Games and Mathematics (7 page)

The Japanese Rules of Go

This is a simple version. The official rules complete with explanatory notes are at:
www.cs.cmu.edu/∼wjh/go/rules/Japanese.html
on which this epitome is based.

Two players compete on a board, to see which can take more territory.

The board is a grid of 19 horizontal and 19 vertical lines forming 361 intersections. A stone can be played on any unoccupied intersection.

The board is initially empty.

The players can alternately play one move at a time, one player playing the black stones, his opponent the white stones.

After a move is completed, a group of stones belonging to one player exists as long as it has a horizontally or vertically adjacent empty point, called a ‘
liberty
’.

If, after a player's move, a group of his opponent's stones has no liberties, the player removes all these stones, as
prisoners
.

A shape in which the players can alternately capture and recapture one opposing stone is called a ‘
ko
’. A player whose stone has been captured in a ko cannot recapture in that ko on the next move.

A group of stones which cannot be captured by the opponent is alive. A group which can be captured is dead.

Empty points surrounded by the live stones of just one player are called ‘
eye points
’. Other empty points are called ‘
dame
’. Stones which are alive but only possess dame are said to be in ‘
seki
’.

Eye points surrounded by stones that are alive but not in seki are called ‘
territory
’, each eye point counting as one point of territory.

When a player passes his move and his opponent passes in succession, the game stops.

After agreement that the game has ended, each player removes any opposing dead stones from his territory and adds them to his prisoners.

Prisoners are then placed in the opponent's territory, and the remaining points of territory are counted. The player with more territory wins.

If both players have the same amount the game is a draw, which is called a ‘
jigo
’.

Combinatorial game theory (CGT)

Some games are so simple, unlike chess and Go, that they can actually be solved completely by mathematical arguments. Nim is the most famous example. Your start with several piles of stones or other objects. Players take turns to select one of the piles and take any number of the objects in it, from a single object to the whole pile. Two rules exist for deciding the winner. Either the winner is the player who does
not
take the last object, or the player who does.

Nim was completely ‘solved’ by Charles Bouton in 1901, meaning that he showed which player should win, given a certain set of piles to start with, and how that player should play to guarantee the win. Like the Tower of Hanoi
, the solution involves binary numbers.

Nim has the important feature, shared with all the games we have presented, that the position is completely open to both players, unlike card games or poker-type games whose analysis therefore involve probabilities. The latter have been analyzed by mathematical game theory which has been applied to economic competition between firms, a nuclear arms race, the sale of licences for mobile phones operators and other more-or-less realistic scenarios.

Nim is important for another reason. In the 1930s Roland Sprague and Patrick Grundy proved independently that all impartial games – meaning that for every position of the game, the same moves are available to both players – are equivalent to some game of nim, so the variety of impartial games is less than it might seem.

The extraordinary book,
Winning Ways for Your Mathematical Plays
, by John Horton Conway, the inventor of the game of Life, and his colleagues Elwyn Berlekamp and Richard Guy [Berlekamp, Conway & Guy
1982/2001
] showed how some very simple games which were not impartial – but not including chess and Go which are far too complex – could also be solved mathematically.

Conway was inspired in his early work on combinatorial games by the endgame at Go which, as we shall see, is much more mathematical than the rest of the game, so it is appropriate that one of the highlights of subsequent work on CGT was
Mathematical Go: Chilling Gets the Last Point
by Elwyn Berlekamp and David Wolfe [Berlekamp & Wolfe
1994
] which reduces the final stages of any Go game to a (complex) calculation.

Three blind mathematicians

Blind mathematicians are rare, but they do exist and they have reached to the very top. (Are there blind theoretical physicists or chemists?) No doubt psychologists could learn much by studying how they think. Ironically, they have been as distinguished at geometry as at algebra, partly by exploiting their sense of touch.

Nicholas Saunderson (1682–1739) was blinded by smallpox at the age of 12 but became Lucasian Professor of Mathematics at Cambridge University. He wrote a textbook,
Elements of Algebra
, and lectured on optics. He also invented the pin-board, much used today by primary school pupils, to create geometrical figures that he could feel with his fingers, a sort of Braille geometry.

Lev Pontryagin (1908–1988) lost his sight in an accident at the age of 14. His devoted mother read books to him, wrote his notes and even learnt to read foreign languages for his sake. At 25 he entered Moscow University, blind but able to remember all his lectures. He published his first original work at the age of 19, like Euler, and went on to become one of the great mathematicians of the twentieth century.

[O’Connor & Robertson 2006]

Bernard Morin (1931–) has been blind since the age of 6, but became a brilliant geometer who discovered how to turn a sphere inside out – yet another ironic achievement for a mathematician who can feel but not see – and what is now called Morin's surface.

A deep philosophical error

Philosophers have long distinguished between two basic types of objects (only), universals and particulars. Particulars, such as an apple or gate post, were no problem, and neither were the ideas of an apple or the idea of a gate post since everyone could agree that my idea of an apple might be subtly different from yours.

Universals, such as the number 2 were a different matter. They seemed to exist independently of individual human minds, but how was this possible? The Greek philosopher Porphyry
asked: ‘Are they beings with independent existence or do they exist only as human concepts?’

The puzzle was: if the number 2 has an ‘independent existence’, then where is it? And if it does not have an existence independent of human minds, how can it be ‘the same’ in every mind? Surely the number 2 is the same for you as it is for me, and everyone else who is very basically numerate?

According to the philosopher Bertrand Russell:

 

The argument that 2 is mental requires that 2 should be essentially an existent. But in that case it would be particular, and it would be impossible for 2 to be in two minds, or in one mind at two times. Thus 2 must be in any case an entity, which will have being even if it is in no mind.

[Russell
1903
: 451]

 

No wonder that, according to Russell,

 

mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true,

 

or that the number 2 is,

 

a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down.

[Russell
1956
: 542]

 

Russell and other traditional philosophers have been mistaken because they have had no conception of the peculiar status of chess pieces and other game-like objects. They have never taken abstract games seriously as a subject for investigation and so have not realised that there are three fundamental kinds of objects, not two.

[Wells
2010
:
Philosophy and Abstract Games
]