# Chess and fascinating mathematics

## chess, math, and mythology

Legend has it that the game was invented by a mathematician in India who elicited a huge reward for its creation. The King of India was so impressed with the game that he asked the mathematician to name a prize as reward. Not wishing to appear greedy, the mathematician asked for one grain of rice to be placed on the first square of the chess board, two grains on the second, four on the third and so on. The number of grains of rice should be doubled each time.

The King thought that he’d got away lightly, but little did he realise the power of doubling to make things big very quickly. By the sixteenth square there was already a kilo of rice on the chess board. By the twentieth square his servant needed to bring in a wheelbarrow of rice. He never reached the 64th and last square on the board. By that point the rice on the board would have totalled a staggering 18,446,744,073,709,551,615 grains.

Playing chess has strong resonances with doing mathematics. There are simple rules for the way each chess piece moves but beyond these basic constraints, the pieces can roam freely across the board. Mathematics also proceeds by taking self-evident truths (called axioms) about properties of numbers and geometry and then by applying basic rules of logic you proceed to move mathematics from its starting point to deduce new statements about numbers and geometry. For example, using the moves allowed by mathematics the 18th-century mathematician Lagrange reached an endgame that showed that every number can be written as the sum of four square numbers, a far from obvious fact. For example, 310 = 172 +42 + 22 + 12.

Some mathematicians have turned their analytic skills on the game of chess itself. A classic problem called the Knight’s Tour asks whether it is possible to use a knight to jump around the chess board visiting each square once only. The first examples were documented in a 9th-century Arabic manuscript. It is only within the past decade that mathematical techniques have been developed to count exactly how many such tours are possible.

It isn’t just mathematicians and chess players who have been fascinated by the Knight’s Tour. The highly styled Sanskrit poem Kavyalankara presents the Knight’s Tour in verse form. And in the 20th century, the French author Georges Perec’s novel Life: A User’s Manual describes an apartment with 100 rooms arranged in a 10×10 grid. In the novel the order that the author visits the rooms is determined by a Knight’s Tour on a 10×10 chessboard.

Mathematicians have also analysed just how many games of chess are possible. If you were to line up chessboards side by side, the number of them you would need to reach from one side of the observable universe to the other would require only 28 digits. Yet Claude Shannon, the mathematician credited as the father of the digital age, estimated that the number of unique games you could play was of the order of 10120 (a 1 followed by 120 0s). It’s this level of complexity that makes chess such an attractive game and ensures that at the Olympiad in Russia in 2010, local spectators will witness games of chess never before seen by the human eye, even if the winning team turns out to have familiar names.