This document describes a chess variant called Hex Chess that is played on a board with an hexagonal shape containing 96 equilateral triangles. It summarizes the movements of the different pieces, which generally follow standard chess movements but adapted to the triangular geometry. It also discusses the development of distance formulas specific to this variant to calculate distances traveled by kings, queens, and potentially other pieces. Finally, it outlines some open problems for further developing and analyzing the geometry and rules of Hex Chess.

1. Hex Chess
December 18, 2014
Villanova University
MAT 7290 Geometry
A chess variant on a hexagonal lattice created by Dan Freeman

2. Project Background
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• At this point, we are all aware that geometric systems exist beyond the standard
Euclidean geometry to which we are all very accustomed
• In this class, we have studied non-Euclidean geometries such as hyperbolic,
spherical and taxicab
• In particular, taxicab geometry relates most closely to my semester project
• In addition, a Chinese Checkers geometry has been developed, which, like
taxicab geometry, allows for travel along lines parallel to the x-axis or y-axis,
but, unlike taxicab geometry, allows for travel along lines parallel to the line y =
x (that is, lines that make an angle of 45° with respect to the x-axis or y-axis)
• The addition of diagonal movement results in Chinese Checkers geometry
having a different distance formula than that of taxicab geometry
• The nuances of Chinese Checkers geometry helped inspire the idea for my
project, a chess variant played on an equilateral triangle grid

3. Taxicab Geometry vs.
Chinese Checkers Geometry
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• Given two points P = (x1, y1) and Q = (x2, y2), we have learned that the distance
formula for taxicab geometry is DT(P, Q) = |x2 - x1| + |y2 - y1|
• By contrast, the distance formula for Chinese Checkers geometry is DCC (P, Q)
= max{|x2 - x1|, |y2 - y1|} + (√2 - 1)*min{|x2 - x1|, |y2 - y1|}
• However, Chinese Checkers geometry does not correspond to the geometry of
an equilateral triangle grid because equilateral triangles have 60° angles
between each pair of adjacent edges, not 45° angles
• So I decided to develop a distance formula for a new geometry that I call
Equilateral Triangle geometry. I found that the distance formula for this
geometry is DET (P, Q) = max{|x2 - x1|, |y2 - y1|} + (2 - √3)*min{|x2 - x1|, |y2 - y1|}
• An important consequence of my discovery of this Equilateral Triangle
geometry is that one can develop an infinite number of geometries simply by
changing the angles between the allowed axes of movement
– For example, I could have just as easily derived a distance formula for a geometry
that has a direction of travel 20° from the x-axis

4. History of Triangular Chess Games
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• In 1986, George R. Dekle, Sr. invented chess variants played on an
equilateral triangle grid called Triangular Chess and Tri-Chess
– Both games consist of a hexagonal-shaped board with 96 equlateral triangles as
cells
• Several other chess variants have been developed, including ones
with regular hexagons as cells
• I have not come across a game by the name of Hex Chess so I
decided to give my game that name due to the hexagonal shape of the
board on which my game is played
• While the board of my game has the same size and shape as
Dekle‘sTriangular Chess and Tri-Chess, I started developing my game
before I even knew about Dekle‘s variants and so the movements of
the pieces in my game are completely different

5. Board Specifications
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White Player’s Side
Black Player’s Side
• In Hex Chess, the board has
a hexagonal shape and
contains 96 triangles on which
the pieces may move
• The triangles alternate
between black and white
colors from row to row and
from column to column
• Like chess, Hex Chess is a
2-player game (white and
black) with each player’s
pieces starting on opposite
ends of the board, as shown
on the next slide

6. Row and Column Definitions
White Player’s Side
Black Player’s Side
A
B
C
D
E
F
G
H
I
J L
K
N
M O
8
7
6
5
4
3
2
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• In Hex Chess, there are 15
columns (files) and 8 rows
(ranks)
• The letters A through O
denote the columns, from left
to right, and the numbers 1
through 8 denote rows, from
bottom to top
• The diagram to the right
labels the rows and columns
according to the white
player’s perspective
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7. Piece Starting Positions
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White Player’s Side
Black Player’s Side
King
Queen
Rook
Bishop
Knight
Pawn

8. Pawn Movement
White Player’s Side
Black Player’s Side
• Pawns move vertically forward
one row when not attacking
another piece (denoted by the
blue circle in the diagram to the
right)
• Pawns move diagonally forward
when attacking another piece
(denoted by the red circles)
• Pawns have the option to move
two rows forward but can only do
so on their first move of the game
(denoted by the green circle)
• Pawns may not move backward
• Once a pawn reaches the
opposite end of the board (row 8),
it gets promoted to either a knight,
bishop, rook or queen, as in
traditional chess
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9. Knight Movement
White Player’s Side
Black Player’s Side
• Knights move either two rows and
one column or one row and two
columns, similar to regular chess
• Knights can move to one of at
most 8 different triangles in a
single move, as denoted by the
blue circles in the diagram to the
right
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10. Bishop Movement
White Player’s Side
Black Player’s Side
• Bishops move any number of
triangles diagonally, similar to
regular chess
• Each player starts with a bishop
that moves on the white triangles
and a bishop that moves on the
black triangles
• In the diagram to the right, the
white triangle bishop can move to
any of the triangles with a blue
circle and the black triangle
bishop can move to any of the
triangles with a green circle
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11. Rook Movement
White Player’s Side
Black Player’s Side
• Rooks move any number of
triangles either vertically or
horizontally, similar to regular
chess
• In the diagram to the right, the
rook can move to any of the
triangles with a blue circle
• As in regular chess, the distance
rooks need to travel from one
triangle to another follows the
taxicab metric
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12. King Movement
White Player’s Side
Black Player’s Side• Kings move to any triangle which
shares a vertex with the triangle
they currently occupy
• Due to the triangular geometry, if
near the center of the board, a
king can move to one of 12
triangles, as opposed to only 8
squares in regular chess
• In the diagram to the right, the
white king can move to any of the
12 triangles with a blue circle
• The black king to move to any of
the triangles with a green circle
• Even though the black pawn is
adjacent to the black king, the
king’s movement allows the black
king to move “beyond” the pawn
to the white triangle to the left
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13. Queen Movement
White Player’s Side
Black Player’s Side
• Queens have the combination of
bishop movement, rook
movement and king movement
• In the diagram to the right, the
queen can move to any of the 36
triangles with a blue circle
• Unlike regular chess, king and
queen movement do not follow the
Chebyshev metric, which is simply
DC(P, Q) = max{|x2 - x1|, |y2 - y1|}
– They follow a completely different
metric, as explained on the next
slide
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14. King and Queen Distance Formula
• The only formula I could come up with for determining king and queen distance
involves two main cases each of which contains two sub-cases for a total of four
cases
• The stepwise formula is contained in the table below:
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Formula Case Sub-case
ceiling{(|x2 - x1| + |y2 - y1|) / 2}
x1 is even
y2 ≥ y1
floor{(|x2 - x1| + |y2 - y1|) / 2} y2 < y1
floor{(|x2 - x1| + |y2 - y1|) / 2}
x1 is odd
y2 ≥ y1
ceiling{(|x2 - x1| + |y2 - y1|) / 2} y2 < y1

15. Problems for Potential Future Research
• Develop a better, more concise formula for the king and queen distance metric in
Hex Chess
• Develop a distance metric for bishops in Hex Chess
• Develop a distance metric for knights in Hex Chess (seems very difficult)
• Create a chess variant played on a board with hexagonal cells and investigate the
geometry associated therein
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16. Sources Cited
• http://www.m-hikari.com/ijcms-password2009/1-4-2009/colakogluIJCMS1-4-
2009.pdf
• http://en.wikipedia.org/wiki/Triangular_Chess#Tri-Chess
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