# Mathematics and Chess

### John MacQuarrie

### Introduction

It is difficult to put an age on the game of Chess. A game of such complexity cannot be dreamed up overnight; it must be developed over time. To put a birth date on chess is akin to putting a date on when in our evolution our race stopped being apes, and started being humans. To answer this second question, one would have to draw an arbitrary line between apes and humans so that a relatively precise date could be attached. The origin of Chess is stated with this analogy in mind. The first known form of chess is called Chaturanga, was devised in the Punjab and was played in India from around the middle of the Sixth Century A.D. Even dating as vaguely as this, one must be cautious; there is considerable controversy about many aspects of the historical evidence on which the claim is made (Eales, 1985). For our purposes however, the common conception will suffice. There are of course some differences between Chaturanga and chess: The board on which Chaturanga is played is uncheckered, the Kings do not sit facing each other at the beginning of the game, Elephants tower on the squares where modern bishops now preach and the King's companion is a man even weaker than him. There are also some differences in the rules, but in spite of them, the games of Chaturanga and chess are very similar. Notably, the game is won in the same way as chess, by checkmating the King. (Murray, 1913)

'Modern Chess' is thought to have begun towards the end of the fifteenth Century. From this point on it is easier to trace the development of the game through the centuries that followed, by examining records of individual changes to rules or game pieces. Since that time, the rules of Chess have been tinkered with regularly, and continue to be. In fact, with Chess computers changing how professionals view the game, the Fédération Internationale des Échecs (FIDE) has been changing (and unchanging) the rules of the game even within the last decade.

Chess is unarguably mathematical; it is simply a finite set (of pieces) on which a finite set of restrictions (or rules) are imposed. Within these restrictions, a variety of possible states (or positions) occur, and certain ordered subsets of these possible states form 'games'. This will be formalised and discussed later. With the simplicity of the rules, coupled with the enormous size and complexity of the set of possible games that result, it is of little wonder that Chess is of interest to mathematicians. What follows is a consideration of how mathematics has been applied to Chess (the term being used rather loosely at times) in several different ways.

Firstly, the notion of Chess Problems will be discussed, giving examples ranging from the very 'Chess-like' to the more abstract. The mathematics behind these problems and their solutions, as well as the mathematical minds involved, will be discussed.

Secondly, an example of a Chess puzzle is considered. A Chess puzzle is less Chess-like than a problem, though the connection between the puzzles and Chess is clear. The puzzle considered is extremely old, but has only been fully solved very recently.

Thirdly, the employment of Set Theory and abstract Game Theory is turned to the task of considering the question 'Is Chess Solvable?' which may be the most fundamental problem in Chess.

It is hoped that it will be clear from the examples and considerations provided that Chess has in history been an extremely fertile bed for a huge variety of interesting mathematics, from simple logical thought, to tremendous levels of abstraction.

'Modern Chess' is thought to have begun towards the end of the fifteenth Century. From this point on it is easier to trace the development of the game through the centuries that followed, by examining records of individual changes to rules or game pieces. Since that time, the rules of Chess have been tinkered with regularly, and continue to be. In fact, with Chess computers changing how professionals view the game, the Fédération Internationale des Échecs (FIDE) has been changing (and unchanging) the rules of the game even within the last decade.

Chess is unarguably mathematical; it is simply a finite set (of pieces) on which a finite set of restrictions (or rules) are imposed. Within these restrictions, a variety of possible states (or positions) occur, and certain ordered subsets of these possible states form 'games'. This will be formalised and discussed later. With the simplicity of the rules, coupled with the enormous size and complexity of the set of possible games that result, it is of little wonder that Chess is of interest to mathematicians. What follows is a consideration of how mathematics has been applied to Chess (the term being used rather loosely at times) in several different ways.

Firstly, the notion of Chess Problems will be discussed, giving examples ranging from the very 'Chess-like' to the more abstract. The mathematics behind these problems and their solutions, as well as the mathematical minds involved, will be discussed.

Secondly, an example of a Chess puzzle is considered. A Chess puzzle is less Chess-like than a problem, though the connection between the puzzles and Chess is clear. The puzzle considered is extremely old, but has only been fully solved very recently.

Thirdly, the employment of Set Theory and abstract Game Theory is turned to the task of considering the question 'Is Chess Solvable?' which may be the most fundamental problem in Chess.

It is hoped that it will be clear from the examples and considerations provided that Chess has in history been an extremely fertile bed for a huge variety of interesting mathematics, from simple logical thought, to tremendous levels of abstraction.