Matrix Applications in Chess
Douglas Steiner
November 18, 2009
Abstract
This report illustrates basic applications of linear algebra for solving simple
endgame positions in chess.
1
Introduction
The game of chess has attracted mathematicians for generations. With its
geometric board and vector like piece movement, the proposition that chess
can be solved mathematically has caused centuries of man hours to be spent
searching for the elusive formula which will reduce chess to a precise formula
which if followed, will inevitably result in victory. Deriving this formula by
hand is generally accepted as impossible, but it is hoped (by some) or even
believed (by a small but vocal minority) that the computer, pointed in the right
direction, will one day do just that: derive the formula that will solve chess. The
use of linear algebra to achieve this end, specifically in the endgame of chess, is
the subject of this paper.
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2.1
Basic Matrix applications in chess: positions,
pieces, and movements
Standard Chess notation
Figure 1: Starting position for chess
Playing chess beyond the occasional game with Grandpa will eventually expose the player to the concept of chess notation. The necessity of chess notation
arose from the desire to analyze why one player lost or another won (or failed to
win what he thought was a won position) a particular game. Unlike many other
games involving piece movement and strategy, chess lends itself well to endless
post-mortem analysis, arising from its square board of 64 alternating black and
white squares and its rigid rules for the movement of the pieces. There are
literally millions of recorded games played by the masters (and grandmasters)
of the sport, available for perusal in printed and electronic databases. The need
to record these moves systematically eventually gave way to what is know as
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algebraic notation. This term algebraic notation is misleading, as the system
does not use algebraic formulas to describe movements but rather alphanumeric
designations which appear to be mathematical equations.
Algebraic notation is can be understood this way: the 8x8 chess board is
numbered and lettered from the perspective of the player with the white pieces,
with the letters from a-h flowing from left to right across the horizontal axis and
numbers from 1-8 proceeding from the bottom to the top of the vertical axis,
as shown below:
Figure 2: chess board with rank and column labels
The chess pieces themselves are assigned a letter designation as follows:
Figure 3: piece notation
Although pawns are in the table, their moves are given by entering only the
destination square, based on the corresponding alphanumeric designation of the
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square. Piece movements (knights and higher) also contain this information,
but also include the capital letter given in the above table.
How does this all come together? Look at the chess position below:
Figure 4
Obviously, the chess pieces are not at their starting positions. The chess
notation which explains how the pieces arrived at this position is as follows:
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5
The movements are recorded sequentially, with the white move given first
and the black move given second. 1.e4 e5 means that white moved his pawn
from square e2 to e4, while black responded with the pawn move e7-e5. 2.Nf3
designates white moving his knight on a7 to the f3 square, while black’s response
2...Nc6 is a knight move from b8 to c6. This form of recording moves using only
the destination square is called short algebraic notation.
While useful for post game analysis by human players, using this form of
notation for mathematical analysis and manipulation of chess positions breaks
down immediately, as the alpha designators are not variables, but constants
which do not change, both for the pieces and for the board.
2.2
Assigning numeric values to the chess board
The task of assigning numeric values to specific squares on the chess board (so
that it can be more easily manipulated with matrices) can be as simple as using
the row column approach, or by assigning binary values with a 10 digit series.
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Figure 5: Row column chessboard
Figure 6: Binary chessboard
The row column approach, it should be noted, starts at the top left, while
algebraic chess notation starts at the bottom left, which means the natural
method of numbering inherent in matrices is upside down and backwards to the
chess method. The cartesian coordinate system is more useful for this reason
as numbering begins in the lower left and proceeds to the upper right, just like
chess notation. In this approach, the x and y axes are off the board.
Figure 7: Cartesian coordinate chessboard
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2.3
Assigning numeric values to the chess pieces
Giving distinct numerical values to chess pieces is straightforward. The table
below gives a value to each chess piece, beginning with the white pawn and ending with the black king. Remember that these numerical values are constants,
and not necessarily a measure of the importance of an individual piece. Also,
white is numbered with single digit numbers, and black with double digits, to
avoid confusion as to what color piece is on each square. The bishops are each
given a distinct number, as one is always on the light squares, and the other
always on the dark squares. As rooks and knights can of course be on either
color square, individual designations for each rook or knight are not needed.
Figure 8: Chess piece ”values”
Again, potential confusion with chess norms requires clarification. For chess
aficionados, the ”values” assigned to chess pieces are of a strategic nature, and
refer to a pieces tactical strength. A queen has a value of 9, a rook 5, a bishop
3, a knight 3, and a pawn 1. There is no value attached to the king, as his
imminent captures ends the game, his value is unlimited. However, his tactical
strength is usually viewed as between a bishop and a rook. For our purposes,
the above chart of values should be seen as piece designators, not actual values
2.4
Matrix representation of chess positions
The simple construction of a chess position using an 8x8 matrix can be accomplished by using the values for individual pieces given above, and by using a zero
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value for unoccupied squares. The starting position on a chess board would be
rendered this way:
Figure 9
Experienced chess players would see big problems with attempts at analyzing positions represented this way. The obvious lack of dark and light square
distinction makes the system unusable, but as it is designed to be manipulated
mathematically by a computer, rather than by a human player, this is not an
issue.
2.5
Vectors and movements of chess pieces
For the author, conceptualization of actual piece movement was the most difficult task. How does one account for both direction and magnitude of all the
move options available to a piece on the board? For a pawn, other than its first
move, all moves will be one space, but how to account for the variety of moves
available to a bishop, for instance, which depending on position can move in 4
separate directions anywhere from 1 to 7 squares?
The solution is to use a relative bearing approach, in which the actual square
a piece occupies becomes the origin of a cartesian coordinate system. In a sample
position with the white king at b2 (chess notation) below moves to the square
c3, his move could be represented as (1,1):
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Figure 10
Figure 11
By this method, each piece would have a separate ”movement matrix” for all
64 squares of the chess board, as the position on the board would dictate how
far each piece could move in each direction. The chart below shows movement
vectors for each piece and for pawns, for simplicity sake constraints such as the
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edge of the board and other pieces are omitted.
Figure 12: Movement vectors
Below are a few sample matrices for different pieces in a few locations around
the board, using the movement vectors from above.
Figure 13
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Figure 14: King on b2 with movement vectors
Figure 15: Numeric representation of black’s light square bishop on d5, with
movement vectors
Another approach for expressing piece movements is one used to describe
movement vectors in a program developed to ”detect” draw by repetition movements (which is where the same position is arrived at three times in the same
game) is to use a variation of long algebraic notation in which the piece designation is omitted, and only the origin and destination squares are called out. In
this case, the movement of the white king from b2 to c3, shown in figure 10 and
11 above, which would be Kb2 in shorthand, would be A1B2. The advantage
of this approach is that the alpha designator can be removed and replaced with
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a number, in which case all moves can be described using a 4 digit string. For
this technique, the first 3 moves of the position shown in figure 4 would be as
follows:
1. 7163 4745 2. 7273 5756 3. 6172 7866
While the simplicity of this system has a certain attraction, en passant, pawn
promotion, and castling movements would need special codes, which may not
be readily intelligible in a four digit string.
It should be noted that in approaching chess with linear algebra, movement
of chess pieces by matrix manipulation cannot be accomplished in one operation. The simple opening move 1.e4 (algebraic notation), shown below, involves
changing the location of the 1 in row 7 column 5 to row 5 column 5, while
leaving all other values constant on the board, and returning a zero to row 7
column 5.
Figure 16: Opening position
Figure 17: White Pawn on e4
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If we call the start position A and the pawn at e4 B, there is no matrix
such that Ax=B. However, if we proceed with simple matrix addition and subtraction, this move can be accomplished, but the method is highly manual and
cannot be integrated into an algorithm to actually solve chess. The mathematical technique used to do this will be sketched out briefly further on.
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3.1
”Solving” chess: matrices and the endgame
Six piece Tablebases
At the start of a chess game there are 32 total pieces on the board, 16 each
for white and black. Chess games which continue to the endgame often arrive
at a position where there are no pawns on the board and a total of six pieces
(including kings). In the quest to ”solve” chess, masters as early as the 17th
century would analyze these 6 piece endgame positions for months and months,
attempting to determine best play for both sides, and any possible wins or
draws. This of course had to be done by hand, but even so the results of
these analyses were tabulated into something called tablebases. A tablebase is
a database of all possible positions with certain set of pieces, which allows a
determination of whether a position is won, lost, or drawn. Chess grandmasters
study these table bases and can therefore play perfect moves (theoretically: in
a real game move sequences are sometimes forgotten or transposed, turning a
win into a draw or worse). These table bases can run into 200 or more moves,
although in actual play such a game would be declared a draw due to the fiftymove rule (which declares a draw after 50 moves with no captures or pawn
moves). There are thousands of different tablebases, accounting for all possible
6 piece combinations and their myriad starting positions. A example would
be the tablebase for KRNKNN which is translated white has king, knight, and
rook; while black has king, knight, and knight. The starting position for this
example is given below, along with the move sequence in symbolic (the chess
piece symbol, rather than the alpha designator, is given) notation:
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Figure 18: Start position of tablebase below (white to move)
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Figure 19: Sample tablebase
As long as this move sequence appears to be, there are 61 additional moves
not shown! Obviously, the 50 move rule would mercifully end this contest, but
the point of the exercise is to show how the combination of matrix representation
of chess positions and computer processing power can solve positions previously
seen as unwinnable.
The impact of the computer on table bases was to develop far more of them,
and to disprove or prove previous results generated by hand. It produced a
subtle yet fundamental shift in chess understanding, and made table bases accessible to millions more chess players. It also allowed the development of 7 and
8 piece table bases, along with piece and pawn configurations.
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3.2
Use of matrices in Tablebases to solve the endgame
As illustrated below, the math involved in solving even basic chess positions is
quite complicated. It is interesting to note that although chess is played in 2
dimensional space (or R2), it is actually solved in 3 dimensional space (or R3).
The basic concept is that matrices are manipulated in layers, so that a single
chess position may have hundreds of submatrices associated with it, in 4 or more
layers. The position below has only the white king and rook, so it is far simpler
to conceive.
Figure 20
The math associated with this position looks like this:
Figure 21
The relevant component of this math, the application of which of illustrated
above, is the use of tensor products, which in this case involves matrices which
are composed of matrices, being multiplied by other matrices composed of matrices. When enough of these tensor products are combined (and multiplied by
each other), the correct chess move for the position is produced. For more on
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this application, see Multilinear Algebra and Chess Endgames by Lewis Stiller,
the publication from which the above illustrations was taken.
3.3
Application: an example of matrix treatment of a
three piece endgame
Below is a typical endgame position, with white having a rook, and black having
only the king. This position is a win for white, if time is not an issue. For simplicity, captures, stalemates, checks, and checkmate will be disregarded unless
otherwise stated. This will be a very simple process.
Figure 22: Sample endgame position
Below are the movement matrices for all three pieces in this position, they
are labeled according to the cartesian coordinate system, with the numeric piece
constants shown earlier. Note, for example the black king on the f2 (chess
notation) square has a movement matrice of 6217. Note also, that because the
kings have restrictions on their movements due to their proximity, they have
single digit -1 values on their movement matrices. This denotes a square under
attack by another piece, which of course the king cannot move to.
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Figure 23
Figure 24
Figure 25
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White can move to any vector for either piece that does not have a negative
value, however, the best choice is the square h2 (chess notation) with the rook
(there would also be a checks matrix in a full blown matrix array for chess,
which would have let us know about the existence a checking square), which in
our case is vector 0,-1 for the 8305 matrice.
Figure 26: Position after 1. Rh2+ (chess notation)
White has delivered check, and effectively cut the black king off from all but
the first rank of the board. Black has no choice but to retreat. Here are the
movement matrices for the current position:
Figure 27: New matrix 6217
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Figure 28
Figure 29: Matrix 8206
Notice that both the white king and the black king still have restrictions on
their movement (negative values represent squares currently under attack from
the other side) due to the proximity of the opposite king, but the black king
now has a -5 on both sides of him on the 2d rank, as well as on his own square.
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This means he is currently in check, and cannot move to any square with a
negative value. His available moves went from 5 in the previous position to 3.
Depending on the philosophy of the player, black can either threaten the rook
with a move to g1, or stay away.
Figure 30: Position after 1...Ke1
We see in the above position that rather than indeed attacking the rook with
a move to g1 (which is easily thwarted by white), the black king flees towards
the middle of the board, hoping white will play imprecisely and let him off the
back rank. The updated movement matrices for the new position appear as
follows:
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Figure 31
Figure 32
Figure 33: Matrix 5117
Notice that the white king no longer has any restrictions on his movement,
while the black king now has only 2 moves available. Whites best strategy is
to move forward towards black with his king (vector 1,0 on board 6417), to
provide support for his rook and close in a mating net. A table base search
would confirm this plan as best for white.
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Figure 34: position after 2.Kf2
With this move, white has forced black to flee towards the opposite end of the
board, as (2Kf1) leads to an immediate mate (3.Rh1). Here are the movement
matrices for all three pieces:
Figure 35: Matrix 6307
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Figure 36
Figure 37: Matrix 5117
Advancing towards the black king did give white less freedom of movement,
but also serves to hem black in. Black is at most 6 moves from mate. The
one thing he cannot do is move towards the rook, as then he will have nowhere
to retreat as any route away from the 1st rank would be cut off by the king
directly in front of him. He has only 2 moves available, so the right one is an
easy choice: Kd1 (vector -1,0 on board 5117).
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Figure 38: Position after 2...Kf1
Black has committed chess suicide! He fails to move away from the threat,
which means he either he has a previous engagement to get to, or he does
not know any better. At any rate, he has just ended the game, as 3. Rh1 is
checkmate. The movement matrices are below:
Figure 39
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Figure 40: Matrix 6307
Figure 41
-A quick check reveals black still has no mobility, the white king is still
somewhat restricted, and the rook has any move available to him. However,
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the obvious move is Rh1 which wins the game. Is this indicated on our matrix?
Not on this one, but it would be on one of the 14 additional matrices for the
white rook, each based on one of the possible moves available to him, and
displaying the results of each. These sub matrices get closer to the heart of
real chess, as they give values for the rook at each position, based on threats,
attacks, captures, checks and checkmate, sacrifices, blunders, etc. Depending
on the game situation, those sub matrices will also have sub matrices, and may
interact with the many other matrices in the many other layers. In this example,
we have greatly simplified the process, but only in the interest of grasping its
complexity. White’s move here is 1,0 from matrice 8205.
Figure 42: Final position after 3.Rh1 checkmate
Game over. Black has nowhere to go, with the white King right in front of
him, he cannot escape the wrath of the white rook. Now look at the final 3
”movement” matrices:
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Figure 43: Matrix 6307
Figure 44
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Figure 45: New matrix 6117
Why look at another matrix after the game is over? Because it is identical
to one of those matrices referred to just a second agoit is just like one of those
fourteen sub matrices of every available move for the white rook. What value
would it return that would have helped us to see the right move? A zero value
for the square occupied by the black king! It is somewhat analogous to the
solution Ax=0all game each player is searching for THIS matrice, the one with
the value of the opponents king at 0. This of course means that the king is under
attack, and cannot escapeits checkmate. In this case our solution to Ax=0 is
matrice 8105, which is the white move that results in the black king matrice
shown above.
Again, each movement vector submatrice would have its own submatrices,
based on outcomes of all of those possible moves. Imagine manipulating the
sub-sub matrices for all 32 pieces at the start of the game! This hints at the
enormity of the problem with finding a solution to chess.
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Conclusion
To fully understand how matrices are used to efficiently solve endgame positions,
one must study a great deal more math at the graduate level. Even so, regardless
of how much math has been applied to date, the elusive solution to an entire
game of chess has yet to materialize. However, it is certain that if the solution
is to be found, it will be rooted in linear combinations of vectors, and matrix
manipulation.
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5
Bibliography
[1] Strang, Gilbert. Introduction To Linear Algebra. 3rd ed. Wellesley, MA:
Wellesley-Cambridge Press, 2003. 363-378. Print.
[2] Siller, Lewis Multilinear Algebra and Chess Endgames.
http://users.rcn.com/lstiller/msri-paper.pdf
[3] Vuckovic, Vladan and Vidanovic, Djordje A New Approach to Draw Detection by Move Repetition in Computer Chess Programming
http://arxiv.org/ftp/cs/papers/0406/0406038.pdf
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