“A chess problem is an exercise in pure mathematics.” The British mathematician Godfrey H. Hardy wrote this in 1940 essay A Mathematician’s Apology. Also The Chess Revolution (2024) by the Dutch chess journalist Peter Doggers is a lot about mathematics. For example, we read about the mathematician and code expert Alan Turing, who designed the first chess computer around 1948. But what mathematics is hidden in this board game?
Anyone who googles this often finds mathematical riddles. For example the ‘horse jumping problem‘. In chess, the knight is only allowed to make L-shaped jumps: one square to the side and two forward, or two to the side and one forward. How can you then make the knight jump so that it visits all 64 squares of the chessboard – but all only once? In fact a ‘traveling salesman problem‘, a well-known concept in mathematics.
Another classic is the ‘eight-queen issue‘. In chess, ladies may move along entire rows, columns and diagonals. How can you place eight queens on a chessboard without one of them being able to capture another?
“Great fun, but these are not problems you encounter in a real game,” responds Daniël Kuijper. He is a master’s student in theoretical physics, “loves mathematics”, and a not without merit. “If only because you never have eight ladies.”
What Kuijper finds more interesting is the complexity of chess, which can easily be described mathematically. Now take the number of possible different chess games. On each turn, a player has about twenty to thirty possible moves. The number of possible games therefore grows exponentially with each turn. The American mathematician Claude Shannon calculated this around 1950. If both players have had a turn four times, the number of possible paths has already increased to almost 85 billion. In total, Shannon estimated the number of possible chess games at 10 to the power of 120: the ‘shannonnumber‘, which is trillions of times greater than the number of atoms in the observable universe.
“In principle, mathematics can of course describe everything, because that is the purpose of mathematics,” says Kuijper. “But there is no specific piece of math that is really useful for learning to play better chess.” The way of thinking you need is similar. For example, you always make decisions based on what your opponent might do. “If I move my bishop there, he can, for example, move his rook there or here. Then you can logically determine which would be more favorable for you.”
If you practice that a lot, you will get better at it. “A matter of pattern recognition,” says Kuijper. “Chess computers can do that too. They are programmed to calculate dozens of moves ‘deeply’ into what the outcome of a move could be.” The computer can express whether something is advantageous or disadvantageous in a number, based on how valuable the piece is (for example, a queen is more powerful than a knight) and whether it is in a ‘powerful’ place on the board. “But he does not calculate all those trillions of paths. That would be impossible. No, he makes a selection of logical scenarios. Again based on pattern recognition.”
Chess and mathematics certainly have similarities, Kuijper concludes, but he would not argue that chess is based on mathematics. “A good chess player is not necessarily a good mathematician, or the other way around. It just depends on what you practice. Einstein was less good at chess than I was.”
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