Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.

tex page 1

Polychromatic chess
Gabriel López Garza

Abstract. This article explores how to extend the geometry and rules of classic chess to hexag-
onal boards using sequences of colors which alow new movements for the pieces. By dividing
the plane with regular hexagonal mosaics and coloring the resulting map with three and four
colors, new geometric and topological configurations do appear which suggest a natural way
of generalizing the game of chess. Behind these configurations of hexagonal boards lie two
old and beautiful mathematical problems: dividing the plane with a regular tessellation and the
four colors problem. Here we study the relationships that appear between these fundamental
problems and the idea of generalizing quadrangular chess to boards with different tessellations
and colorings.

1. INTRODUCTION Is it possible to enrich the chess game? Is it possible to find

a different tiling of the chessboard with higher degrees of freedom for the pieces? In
answering this questions two old and beautiful mathematical problems are involved:
the problem of the regular tiling of the plane and the problem of coloring a map not
allowing two neighbors to have the same color. In this article we will explore how
to extend the possibilities of a chessboard and thus create a new rules of the game
departing from the two famous problems.
The problem of tiling the plane with regular polygons may be stated formally as
Theorem 1 (Regular tessellation). The only possible periods for a rotational symme-
try of a lattice are 2,3,4,6.
On the other hand, the problem of coloring a map has a formal formulation as the
following
Theorem 2 (Four color theorem). Every map can be colored with at most four colors
in such a way that neighboring countries are colored differently.
The problem of a regular tessellation is fundamental in geometry. An excellent and
classic approach to tessellation is in the book of Coxeter [3] where the definitions of
the terms involved and proof of theorem 1 are in Chapter 4. For the reading of the
history of the four color theorem, it is advisable the delightful book of Robin Wilson
[7]. The proof of theorem 2, as is well known, opened a new world of possibilities for
proving mathematical results with the assistance of computers. The arguments of the
proof of theorem 2 are complex and lengthy, but, contrarily, a light and folk form of
the statement of the theorem 1 may be proved very easily with elementary properties
of triangles. The folk version of theorem 1 can be formulated as follows:
Dividing the plane with non-overlapping regular polygons of one kind is possible only
by using equilateral triangles or squares or hexagons1 .
If we denote with n the number of sides of a regular polygon and with m the number
of polygons in each vertex, the classic chess board is constructed with n = 4, m = 4.
On the other hand, in the chessboard, two is the minimum number of colors required to
avoid neighbor squares to have the same color. In this paper, we explore the possibility
of the solution n = 6, m = 3, i.e., covering the plane with regular hexagons with
1A simple proof of this statement is in the appendix.

January 2014] CHROMATIC CHESS 1

Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 2

three neighbors in each vertex. In doing this, two novel possibilities arise, coloring
with three colors (minimum number of colors for this mapping), and coloring with
four colors (minimum number of colors for every existing or thinkable map); giving
rise to two different sets of boards. Accordingly, with the idea of keeping as possible
the logic of the classical chess game, we will obtain a new set of pieces and rules
for hexagonal chess. In this article, we deal with some of the problems that arise in
designing new games. Some basic and interesting problems are included in section 4.
A full set of proposed new rules are included in the Appendix.

2. BACKGROUND As mentioned in the previous section it is possible to cover the

plane with equilateral triangles, squares, and hexagons. Classic chess presupposes the
division of the plane into squares where the pieces move through the sides and the ver-
tices of the squares in fixed directions. Historically some players considered other pos-
sible divisions of the plane different from squares and constructed boards and pieces
that move in the same way as in chess, but on chessboards with different geometries.
Indeed, chessboards formed with triangles are shown for the interested reader in [2,
Ch. 23] 2 , among which we can mention the inventions of Dekle, Berard, and Loyd
among others, as well as triangular boards for three players, for example in [6] and
references therein. Boards formed with hexagons for two and more players have been
constructed too, [2]. The most famous and played, in the author opinion, are those of
Glinski, Reid, and McCooey and Honeycutt whose apparatus descriptions and rules
of the games can be found in [2, Ch. 22] and at http://www.chessvariants.com/
hexagonal.dir/hexagonal.html. 3 The chessboards divided with hexagons4 rec-
ognized by International Hexagonal Chess Federation (IHCF) use only three colors,
and utilize the same chess pieces that the traditional chess game, so that only a new
bishop is required for the new color. The hexagonal chess players in the IHCF do not
define pieces beyond the scope of traditional chess. Nonetheless, adding colors to the
chessboards increases the possibilities of movement, and thus, is natural to create new
chess pieces for the enriched board, as shown in this article.

3. COLORED CHESS BOARDS In the history of chess, the color in the boards did
not appear in the beginning. In the old chaturanga game [5], there is only one piece
equivalent to the bishop so that a checkered board with two colors is not required.
Clearly, in the hexagonal chess board, the four color theorem is trivially satisfied with
only three colors, for instance, in Glinski’s chess. A third color is required to avoid
having two neighbor hexagons with the same color. Hence the Glinski’s game has
three bishops, and this is the main difference with the traditional chess concerning the
number of pieces. However, a natural way to move the bishops through hexagons of
the same color leads to the authors of Hexagonal chess games to establish that bishops
2 The book of John and Sue Beasley cite B is at https://www.jsbeasley.co.uk.
3 On this page you can also find games and simulators of some variants, in addition to dates of champi-
onships, games’ databases and problems.
4 Moreover, chess games with hexagonal boards or with truncated triangular boards are known in the prior

art. Chessgames using flat, stackable pieces are also known. Chess games with a board with spaces that are
taken out of play during the course of the game are also known in the prior art.
Several chess sets use a hexagonal board or hexagonal spaces to create the board. Examples of strictly
hexagonal boards are US. Pat. Nos. 3,920,247 (Jenkins), 3,964,747 (Balmforth) and 5,582,410 (Hunt) showing
hexagonal boards with seven spaces per side, 6,070,871 (Wilson) with eight spaces per side, and 4,580,787
(Baker) with nine spaces per side. Examples of boards With six sides where the sides alternate in length
(truncated triangles) are 3,744,797 (Hopkins) with alternating sides of five and ten spaces, 3,778,065 (Hale)
with alternating sides of six and eight spaces, 5,158,302 (ReWega) with alternating sides of four and eight
spaces, and 6,170,826 (Jones) with alternating sides of eight and nine spaces. (Schroeder et al) 2005/0179203
A1.

2
c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121
Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 3

Figure (1). The figure shows the possible movements of a bishop-like piece labeled with light-green. The stars
show all the possible moves for a piece in that specific position. The reader may notice the labels in the board,
the mentioned bishop is sited in E6 according to this labeling.

must jump, since hexagons of the same color are isolated. In Figure 1 is shown an
example of how a bishop may move in a three-color chess board.
However, hexagonal chess authors did not dare to go beyond creating new pieces
and exploring the new topological properties of a colored board that are apparent.
For example, why not to create a piece able to move through three consecutive and
different colors? In figure 1 is clear that one natural movement is from vertex to vertex
of the board, but such a piece must be allowed to move through three colors sequences.
Also, what about moving through only two colors? Moreover, what about inventing a
queen like piece moving as all these pieces together? What about a piece like the knight
to attack such a powerful new queen? In short, what we do in this article is precisely
to dare to explore new possibilities of pieces in a richer topology that emerged by
coloring a hexagonal tile board. We will focus on the four-color chess board, but its
rules can be quickly reduced to a three-color board as explained in the Appendix. We
begin by introducing new possible pieces.

New pieces for richer topology The most basic rule of movement of chess pieces
can be described by saying that the pieces move through sides of the squares (rooks)
or move through vertex (bishops) or either (queen and king), or combination of sides
and vertex (knights and pawns). This geometric point of view of the movements has
a correlative, let us say, topological standpoint, given the coloring of the chess board
some pieces move through one color (bishops), two colors (rooks), both possibilities
(queen and king), and alternating one color or two color movements (all the  other

2
pieces). From this point of view chess is redundant. Let me explain, there are =
1
 
2
2, ways of choosing pieces of one color (bishops), but only = 1 form of choosing
2
pieces of two colors (rooks). So if one conjectural chess inventor wanted to keep chess
as economical as possible, traditional chess would have only one rook! The idea of
duplicating rooks is to maintain certain symmetry, and (or) can be chosen to keep a
specific number of squares in each side (eight was somehow a sacred number in ancient
India where allegedly chess was invented). Since we do not have yet any preconception

January 2014] CHROMATIC CHESS 3

Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 4

of the number of hexagons in each side of the board that we are inventing right now,
we keep this number free by the moment, and we leave open the possibility of the
existence of minimal or redundant boards. We call a minimal chess set a set which
includes only one piece of any type given the combinatorial possibilities of choosing
from one, two, and et cetera colors from the available three or four. Also, we call
symmetric chess set a set which allows duplicating pieces for the sake of symmetry.
Consequently, in a hexagonal board, we obtain two sets of different kind of pieces,
one for the three colors, and other for the four color hexagonal boards. We show the
different minimal sets in the following table, keep in mind that the two colors set, is
equivalent to the classic chess set.

Table (1). Two, three and four colors minimal boards. The pieces named “1-color” correspond to bishops
in the old chess, the “2-colors” pieces correspond to rooks, the “1&2” correspond to the queen, the “anti-2”
pieces are equivalent to knights.

Piece old chess three color set four color set

     
2 3 4
1-color =2 =3 =4
1 1 1
2 3 4
2-colors =1 =3 =6
2 2 2
3 4
3-colors none =1 =4
3 3
4
4-colors none none =1
4
1&2-colors 1(Queen) none none
1&3-colors none 1 none
1&4-colors none none 1
anti-2 1 (knight) none none
anti-3 none 1 none
anti-4 none none 1
aim 1 (King) 1 1
Total 6 10 18

In Table 1 the pieces which are equivalent to bishops of the square chessboard are
the 1-color. The equivalent to rooks are the 2-colors and 3-colors. The aim piece cor-
respond to the king. We will clarify the movements of the pieces after introducing the
hexagonal chessboard in the following section. The equivalent to a queen are the pieces
1&2, 1&3, and 1&4-colors, which include the possibility of moving as a bishop (i. e.
as a 1-color piece) as well as a rook (2-colors piece), and et cetera. Clearly, the queen,
the most powerful piece in each board is the 1&2, 1&3 and the 1&4, respectively. The
equivalent to the knights are the anti-2, anti-3 and anti-4 which can be defined as the
pieces which may attack the 1&2, 1&3 and the 1&4 pieces without being attacked.
Note that the conjectural minimal two colors chess would have a six times six board
since it has only six pieces and not an 8 × 8 board as traditional chess. Observe, as
a curiosity, that the last row of table 1 in the fourth column, shows that a minimal
chromatic chessboard must have 18 pieces, a mystic number already mentioned.
The symmetric chess set of two colors is the actual 8 × 8 chessboard. The idea of
the symmetric chess is to keep the equilibrium of forces as possible and in an economic

4
c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121
Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 5

way: in the classical chess, rooks and knights are duplicated, but do not queens. With
the same scheme in mind we introduce the pieces in the following Table 2.

Table (2). Two, three and four colors symmetric boards

Piece old chess three color set four color set

1-color 2 (bishop) 3 4
2-colors 2 (rook) 6 12
3-colors none 2 8
4-colors none none 2
1&2-colors 1 (queen) none none
1&3-colors none 1 none
1&4-colors none none 1
anti-2 2 (knight) none none
anti-3 none 2 none
anti-4 none none 2
aim 1 (king) 1 1
Total 8 15 30

New boards Now with the considerations of the possible number of pieces for col-
ored boards in mind we have to choose the number of hexagons in each side of the
hexagonal board. For a four-color board, even though 18 is also a sacred number (ash-
tadash) in ancient India5 , the idea of having a hexagonal board with 18 hexagons by
side (as in the symmetric board of Table 2) is frankly overwhelming. Nevertheless if
we want to be objective, we may try to introduce some kind of measure of the boards
relative to the pieces’ movements possibilities. For instance we can compare the size
of the board compared with the number of hexagons attacked by the most powerful
piece (the queen-like piece) placed in the center of an empty board. For the 8 × 8
square board, this number is q = 27/64 ≈ 0.4218. For the 1&3-colors piece, the
most powerful piece in a hexagonal board of 18 hexagons by side, this number would
be q = 150/991 ≈ 0.15. So that in this sense, the 18 hexagons by sideboard may be
considered big. A finer measure of relative size may be the maximum number of the
most powerful pieces which fit the board without attacking each other. For the 8 × 8
square board this number is 8, and Gauss [1] showed that there are 92 variants6 of this
kind of distributions in the board. I do not intend to calculate this number for the 18
by side hexagonal board, but the reader may try if she wish.
Given the consideration of the last paragraph we do not try to deal with boards of 18
hexagons by side nor the 30 of Table 2. But there exist a possibility of having almost
18 pieces if we allow to place the pieces in the second row in the initial setting. For
instance by maintaining 8 hexagons in the first row we would have 9 in the second,
and we will have a total of 17 pieces without including the pawns. Which one may be
sacrificed? But wait a minute! Placing mayor pieces in a second row? Does it make
5 The Mahabharata is divided into 18 parvas or books while the sacred text, Bhagwat Gita is also divided

into 18 chapters, moreover Shri Krishnas caste, Yadava, also had 18 clans.
6 May be a better measure for the relative size of a board, could be dividing the maximum number number

of the most powerful pieces that fit the board without attacking each other, divided by the number of possible
variants of these configurations, for instance for the square board this number is 8/92.

January 2014] CHROMATIC CHESS 5

Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 6

Figure (2). Eight by side hexagonal board with four colors and an algebraic label of the board.

sense? To answer this questions first we need a method to label each hexagon as shown
in Figure 1. For a four-color chess, in Figure 2, we introduce a so-called algebraic
labeling of the board. The algebraic labeling for the three-colors board is detailed in
the Appendix. We must warn the reader that many different boards with four colors
may be obtained (how many?). To avoid naming specific colors, we call them c1, c2,
c3, c4, departing from this point.
After labeling the entire board we are in the position of showing some of the most
conspicuous pieces, the complete set is depicted in Figure 6, and the full set of rules is
in the Appendix. First we emphasize that in considering our democratic times the King
is named the aim piece; the keen, the 4-color piece, and the knight the anti-4 piece.
The 4-colors piece. This piece is equivalent to the queen in regular chess, since q as
defined in section 3 is q = 72/169 ≈ 0.4260. The 4-colors piece may move following
the sequences c1, c2, c3, c4 or c2, c3 ,c4, c1, or c3, c4, c1, c2 or c4, c1, c2, c3, and
their inverses of each one, and by keeping a constant edge or vertex direction, in Figure
3 this movements are shown with the symbols: , , 4, and . This piece also may
move as a 2-colors (castle) in the color in which is placed and the non-neighboring
color in the four color sequence in which is moving. Finally, this piece may also move
as a 1-color (bishop) piece in the color in which the piece is placed and not included in
the other options. Note that in adding one 2-colors direction and one 1-color direction
we keep the symmetry of the 4-colors piece movements. When moving this piece, the
player must announce in advance what kind of possibilities would be using: 4-colors
or 2-colors or 1-color, to avoid misunderstandings. The reader must notice that we do
not allow the 4-color to move as 3-color piece. The reader can calculate the number q
for such a piece if we allow the four piece to move in such way.
The anti4 piece. The nonlinear move of the knight is recovered with this piece. This

6
c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121
Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 7

Figure (3). Example of a 4-colors piece movements.

Figure (4). Example of the anti4 piece movements for the four colors board.

piece may attack the queen without being attacked, regardless of the existence of pieces
in between. An example of this piece possible movements is in Figure 4.
Last but not least, the soul of the game pieces, the pawns. These humble pieces
reconfigure the board geometry only by their movements. In the proposed rules these
pieces have more complex movements than in the traditional chess, but anyhow they
capture the essence of the popular game.
Pawns. In Figure 5 we show team 1 (white) and team 2 (black) pawns movements.
The black pawn sited in G6 may move through G5 and F 5, i. e., the  symbol in

January 2014] CHROMATIC CHESS 7

Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 8

Figure (5). Example of black team and white team pawns movements. The black pawn in G6 can move to
the black squares and can capture in cross marked hexagons. The white pawn has the possibility of more
movements since it is located in its initial position and it is its first move. The white pawn can capture pieces
in the diamond marked hexagons.

Figure 5, and may capture pieces in H5 and E5 (× symbol). The white pawn sited in
B3 may move to C4 or alternatively to D5 ( symbol) and may capture in A4, D4 or
C5 ( symbol). The general rules of pawns movements, including the en passant rule
are fully described in the Appendix.
A big problem for any chess game is to establish the initial position of the pieces.
It may be taken into account that at the beginning of the game all pieces must be
protected by at least one piece. The symbols and the initial positions for the pieces
are in Figure 6. We recall that the most basic movements are through edges of the
hexagons and vertex each in six different directions. As can be seen in Figure 3 the
4-colors piece placed in H8 may be attacked by a 2-colors piece (green and white) set
in the column H without being attacked. However, this singularity may be considered
as a minor problem since in allowing the queen to move as 2-colors may produce
a monstrous piece, as we already mentioned. As a remark notice that after all, 18
(ashtadash) different pieces are obtained for the four color board! May be the Shri
Krishna’s caste and Yadava clans would recreate their Mahabharata battles in our four-
color hexagonal chess.

4. PROBLEMS Finally, in this section the problems that appeared throughout the
article are summarized, some other issues are also included.

Basic chess checkmates For the following problems the black team has in the board
only the king, and the white team has the king and only the pieces established in each
item.
1. In the four-colors chess are there four bishops (1-color pieces). Is it possible to
checkmate a lonely king of the adversary with the four bishops only?
2. What is the minimum number of 2-color pieces (are there six) to checkmate a
lonely king?

8
c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121
Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 9

Figure (6). Pieces and initial positions for the four colors board

3. Is it possible to checkmate with three bishops and the knight? with two bishops
and the knight?
4. With the black king in the center of the board H8, what is the minimum number
of moves to checkmate with the white king in A1 and the queen in A2?
5. What is the minimum number of pawns to checkmate a king in G15?
6. Find different stalemate positions with pawns alone, with bishops alone, with
2-color, 3-color and 4-color pieces. There exists such positions?
7. Find a sequence of moves departing from the initial position setting, such that
that sequence in minimal. Recall that in the traditional chess there is a three-
movement sequence.
8. Is it possible in any initial position setting to checkmate lonely king with only a
queen and king?

Basic combinator questions

1. What is maximum number of queens that fit in the four-color, and in the three-
color boards without attacking each other? How many different positions are
there in each case?
2. Can a knight sited in its initial position move in such a way that can be positioned
in each of the hexagons of the entire board? If so, what is the minimum number
of movements to accomplish such task?
3. What is the q number defined in section 3 for a piece which moves as all 1-color,
2-color, 3-color, and 4-color pieces sited in H8?
4. Same as the last question for the three-color board. What is the q number defined
in section 3 for a piece which moves as all 1-color, 2-color,and 3-color pieces
sited in H8? From the group of symmetries standpoint are there different four-
colors boards. How many are there?

Basic design of the game questions and possible flaws.

January 2014] CHROMATIC CHESS 9

Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 10

Figure (7). Four-color chromatic chess designed by the author. In the figure, the pawns are of four different
colors suitable for four players, two in each team.

1. In playing one king vs. a king and a pawn, is it possible to stalemate? in which
cases is possible to crown the pawn?
2. It is possible to construct an square board of 7 × 7 filled with hexagons. Cal-
culate the q number defined in section 3 for 4-colors piece. Can this number be
considered big with respect to the hexagonal board?
3. It is possible to place the king in the initial position in a vertex of the hexagonal
board as in the games of Glinski, Reid, and McCooey and Honeycutt. Is it pos-
sible to find a symmetric initial configuration such that any piece is protected by
at least one piece since the beginning of the game? How many pawns are needed
for such a game?
4. For reasons never explained by the author, this paper includes only one of the
possibles four-color possible boards (Figure 2 and subsequent figures). Can the
reader find another better board in considering the number of pieces, a consistent
initial position, and the simplest board to visualize 4-colors piece moves?
5. Compare the complexity of the chromatic chess games with the traditional chess
in terms of information theory complexity. Are chromatic chess games playable
for humans in finite time? Just try!

5. ADDITIONAL MATERIAL Two hexagonal boards of three and four colors with
algebraic labels and the pieces as shown in Figures 2 and 6, can be found at au-
thor’s page —@—– . The reader may reproduce this copyrighted material for non-
commercial, personal use at will. Paper sizes A1 and A2 are strongly recommended.
A four-colors board is shown in Figure 7.

REFERENCES

1. Averbakh, Y., Beilin, M. Journey to the Chess Kindom. ISBN: 978-83-934656-6-8.

2. Beasley, J. & Beasley S. The Classified Encyclopedia of Chess Variants. D. B. Pritchard, ISBN 978-0-
9555168-0-1, 2007.
3. Coxeter, H. S. M., Introduction To Geometry Second Edition. John Wiley & Sons, Inc.
4. FIDE, Laws of chess. https://www.fide.com/FIDE/handbook/LawsOfChess.pdf.
5. Murray, H. J. R., A short history of chess. Ishi Press International ISBN 4-87187-754-X.
6. Treugut et al. Chess game for three people. US Patent 3 963 242, june 15 1976.

10
c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121
Mathematical Assoc. of America American Mathematical Monthly 121:1 January 6, 2020 11:06 a.m. Chromatic˙Chessmt˙NN.tex page 11

7. Robin Wilson, Four Colors suffice, how the map problem was solved. Princeton University Press.

January 2014] CHROMATIC CHESS 11