effectively ».

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 THE OPTIMAL POINT COUNT – CONCLUSION

And such “intuitive” evaluation seems to be common practice for

many and is often comforted by Bridge literature as illustrated in
Mike Lawrence’s 1983 book on hand evaluation « The complete book
on Hand Evaluation in Contract Bridge ».
In this book, Mike Lawrence rarely refers to quantified point
evaluation when assessing a hand and choosing the bid he would
make with it. Here is one of the examples he uses :
« Your partner has opened 1 ª (5-card Major) and you hold :
ª Q 10 x x
© Axx
¨ Jxxx
§ Kx
After commenting on the “pluses” and “minuses” of this hand, with
no mention of any point count, he concludes in favor of a 3 ª raise.
Well, this rather “intuitive” hand assessment is misguided and
leads to the wrong bid. Should opener have the following hand :
ª AKxxx
© Kxxx
¨ Qxx
§ x
3 ª is in serious jeopardy – particularly if the missing diamonds
are 4 – 2 with the lead of ¨ A K followed by a 3rd round ruffed.
But with this opener’s hand :
ª AJxxx
© Kxxx
¨ x
§ QJx
Now, 4 ª should be played. But how could the opener know
which hand partner has on a 3 ª limit raise ?!...
There is a better way… The optimal point count gives the
responder’s hand 15 ½ HDF pts (5 ½ Fit pts in spades and 2 D pts
for the § doubleton with 4 trumps). Therefore, a limit raise is out
of the question! And a bidding system that “burns” 9 bidding steps
to describe… nothing! cannot possibly be the solution.
A 2 NT Jacoby raise would have been more helpful…

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 THE OPTIMAL POINT COUNT – CONCLUSION

On a 3 § bid by the opener, the responder would sign off in 3 ª –

wasted honor pts in clubs and duplication of distribution pts counted
by each hand – while he would bid 4 ª on opener’s 3 ¨, “no wasted
honor pts” in diamonds.
In the following example, from the same book, M. Lawrence gives
the following responder’s hand, which prompted lengthy and
revealing comments on his part – with no mention of point count :
ª Axx
© Ax
¨ K Q J 10 x
§ Qxx
On his partner’s 1 § opening, the auction has been :
1§ 1¨
2§ 2ª
2 NT ?
And M. Lawrence’s comments are : « What now ? this is a hard hand
to bid. Opener has 6 § and 7 NT could be cold. That’s what prompted me
to bid 2 ª to learn more about opener’s hand. But his rebid of 2 NT would
indicate rather minimum values and makes me rule out a grand slam at
this point. My choice would be in favor of a simple sign-off bid of 3 NT
as I do not see any other sensible bid that would help partner make an
educated decision : if I bid 3 §, my partner may fear that I am short in
hearts and, being himself probably short in diamonds, he won’t know
what to do or why he should do it. And for me to bid a “quantitative”
4 NT would be pointless as I already know that he will Pass in most cases,
being minimum. Bidding 3 NT will avoid a nerve-wracking auction
which would not likely help much anyway. Trying “scientific” bidding
with such hands would be overdoing it ».
Wow! But if visualising what partner may have must be part of the
“anticipation” principle, then why not anticipate the simplest
scenario – which could turn out to be rather favourable, with a
partner having, for example, the following hand :
ª xx
© xxx
¨ Ax
§ AKJxxx

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 THE OPTIMAL POINT COUNT – CONCLUSION

A minimum opener’s hand, that yields… a 100 % 7 NT contract !

And the side does have the optimal points for it : 16 HL pts (no
Queen but 2 pts for 3 honors in a 6-card suit) for opener’s hand +
19 ½ HL pts in responder’s hand + 3 Fit pts in clubs = 38 HLF pts.
And how difficult is it to imagine a partner with such a hand ?
But an opener with this hand is not likely to bid again over a 3 NT
sign-off bid! With the proper point count, wouldn’t it have been
rather simple to bid as follows :
1§ 1¨
2 § (6 §) 4 § (RKC Ask, trump is §)
4 ¨ (3 Key cards) 7 NT

One last example from the same book :

ª AQxxx ª Kxxx
© Ax © xxx
¨ KJxx ¨ A Q 10 x
§ Ax § Jx
M. Lawrence comments this deal as follows : « When you and your
partner discover quickly a suit Fit in a Major, it is often difficult to find
a better contract in another suit, particularly in a minor suit. The two
hands above are a good illustration of this : they will easily make 4 or
5 ª but 6 ¨ is a much better contract to find! But how should it be bid
after an auction that started as such : 1 ª 3ª
Hard to untangle, and I’m not sure how to get about it ».
What’s wrong with this picture? Just about everything, starting with
opening a hand worth 21 ½ HLD pts by a non-forcing 1 ª, and then
biding a 3 ª limit raise whith a hand worth 15 ½ HDF pts! (3 Fit
pts for the 9-card ª Fit with the King and 2 D pts for the
§ doubleton with 4 trumps). A 3 ª limit raise is out of the question!
The right bidding has to start with a precise point count! And this
would have allowed responder to start with a 2¨ response
followed by 3 ª, slam invitation.
A 4 § RKC Ask would then allow the opener to get to 6 ¨.
But then, that’s assuming that West’s 4 § bid is not used to identify the
§ control which would force making the RKC Ask at the 4 NT level – too
high to allow West to detect whether East has the ¨ Q, instead of the
heart or club Q or J 10 of spades.

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 THE OPTIMAL POINT COUNT – CONCLUSION

The following would be better bidding, using a strong § system :

1 § (16 + HL) 1 NT (10/12 HL, balanced)
2 § (Stayman) 2 ª (4 spades)
2 NT (forcing) 3 ¨ (4 diamonds)
4 § (RKC Ask, trump is ª) 4 ª (2 Keys, w/o ª Q)
5 ¨ ? (¨ K & Q Ask) 5 ª (¨ Q)
6¨
Sound judgment is not intuitive; it is an educated assessment which
can and should be precisely quantified. Hopefully, this book will have
contributed to demonstrate this in a convincing manner.
In closing, the most important feature of the optimal hand
evaluation point count is its ability to accurately assess the value
not just of one hand, but of the combined hands of one side – through
Fit points, misfit points, “wasted honor points” and others – the true
test of an effective hand evaluation method.
Thus, the “luck” factor is minimized – assuming that the bidding
system used does provide the precise information needed to count
points accurately. The 6 NT example deal shown in this book’s
foreword clearly illustrates this; the hands were the following :
ª Axx ª xx
© xx © Axx
¨ Kxx ¨ AQJxx
§ AQxxx § Kxx
Had the Kings been, instead, in the Major suits, such as :
ª Axx ª Kxx
© Kxx © Axx
¨ xx ¨ AQJxx
§ AQxxx § xx
The bidding would then be :
1 NT (15/17 HL) 2 ¨ (forcing Stayman)
3 § (5 §) 3 ¨ (forcing-relay)
3 NT (¨ doubleton) Pass
This time, with minus 1 point for a doubleton without honor in
partner’s long suit, clubs, instead of + 2 pts for an 8-card § Fit with
the King, and no 8-card ¨ Fit, the side does not have the points for
a slam and East can pass on 3 NT.
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 THE OPTIMAL POINT COUNT – CONCLUSION

The key to the optimal point count’s ability to accurately assess

the value of the combined hands of the same side is that it translates
Fits and favorable honor mesh into additional points, and Misfits
into point deductions.

But then, to fully and effectively exploit the optimal point count,
we need to play bidding systems of equal precision – Distributional
precision as well as Strength precision within 3 HLD point zones.
And the optimal Hand Evaluation point count has major
implications on bidding, and these implications are outlined in
this book’s next and last chapter, titled Epilogue.

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 IMPLICATIONS OF THE OPTIMAL HAND EVALUATION ON BIDDING

Epilogue : Implications of the Optimal

hand evaluation on Bidding

KEY IMPLICATIONS OF THE OPTIMAL HAND EVALUATION

The Optimal Hand Evaluation has serious implications on some

bidding practices and principles. Some of them have already
previously been mentioned, such as :
– The “quantitative” 4 NT jump on a 1 NT or 2 NT opening must
be stricken from bidding, in its current meaning, as it totally ignores
the possibility of Fit points.
– The same applies to an invitational 2 NT response to a 1 NT
opening, for the same reasons. Some other meaning must be
attributed to that bid.
– “Splinter” bids at the level of 4 must have a minimum of 15 HLDF
pts (15/17 point zone, instead of 13/15), as 13 or 14 HLDF pts
opposite an opener with a minimum 13 HLD pts hand will not add
up to 26 pts whenever a singleton (or a void) finds “wasted honor
pts” in partner’s hand.
And there are several others, as well, some being very significant.
Hereunder are some of the most important ones :
– The discovery that Fit points count for all suits and all contracts,
including NT, now rules out “hiding” a 6-card minor single-suit
behind a NT opening. Illustrations :
Readers’ mail, 2010 – Le Bridgeur.
ª A 10 x ª Kxx
© Ax © xxx
¨ xx ¨ xxxx
§ A K 10 x x x § Qxx
« We bid as follows : 1§ 1¨
2 NT Pass
« And missed the 3 NT game. What did we do wrong ? ».
The wrong doing isn’t the players’ fault – It comes from the poor
bidding system played which prevents the appropriate point count.
East, with its 3 ½ pts (- 1 for 4 3 3 3), cannot bid 3 NT without
knowing West’s 6 clubs – a fact well “concealed” by the opener !

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 IMPLICATIONS OF THE OPTIMAL HAND EVALUATION ON BIDDING

Had East known it, he could have added 3 pts for the 9-card § Fit
with the Queen. A different auction could have been :
1§ 1¨
3 § (6 §, 17/19 HLD) 3 © ? (© guarded ?)
3 NT (yes, © guarded) Pass

Regional tournament, 2009.

ª KQx ª xxx
© Ax © xxx
¨ Axxxxx ¨ K 10 x x
§ xx § Axx
An amusing case where all West hands were invariably opened
1 NT. A NT opening which, for some, was 15/17 or 14/16 HL pts,
and for others 13/15 or 12/14 H pts. You guessed it… in all cases,
the response was : Pass !
Well, isn’t it exactly how we have been taught to count points and to bid
for the last 40 years or so ?!...
Instead, the bidding could be :
2 ¨ (6 + ¨, 15/17 HLD) 2 © ? (relay)
2 NT (6 ¨ 3 2 2) 3 ¨ (¨ Fit, invitation to 3 NT
3 NT (Majors guarded) if West guards the Majors)
Right upon West’s opening, East can count 10 HLFit pts in
diamonds (8 H pts - 2 pts for 4 3 3 3 and no Queen + 4 pts for the
10-card ¨ Fit with the King). He “relays” to detect a possible
singleton in West’s hand. Upon West’s 2 NT rebid, 3 ¨ shows the
¨ Fit and invites West to bid 3 NT if he guards the Majors.
– The unequivocal fact that opening hands must be counted in
HLD pts, not in H or HL pts only, dictates that weak 2 openings in
a Major must be 12/14 HLD pts, not 7 to 10 H pts which put the
partnership outside the bidding safety level and are, therefore,
unplayable! One simple example will illustrate this :
North, you hold this hand : ª Kx
© QJxxx
¨ Axxx
§ xx
And your partner opened a weak 2 ª of 7/10 H pts.

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 IMPLICATIONS OF THE OPTIMAL HAND EVALUATION ON BIDDING

Which of these two hands could he have ? :

Hand A Hand B
ª Q J 10 x x x ª AQxxxx
© Kx © Kxx
¨ xxx ¨ x
§ Qx § xxx
If he has hand A, you must not bid on as 2 ª already goes down
one, when the opponents don’t even have a play for a game in a Major –
but then you will miss 4 © opposite hand B.
And that is because hand A only has 10 HLD pts (one point only
for the §Q doubleton, one point only for the two doubletons,
minus 1 point for an Aceless opening hand).
With your 13 ½ HLFit pts in spades in North, the total isn’t close
to 27 pts – if you knew it, you would Pass.
Hand B, on the other hand, has 13 ½ HLD pts + 2 pts for the 8-card
© Fit with the King + your own 13 ½ pts = 29 pts. And when you
can count on an opener having, at least, 12 HLD pts, you can safely
“relay” to discover South’s 6 ª 3 © 3 § 1 ¨ distribution and play
4 ©! Such as :
ª AQxxxx ª Kx
© Kxx © QJxxx
¨ x ¨ Axxx
§ xxx § xx
2 ª (6 ª, 12/14 HLD) 2 NT ? (relay)
3 ¨ (¨ singleton) 4©
Counting opening hands in H pts only is pure heresy! And when
weak 2 ª or © openings “boomerang” against its own side, it is
often the count in H pts that is the primary reason for it.

– And this applies, as well, to balanced hands which must be

counted in HL pts. Thus, it is inconceivable to open the following
hand 1 NT, 13/15 or 14/16 or 15/17 or 16/18 H pts :
ª Ax
© J 10 x
¨ AKQxx
§ J 10 x

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 IMPLICATIONS OF THE OPTIMAL HAND EVALUATION ON BIDDING

The above hand has 20 HL pts! (9 pts for 2 Aces, 2 pts for each J 10,
1 L point for the 5 diamonds + 1 point for 3 honors in a 5-card suit).

– The critical importance of identifying precisely the length of

suits, for Fit pts, as well as singletons (or voids) in order to account
for “wasted honor pts”, prohibits jump raises that do not describe a
short suit. Illustration :
ª AQxx ª Kxxxx
© x © xxx
¨ AKxxx ¨ xx
§ xxx § KQx
If, in traditional bidding, the auction is :
1¨ 1ª
3 ª (ª Fit, 17/19 HL) ?
What is East to do now? He knows nothing about West’s ©/§
residual distribution. If West has a § singleton, he must pass.
But then, he will miss 4 ª if West has a © singleton...
Instead of : 1¨ 1ª
3 © (mini-splinter, ª Fit) 4ª
West’s “mini-splinter” indicates that he has 17/19 HLDF pts with at
least an 8-card ª Fit + a third point for a singleton with 4 trumps.
Knowing West’s © singleton, East can now add 2 pts for “no wasted
honor pts” in hearts for a minimum of 27 pts on his side = 4 ª.
– As well, “trial” bids following a trump raise should no longer be
inquiring about partner’s holding in a suit but should, instead, be
short-suit trial bids in order to assess “wasted honor pts”. Example :
ª Axx ª xxx
© A K 10 x x © Qxxx
¨ QJxx ¨ Kxx
§ x § KJx
If, in traditional bidding, the auction is :
1© 2©
3 ¨ (¨ strength ?) 4©

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 IMPLICATIONS OF THE OPTIMAL HAND EVALUATION ON BIDDING

East will conclude in 4 © as he does have the complementing

diamond honor West is looking for, but it won’t help – as the key
is for West to identify his singleton, enabling East to assess “wasted
honor pts”.
While a 3 § “short suit” trial bid will enable East to stop in 3 ©.

Last, but not least, the most significant implication.

– All opening hands must be counted in HLD pts and opening
bids at the level of 1 must be precisely within a 6 HLD point zone –
since no more than two point zones of 3 HLD pts each can be
described economically, by any system !
This necessarily dictates to use one opening bid to identify hands
just above that limit. The implication is clear and unequivocal :
A strong 1 § opening bid must be used to do that !
Putting the “break” point, say, at 18 HLD pts, would mean that
openings of 1 ¨, 1 © and 1 ª would be limited to 17 HLD pts with
2 HLD point zones of : 12/14 and 15/17 HLD pts, while the 1 §
opening would be in the 18/20 and 21/23 HLD point zones.
So, now, playing a strong 1 § opening is no longer a “systemic
choice or preference” – it is a “must” for any bidding system !
This definitely fixes the most fundamental problem of traditional
bidding : a point range too wide for openings of 1 §, 1 ¨, 1 © or
1 ª, a flaw denouced by H. Schenken as far back as 1963 when he
wrote, in his introduction to his bidding system « Better Bidding in
15 minutes / The Schenken Club », that, in his opinion, « Traditional
bidding is inaccurate and ineffective and has become obsolete…
Its biggest problem being the wide range of the opening of one ».
Note : At the same time, bidding will benefit from the other
benefits such a strong 1 § opening provides :
– Its forcing nature prevents an inappropriate Pass by a weak
partner while enabling him to know immediately the partner-
ship’s minimum strength – a significant benefit when opponents
intervene.
– The most economical 1 § opening gives partner the most bidding
space to describe his hand – and that’s as it should be as it is the
strong hand that should have the weaker hand described.
– It enables partner to make precise and aggressive “limit” bid
responses directly at the levels of two and three with 7/9 HL pts
with the assurance that their side has a minimum of 25 HLD pts.

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 IMPLICATIONS OF THE OPTIMAL HAND EVALUATION ON BIDDING

– It also means that on an opening bid of 1 ¨, 1 © or 1 ª, limited

to 17 HLD pts, partner no longer needs to bid with a weak hand of
5 H pts. With less than 7 HL pts, he can now pass without fear of
missing a game.
It also means that 1© and 1ª openings can now be “light”
openings in the 12/14 HLD point range, with, for example, 5 4 3 1
hands of… 9 H pts! Thus, the following hands can be bid to 4 © :
ª KQxx ª Axxx
© Axxxx © Kxxx
¨ xxx ¨ x
§ x § xxxx
With 12 ½ HLD pts, West can open, and the auction could be :

1 © (5 + ©) 1 ª (7 + HL pts)
2 ª (ª Fit, 13/16 HLDF) 3 ¨ (“trial” bid)
4ª Pass

On East’s 1 ª response, West’s hand has 14 ½ HLDFit pts (+ 1 for

the 8-card ª Fit + 1 for the § singleton with 4 trumps) and, after
East’s 3 ¨ short-suit trial bid (11/13 Fit pts), West can add 2 pts for
“no wasted honor pts” in diamonds = 27 ½ + HLDF pts = 4 ª.

A little better than : Pass Pass wouldn’t you say ?...

_________________________________________________________

As was just illustrated above, the implications of the optimal

Hand Evaluation point count on bidding are significant and
applying them is sure to lead to much improved bidding and
better results.
_________________________________________________________

I would welcome any comments readers may have. Please feel free to
address questions/comments to : patrick.darricades@gmail.com

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 OPTIMAL HAND EVALUATION – INDEX / NOTES

INDEX / NOTES
_________________________________________________________

Note # 1 : The ZAR point count.

In the ZAR count, honors are counted 6 4 2 1 pts for the Ace, King,
Queen, and Jack (that’s the « four Aces » 3 2 1 ½ count multiplied
by two, to avoid counting in half-points) while distribution pts are
the sum of the length of the two longest suits added to the difference
between the longest suit and the shortest suit, i.e. a 5 4 3 1 hand has
13 D pts : 5 + 4 + 5 – 1 = 13.
The total of H pts and D pts equates to twice the traditional point
count i.e. 52 ZAR pts are needed for a game in a Major suit, that is
twice the 26 pts needed in the traditional count.
This method’s distribution point count does differentiate various
distribution shapes, such as : 8 pts for 4 3 3 3, 10 pts for 4 4 3 2,
11 pts for 5 3 3 2, 12 pts for 5 4 2 2, 13 pts for 5 4 3 1, etc. But it has
two major flaws :
1) The points assigned to distribution are disproportionally greater
than those assigned to honors, i.e. the following hand :
ª AKxxx © KQJx ¨ xxx § x
is given 17 H pts and 13 D pts – This gives 43 % of total pts to this
5 4 3 1 distribution, an absurdly high ratio without any possible
statistical validity !
The optimal count gives this hand 13 ½ H pts and 3 LD pts, and
these LD pts account for about 20 % of total pts.
2) Yet, the differentiation of only one point between each of the hand
distributions shown above is way too small to accurately reflect
their differences – equal to only one half point in traditional count.
As a result, the more unbalanced the hand is, the less its true
distributional value is accurately reflected by the ZAR count.
Two example hands, below, will clearly illustrate the ZAR count’s
two major flaws identified above :
Hand 1 ª A K Q J 10 © A K Q J 10 ¨ xxx § ---
In ZAR count, this hand has 26 H pts + 15 D pts for a total of 41 pts
– Far, far short of the 52 pts needed for a game in either Major,
which you would want to bid with just a doubleton in either Major
in partner’s hand. And 8 D pts in partner’s hand (4 3 3 3, meaning
two 8-card Fits in the Majors !) will only add up to 49 pts !

145
 OPTIMAL HAND EVALUATION – INDEX / NOTES

Clearly, this hand’s power comes overwhelmingly from its strong

honors concentrated in two suits but ZAR’s honor pts represent
only 63 % of total pts. Yet, distribution pts are underrated at
15 pts, only 2 pts more than the 13 pts given to a 5 4 3 1 hand.
In the optimal point count outlined earlier, this hand has 21 H pts
+ 6 LD pts = 27 HLD pts – the points needed for a Major suit game.
Its 6 LD pts are double the 3 LD pts given to a 5 4 3 1 hand, yet honor
pts represent 77 % of total pts.
The next example hand is the imaginary hand of 13 cards in the
same suit which no hand evaluation method seems to have been
able to properly quantify – until now :
Hand 2 ª A K Q J 10 x x x x x x x x © --- ¨ --- § ---
In ZAR count, the above hand has 13 H pts + 26 D pts for a total of
39 pts – Fewer points than the 41 pts given to the previous hand !
Barely good enough for… a 1 ª contract! (equivalent to 19 ½ pts in
traditional count). And a partner with a 0 5 4 4 distribution will only
add 14 D pts for a total of 53 pts – barely enough pts for a game !
Pure nonsense !
Here, unlike the previous example, this hand’s power clearly
comes from its extraordinary distribution, yet the ZAR D pts only
represent 66 % of total pts.
As previously mentioned, the more unbalanced the hand, the less
accurate the ZAR distribution count is. Yet again, a point count
seriously flawed…
In the optimal point count outlined earlier, Hand 2 has 10 ½ H pts
+ 16 L pts for length + 12 D pts for 3 voids = 38 ½ HLD pts – The
points for a Grand slam !!
Finally, a hand evaluation method for Bridge hands which does quantify
properly this famous hand distribution !
__________________________________________________________

Note # 2 : The true value of 10s.

D) A ten which accompanies a Queen adds significant value which

should be assessed, on average, one point.
Q 10 x xxx The 10 adds little value.
Q 10 x Jxx The 10 is likely to generate one trick, worth 2.8
pts. The Queen being valued 1 ½ pts, the 10 is
worth 1.3 points.

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 OPTIMAL HAND EVALUATION – INDEX / NOTES

Q 10 x Kxx The 10 allows here finessing the J, succeeding

half the time. It is worth half a trick = 1.4 pts.
Q 10 x A x x The 10 allows here finessing the J, succeeding
half the time. It is worth half a trick = 1.4 pts.
Q 10 x KJx The 10 will not generate an additional trick.
Q 10 x AJx The 10 allows here finessing the K. Succeeding
half the time, it is worth half a trick = 1.4 pts.
Q 10 x A K x The 10 will not generate an additional trick.
Q 10 x x K J x The 10 will generate one additional trick. But
so would Q x x x on a 3 – 3 break (36 % of the
time). The 10 is worth about 2 pts.
Q 10 x x A J x The 10 allows here finessing the K. Succeeding
half the time, it is worth half a trick = 1.4 pts.

Here again, deducting a card or adding a 4th card to either side

will rarely alter the above.
When it accompanies a Q, a 10 often provides significant value
which, on average, can be estimated to be worth one full point.
E) A 10 which accompanies a Jack provides even more significant
value than it brings to the Queen. It should be evaluated, on
average, one full point – and, furthermore, it gives the Jack a value of
one full point in all cases.
J 10 x xxx The 10 adds no value.
J 10 x Qxx J 10 is likely to generate one trick. The J 10
combination is worth 2.8 pts = 1.4 point each.
J 10 x Kxx J 10 is likely to generate one trick. The J 10
combination is worth 2.8 pts = 1.4 point each.
J 10 x Axx The 10 provides defensive value which can
be evaluated 1 point.
J 10 x KQx The 10 will not generate an additional trick.
J 10 x AQx The 10 allows here finessing the K. Succeeding
half the time, it is worth half a trick = 1.4 pts.
J 10 x AKx The 10 allows here finessing the Q. Succeeding
half the time, it is worth half a trick = 1.4 pts.
J 10 x x K Q x J 10 will generate one trick. But so would
J x x x on a 3 – 3 break ( 36 % of the time ). The
J 10 combination is worth about 2 pts = 1 point
each for the Jack and the 10.
J 10 x x A Q x The 10 allows here finessing the K. Succeeding
half the time, it is worth half a trick = 1.4 pts.
J 10 x x A K x The 10 allows here finessing the Q. Succeeding
half the time. It is worth half a trick = 1.4 pts.

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 OPTIMAL HAND EVALUATION – INDEX NOTES

Here too, removing a card or adding a 4th card to either side will
not alter the above. When it accompanies a J, a ten provides
significant value which, on average, can been assessed to be worth
1 full point.
_______________________________________________________________________

Note # 3 : Three small cards opposite a 5-card suit.

First, such assertion implies that, opposite a 5-card suit in partner’s
hand, holding two small cards is better than holding three small
cards as playing a third round in the suit can produce a ruff, which
could not take place with three small cards instead of two.
But a combination such as : A x x x x opposite x x will generate
3 tricks 36 % of the time (missing 6 cards distributed 3 – 3), while
A x x x x opposite x x x will generate three tricks 68 % of the time
(missing 5 cards distributed 3 – 2).
And analysing all possible honor combinations demonstrates that
a third small card does add value in the large majority of cases, not
the reverse. Illustrations :
Opposite 3 small cards, the following 5-card holdings : x x x x x
or 10 x x x x or J x x x x or J 10 x x x or Q x x x x or Q 10 x x x
are not likely to produce one single winning trick (unless it is the
trump suit), but the following holdings will generate tricks which
must be accounted for in points attributed by an appropriate hand
evaluation point count :
Winning tricks = 2.8 pts/trick ------- Optimal points -------
QJxxx xxx 2 ½ tricks = 7 pts 4 HL pts + 1 Fit pt = 5 pts
Q J 10 x x xxx 3 tricks = 8 ½ pts 5 HL pts + 1 Fit pt = 6 pts
Kxxxx xxx 2 ½ tricks = 7 pts 4 HL pts + 1 Fit pt = 5 pts
K 10 x x x x x x 2 ½ tricks = 7 pts 4 ½ HL pts + 1 Fit pt = 5 ½ pts
KJxxx xxx 3 tricks = 8 ½ pts 5 HL pts + 1 Fit pt = 6 pts
K J 10 x x x x x 3 ½ tricks = 10 pts 6 HL pts + 1 Fit pt = 7 pts
KQxxx x x x 3 ½ tricks = 10 pts 6 HL pts + 1 Fit pt = 7 pts
K Q 10 x x x x x 3 ½ tricks = 10 pts 7 HL pts + 1 Fit pt = 8 pts
KQJxx xxx 4 tricks = 11 pts 8 HL pts + 1 Fit pt = 9 pts
K Q J 10 x x x x 4 tricks = 11 pts 9 HL pts + 1 Fit pt = 10 pts

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 OPTIMAL HAND EVALUATION – INDEX / NOTES

The above honor combinations represent 10 holdings where

3 small cards opposite 5 cards add considerable value to the suit –
actually, much greater value than given by the optimal point count.
And 15 other combinations, this time with the Ace, show similar
results and bring the final tally to 25 cases where three small cards
opposite a 5-card suit add value vs only 6 cases where they do not.
The conclusion is clear and unequivocal : the assertion that three
small cards opposite a 5-card suit in partner’s hand is « the worst
possible holding », is totally erroneous and without foundation.

149
INTERMEDIATE/ADVANCED

Accurately assess the value

of your combined hands
Accurate hand evaluation is critically important to reaching the
right contract in bridge. Yet, for over 75 years now, hand evaluation
has essentially relied on Milton Work’s 4-3-2-1 honor point count
— a count known to be seriously flawed and inaccurate. Even in
balanced hands aces are under-valued, queens and jacks are over-
valued when isolated, tens are not factored in, and distribution and
fit are not accounted for.
All in all, this method gives a very unsatisfactory, inaccurate
point count that leads, too often, to the wrong contract.

In this book, the author addresses these flaws by applying corrections

to the 4-3-2-1 honor point count and Goren’s distribution point
count, and introduces revolutionary new findings that lead to
much more accurate hand evaluation. Most importantly, this new
Optimal hand evaluation accurately assesses the value of both
hands combined, the true test of hand evaluation.
Over 100 example hands clearly establish that the Work/Goren
point count is simply erroneous and that various other more
modern point counts do not offer a better alternative.
Reading this book will let you be the judge, and applying the
Optimal point count is sure to substantially improve your bidding.

PATRICK DARRICADES (Canada) learned to play bridge as a

teenager, but quickly became more interested in the theory of the
game rather than in playing it. His studies of the statistical work of
J-R. Vernes (which formed the basis for the Law of Total Tricks) led
to his work on hand evaluation, and the present book.

A N H ONO RS eB O O K FRO M MASTER POIN T PR ESS