Math Dept Colloquium, West Chester University

Mathematical Card Tricks

Colm Mulcahy
www.cardcolm.org
@CardColm
Spelman College,
Atlanta, Georgia

Thu, 17 Nov 2022

Abstract
Amaze and amuse your family and friends armed with just a deck
of cards and a little insider knowledge.

Mathematics underpins numerous classic amusements with cards,

from forcing to prediction effects, and many such tricks have been
written about for general audiences by popularizers such as Martin
Gardner.

The mathematics involved ranges from simple ”card counting”

(basic arithmetic) to parity principles, to surprising shuffling
fundamentals (Gilbreath and Faro) discovered in the second half of
the last century.

We’ll discuss several original and totally different principles

discovered since 2000, as well as how to present them as
entertainments in ways that leave audiences (even students of
mathematics) baffled as to how mathematics could be involved.
Consider a Regular Deck of 52 Playing Cards
Note that 52 = 26 + 26 = 13 + 13 + 13 + 13.

A deck has 26 black (♣ & ♠) and 26 red (♥ & ♦) cards.

There are 13 of each of the 4 suits:

♣ (Clubs), ♥ (Hearts), ♠ (Spades), ♦ (Diamonds)

A♣ (Ace), 2♣,. . . , 10♣, J♣ (Jack), Q♣ (Queen), K♣ (King),

A♥ (Ace), 2♥,. . . , 10♥, J♥ (Jack), Q♥ (Queen), K♥ (King),

A♠ (Ace), 2♠,. . . , 10♠, J♠ (Jack), Q♠ (Queen), K♠ (King),

A♦ (Ace), 2♦,. . . , 10♦, J♦ (Jack), Q♦ (Queen), K♦ (King)

What if you don’t like card tricks?

Spend the next 45 minutes productively. List all

possible rearrangements of:
A♣, 2♣, . . . , K♣,
A♥, 2♥,. . . , K♥,
A♠, 2♠,. . . , K♠,
A♦, 2♦, . . . , K♦.

What if you don’t like cards at all?

List all possible rearrangements of the 52 white

notes on a piano, using no note twice.
Little Fibs
Shuffle the deck well, and have one card each
selected by two spectators.

They remember their cards (value and suit), share

the results with each other, and tell you the sum of
the chosen card values.

You soon announce what each individual card is!

Secret Number 1:

When several numbers are added up, each one

can be determined from the sum.
Secret Number 2:

Totally free choices of cards are offered, but

only from a controlled small part of the deck.

The possibilities are narrowed down by having six

key cards at the top of the deck at the start, in any
order, and keeping them there throughout some
fair-looking shuffles.

The mathematics is about numbers (card values).

Secret Number 3:
You’ve memorized the suits of the top half
dozen key cards.

We use the Fibonacci numbers: start with 1, 2; add

to get the next one. Repeat.
1 + 2 = 3,

2 + 3 = 5,

3 + 5 = 8,

5 + 8 = 13, and so on.

The list of Fibonacci numbers continues forever:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . .

For magic with playing cards, we focus on the first

six of these, namely the Little Fibs, i.e., 1, 2, 3, 5,
8, and 13, agreeing that 1 = Ace and 13 = King.
Cconsider particular cards with those values, e.g.,
A♣, 2♥, 3♠, 5♦, 8♣, K♥ (CHaSeD order).

If any two cards are selected from these, they can

be determined from the sum of their values, because
of the unaddition property:
 Possible totals arise in only one way.

It’s easy to break up any sum into the two Fibs it’s
made up from: just peel off the largest possible Fib,
what’s left is the other one. E.g.,
8 = 5 + 3,
10 = 8 + 2,
14 = 13 + 1,
18 = 13 + 5.
Other lists of numbers work: the Lucas sequence 2,
1, 3, 4, 7, 11, 18, ..., is a kind of generalized
Fibonacci sequence. If we omit the 2, it too has the
desired “unaddition” property.
We don’t need generalized Fibonacci sequences.

The numbers 1, 2, 4, 6, 10 work.

So do 1, 2, 5, 7, 13.

And?

Can we find a set of five or six interesting numbers

such that all subsets of any size generate unique
sums?
The Three Scoop Miracle

Hand out the deck for shuffling. A spectator is

asked to call out her favorite ice-cream flavour; let’s
suppose she says, “Chocolate.”

Take the cards back, and take off about a quarter of

the deck. Mix them further until told when to stop.

Deal cards, one for each letter of “chocolate,”

before dropping the rest on top as a topping.

This spelling/topping routine is repeated twice

more—so three times total.
The Three Scoop Miracle
Emphasize how random the dealing was, since the
cards were shuffled and you had no control over the
named ice-cream flavour.

Have the spectator press down hard on the final top

card, asking her to magically turn it into a specfic
card, say the 4♠. When that card is turned over it
is found to be the desired card.

Published 21 Oct 2004 at MAA.org as ”Low Down

Triple Dealing” as the first Card Colm, dedicated to
Martin on the occasion of his 90th birthday. See the
“Little Fibs” video on Numberphile.
Low Down Triple Dealing
The key move here is a reversed transfer of a fixed
number of cards in a packet—at least half—from
top to bottom, done three times total.

The dealing out of k cards from a packet that runs

{1, 2, . . . , k − 1, k, k + 1, k + 2, . . . , n − 1, n} from
the top down, and then dropping the rest on top as
a unit, yields the rearranged packet

{k + 1, k + 2, . . . , n − 1, n, k, k − 1, . . . , 2, 1}.

True, but it hides what is really going on!

Low Down Triple Dealing

When k ≥ n2 , doing this three times brings the

original bottom card(s) to the top.

Given a flavour of length k and a number n ≤ 2k,

the packet of size n breaks symmetrically into three
pieces T , M, B of sizes n − k, 2k − n, n − k, such
that the count-out-and-transfer operation (of k
cards each time) is

T , M, B → B, M, T ,

where the bar indicates reversal within that piece.

Low Down Triple Dealing Generalized?

Using this approach, the Bottom to Top (with three

moves) property can be proved. Actually. . .

The Bottom to Top Property is only 75% of the

story. Here’s the real scoop:

The Period 4 Principle If four reversed

transfers of k cards are done to a packet of
size n, where k ≥ n2 , then every card in the
packet is returned to its original position.
Proof without words
Poker Hand Control

Hand out the deck to a spectator for shuffling, the

more jumbled it is the better. Deal out two piles of
five cards in an alternating way, remarking, “Let’s
deal out two poker hands. In a moment I’m going
to let you decide who gets which cards. I just want
to take a peek at what we have here.”

Pick up one hand of cards, glance at the faces, then

pick up the other hand, and look at those faces,
tucking one hand of cards behind the other. Mutter
vague (true!) things about what you see, such as,
“Interesting, a pair of 4s and a Jack, Queen and
King.” Then turn the ten card packet face down.
Poker Hand Control
Declare, “As I said, you get to decide who gets
which cards.”

Holding the cards face down, take the top two off
and say, “One of these is your hole card, the other is
mine. Which do you want, top or bottom?”

Whichever card is claimed, place it face down in

front of the spectator, and place the other face
down in front of yourself. Say something like, “Did I
mention you’d get the pair of 4s? Maybe I shouldn’t
have said that, I wouldn’t want to influence your
decisions in any way.”
Poker Hand Control

Hold up the next two cards from the packet, and

ask, “Which do you want, top or bottom? The
other goes underneath.” Whichever card is claimed,
add it to the spectator’s face-down pile and tuck
the other one underneath the packet in your hand!
There has been a subtle change in procedure here,
and it is repeated five more times, by which time
the spectator may have forgotten the different
nature of the first choice offered. Once more, hold
up the next two cards from the packet, and ask,
“Which do you want, top or bottom? The other
goes underneath.”
Poker Hand Control

Act accordingly. The spectator now has three cards

in her pile.

Then say, “You also get to pick which cards I get; it

couldn’t be more fair.” Hold up the next two cards
from the packet, and ask, “Which one do I get, top
or bottom? The other goes underneath.”

Whichever card is indicated, add it to your pile and

tuck the other one underneath the shrinking packet
in your hand. Do the same thing one more time.
Poker Hand Control

At this stage, you have four cards in your hand and

each pile on the table has three cards in it. Say,
“Back to you,” as you once more offer the spectator
one of the top two cards, placing the other
underneath. Then say, “One more for me,” as the
spectator selects a fourth card for your pile from the
top two of the three left in the packet in your hand.

Tuck the other underneath the sole remaining card

and casually say, “One final one for each of us,”
putting the top one on the spectator’s pile and the
other one on yours.
Poker Hand Control

Now have the spectator pick up her five cards and

look at their faces, as you do likewise with yours.

Ask, “Did you get those two 4s like I predicted?”

Have her display her hand face up, she will indeed
have the cards mentioned.

Congratulate her, and add, “Unfortunately, you gave

me a pair of 9s, so I guess I win this time!”

Throw your cards down for all to see.

Repeatable Poker Hand Control

Similar results can be obtained with the next ten

cards from the deck, and in our experience the
spectator is usually happy to give it another try.

If you wish to be nice, you can let the spectator win

this time, and even predict that outcome.

It’s quite baffling, you seemingly controlling two

poker hands with ten random cards, while the
spectator makes almost all of the decisions.
Explanation Part I: Birthday Card Matches

Two unrelated concepts make this effect possible.

First, The Birthday Card Match Principle:

Given ten random cards from a regular 52-card

deck, then it’s very likely (98%) that there are at
least two cards of the same value among them.

This says that most of the time, there will be at

least a pair (in the poker sense) somewhere in the
two hands first dealt out.

Why?
The inevitability of birthday coincidences
With over 60 randomly selected people, the chances
of a birthday match are 99.4%, though you can
handpick 366 people with different birthdays!
With just 23 randomly selected people, the chances
of a birthday match are a little over 50%.
(“The Birthday Paradox”)
The key to estimating such probabilities is to turn
things around, and focus on the chances of there
being no match, noting that

Prob(≥ one match) = 1−Prob(no match).

Poker with Ten Random Cards

If k cards are picked at random, then since there are

four cards of each value, the chance of getting at
least one matching value (i.e., a pair or better) is:

52 48
1 − 52 × 51 × 44 52−4k+4
50 × . . . × 52−k+1 .

For 5 cards, this comes out to be about 50%. For 8

or 9 cards, it’s about 89% or 95%, respectively.

For 10 cards, it’s about 98%.

Only 2% of the time will you be unlucky!

Explanation Part II: Bill Simon’s contribution (1964)

It’s easy to give the illusion of free choices

while really controlling exactly how to split a
packet of 8 cards into two piles of 4.
In fact, you retain control of the division in one key
sense: the bottom 4 cards end up in the second pile.

E.g., if you start with 4 red cards of top of 4 blacks,

the piles maintain colour separation, with the reds
in the first pile and the blacks in the second pile.

For poker purposes, it’s winning and losing cards

that are controlled, not reds and blacks.
The inevitability of poker hand control

Given 10 random cards, assuming that there is at

least one matching pair, the poker hand control is
based on your making sure that the “winning cards”
are in the bottom four, with as little suspicion-
arousing rearrangement as possible.

Those are positions seven to ten, from the top.

The “losing cards” are in positions three to six.

The top two cards must not impact which of you

wins (i.e., they aren’t “deal-breakers”).
Chapter 10 (Mathematical Card Tricks) of Martin Gardner’s “The
Scientific American Book of Mathematical Puzzles and Diversions”
(originally 1959, later republished as “Hexaflexagons and Other
Mathematical Diversions”) opens with this cautionary exchange
and commentary, before moving on to an expository tour de force.

Somerset Maugham’s short story “Mr Know-All” contains the

following dialogue:
“Do you like card tricks?”
“No, I hate card tricks.””
“Well, I’ll just show you this one.””

After the third trick, the victim finds an excuse to leave the room.
His reaction is understandable. Most card magic is a crashing bore
unless it is performed by skillful professionals. There are, however,
some “self-working” card tricks that are interesting from a
mathematical standpoint.
Cool, Colm & Collected

Mathematical Card Magic: Fifty-Two New Effects

AK Peters/CRC Press, 2013. Full colour, 380 pages.
Endorsed by Max Maven, Ron Graham, Persi Diaconis, Art
Benjamin & Lennart Green.
13 main chapters, each with 4 effects. Largely original material.
Made that list of rearrangements of 52 items?

Checked it twice? It should have 52! items on it, i.e.,

80658175170943878571660636856403766975289505440883277824000000000000.

That’s approximately

8 × 1067

This number is larger than the current estimates of the total

number of elementary particles in our galaxy.

That’s a very big deal indeed.

Imagine two seeds released into the air from the top
of a windy mountain on different days.

What are the chances that the seeds end up in the

exact same place?

It’s very very very very small.

But it’s much much much much bigger than the

probability that two well-shuffled decks of cards are
in the same order.
Let’s wrap it up (from Cut-the-Knot site)

Wrap the gold cube completely with the blue paper!

All cutting and folding must be along grid lines.

The paper must remain in one piece.