I have been playing chess since I was seven years old and somehow the game has never failed to amaze me. Being the Head of the Chess Society in my school, I have had the opportunity to mentor numerous peers and juniors and each interaction has led to endless discussions and questions around the possibilities of every single move in the game . Given that I am mathematically inclined, these discussions left me curious to understand chess through the lens of mathematics. The article below summarizes my learnings

**The Game**

The game of chess has been around for over hundreds of years. Today, it is played by millions of people around the world. The game requires strategy, patience, and above all, problem solving. Chess is quite complex yet the objective is beautifully simple. The game is played on a square board made up of 64 smaller squares. Half of these small squares are black and the other half are white, making a checkerboard pattern. At the beginning of the game, there are 32 pieces, 16 black and 16 white. Each player has eight pawns, two castles, two knights, two bishops, one queen, and one king.

Mathematics, particularly the topics of probability, statistics, and combinatorics fit into the world of chess. Geometry also plays a role in the game.

**Ernst Zermelo’s Work**

Ernst Zermelo, a German Mathematician pioneered research game theory and set theory. In 1913, he published an article titled, “On an Application of Set Theory to the Theory of the Game of Chess.” In this article, Zermelo discusses two player games without chance moves where each player is trying to beat the other.

Zermelo presents two questions First, when is a player in a “winning” position and can this be defined in a mathematical sense? Second, if a player is in a winning position, can we determine the

number of moves needed to force a win? To answer the first question, Zermelo states that there must exist a nonempty set that contains all sequences of moves that produce a victory for one player, say white, no matter how the opponent plays. If this set is empty, then the best white could do would be to force a draw. So Zermelo defined another set as the set containing all sequences of moves such that white can infinitely postpone a loss. However, a chess game cannot last forever. This is because a draw can be demanded if an exact position of the pieces is repeated three times. Therefore, this set must be empty. So white would only be able to postpone a loss for a finite amount of moves. This is equivalent to saying that black can force a victory. So theoretically, one player should be able to force a win or force a draw.

Moving to the second question, if a player is in a winning position, can we determine the number of moves needed to force a win? Zermelo reasons that it would never take more moves than the amount of positions in the game. He proved this by contradiction: Assume that white can win in a number of moves greater than the possible positions. Then a winning position would have been repeated at least once. If white had of played the winning move the first time, he would have won in fewer moves than the amount of possible positions.

**The Total Number of Chess Games and Shannon’s Number**

Many of you may have wondered how many games of chess can be possibly played on the chessboard? This number of games is known as **Shannon’s Number** (greater than the number of atoms in the observable universe!).

Claude Shannon came up with an estimate that the number of games of chess would be 10120.

Now, let us try to understand how exactly he arrived at this huge number.

After some deep thinking and observing games, he noticed that on an average, in any position, there are about 30 legal moves one can make.

For example, in this random position, Black has 37 possibilities to choose from.

You can try this yourself. Take a random position from any of your chess games and count how many legal moves each side has and the number will be close to around 30.

Now if both players make their moves, this will give rise to 302 possible games which is 900 games and when they do that again we will have 304=810,000 games. The pattern is now easy to recognize.

According to Shannon, an average single chess match is worked out to be 40 moves long. Since a single person can make on an average 30 moves, we have the simple representation of the number of possible games as:

308010120 games.

English Mathematician by the name Godfrey Harold Hardy had a different estimate. His number read which reads 10 to the power 10 to the power 50. This approximation makes Shannon’s number absolutely miniscule. The reason for this was that Shannon had not taken into account unrealistic games. For example, even if a player has a chance to checkmate in one move, he may not opt for it and the game could carry on further.

**The Eight Queens Puzzle**

This puzzle is the famous problem of placing eight queens on an 8×8 chessboard so that no two queens are attacking each other. This means that the queens must not share the same row, column, or diagonal.

Below is an example of how a correct solution looks:

Now if you think about it, rotating the position 90 degrees about the center of the board will produce a new solution. This position has 8 variants obtained by rotating 900, 1800, 2700 and then reflecting the rotational variants in a mirror.

There are 12 fundamental solutions with 8 variants each which means 12 x 8 = **92 distinct solutions**

**The Rule of the Square**

In the endgame, a pawn has the objective of being promoted to a queen. Using basic geometry, if a King is inside this area, he will be able to catch the pawn. If he is outside however, he will not be able to do so.

You can draw a diagonal from the pawn to the closest square on the last rank and form a square. This is an example of the geometrical application used in chess.

The connection between mathematics and chess is deep and there is a wide-ranging correlation that can be established. Researching about the mathematics associated with this game has also sparked my interest in combinatorics. Currently I am thoroughly enjoying a course on Combinatorial Game Theory and hoping to understand and explore more applications of mathematics in chess.

References

How Many Chess Games Are Possible? This Will Blow Your Mind!

https://news.stlpublicradio.org/arts/2019-03-28/on-chess-chess-and-mathematics

https://firescholars.seu.edu/cgi/viewcontent.cgi?article=1096&context=honors