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Combinatorial Games (Part I): The World of Piles of Stones

1. Introduction

When one thinks about mathematics, one thinks of numbers and shapes. In the somewhat more than 2000 year history of the organized study of mathematics, surely we know what there is to know about numbers? Yet only about 30 years ago John Horton Conway showed how little we knew about numbers. He did this by pointing out a deep relationship between numbers and the seemingly more frivolous pursuit of human beings - when compared with doing mathematics - of playing games. To many people, games are merely a way of pleasantly spending time in play. However some of the most distinguished mathematicians of the 20th century, John von Neumann (1903-1957), John Nash, and John Conway showed the value to mathematics of taking games seriously.

Picture of John von Neumann Picture of John Nash Picture of John Conway

There are actually two major mathematical theories of games. One of these theories, now usually referred to as Game Theory, tries to help get insight into making the wisest decision in situations where various amounts of information are known. This information is in the form of what payoffs might accrue to the different players (either human opponents or a passive opponent called nature) and the extent to which they know what courses of action are available to their opponents. One example of such a game would involve a farmer who has a choice of what crops to plant. The farmer must make her decision without being certain what the weather will be like during the growing season and without knowing what prices she will get for her harvested crops. A second example would involve two political leaders deciding among various political choices of action, where each knows exactly what actions the other can take and the payoffs to each depending on which action is taken. Games such as Prisoner's Dilemma and Chicken are of this kind.

The other theory of games is often referred to as Combinatorial Games. One classical game of this kind is Nim, where two players move alternately by selecting a subset of the stones (any number of stones: from one stone to the whole pile) from a single pile of stones in a collection of piles of stones. A player who can not move loses. Games such as Nim lead in many unexpected directions and we will see that one of these directions is what inspired John Conway to develop a rich world of new numbers, now generally referred to as surreal numbers.

Joseph Malkevitch
York College (CUNY)

Email: malkevitch@york.cuny.edu

  1. Introduction
  2. What is a combinatorial game?
  3. Finding the winner of a game
  4. Solving Nim
  5. Towards a theory for combinatorial games
  6. Towards the surreal
  7. References

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