Maths and Chess

Maths and Chess

Magnus Carlsen, the 22 year old chess prodigy from Norway (pictured above), has just been crowned World Chess Champion, winning £1.4 million in the process.  He beat the Indian Grandmaster Vishy Anand in a 12 match series in India with 2 games to go.  Anand has been the World Champion since 2007 – and by beating him Carlsen has become only the second youngest player ever to become World Champion.

There is a close connection between maths and chess – both require an ability to solve problems, a dedication to single minded concentration and an aptitude for thinking in three dimensions.  Indeed, there is some research that suggests that regularly playing chess can actually improve maths scores.

Even though chess has somewhat declined in international popularity since the glory days of Kasparov, Karpov  and Fischer, it remains one of the world’s most popular games – with millions of players across the globe.

Chess is in theory “solvable” using game theory.  Game theory is a subset of mathematics which uses combinatorial mathematics to analyses “perfect” moves.  Noughts and crosses, for example, is solvable.  You can predict the outcome of the game prior to any moves being made as long as both players use optimal strategies.  In the case of noughts and crosses the game should always end in a draw.

With a large enough computer, it should one day be possible to solve chess – and to calculate without a piece being moved whether optimal moves lead to either a draw or a win for white.  The reason why this hasn’t been done so far is that the number of potential moves which need to be analysed are astronomical.   Shannon’s Number, 10¹²⁰ was derived as an conservative approximation for the number of distinct  possible moves in an average game of chess.  To put this into perspective, the number of atoms in the observable universe is only thought to be of the order 10⁸⁰. With numbers this big chess is unlikely to be solved any time soon.  Indeed, the mathematician Jonathan Schaeffer, who helped to solve checkers, predicts that it will require the advent of quantum computing to solve a game as complex as chess.

There are many different chess problems to study.  The Knight’s Tour (covered in more detail here) is one of the most famous – studied by the great mathematician Euler himself.  Other puzzles require that a set number of pieces attack all 64 squares of a standard chess board.  For example the board above shows how 5 queens can achieve this.

If you enjoyed this post you might also like:

Game Theory and Evolution – does it pay to be nice?

The Knight’s Tour – A history behind the 100o year old chess puzzle that remains unsolved.

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.