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Combinatorial Games (Part II): Different Moves for Left and Right

Posted March 2003.


Feature Column Archive

1. Introduction

Conflict and aggression are widespread in the natural world. Those of us lucky enough to have seen animals such as lions, rhinos, and wild dogs in a natural environment or on television have seen these animals fight for mates and food. Some of these battles end in death. Humans display their aggression in variety of ways ranging from bullying to war. However, we have also developed a stylized form of aggression that takes the form of playing sports (football and rugby are pretty rough) and games (chess, checkers, and poker). Sometimes games take the form of each player making a move from a common set of pieces (e.g. Nim and its look alike games). However, more commonly, each player has his own forces or pieces which he/she plays. Such games, known aspartizan games, are not only fun to play but lead to surprisingly rich mathematical material. From a mathematical perspective games deserve to be viewed as much more than a part of recreational mathematics. The analysis of combinatorial games often draws on many different parts of mathematics ranging from analysis to knot theory to graph theory.

Joseph Malkevitch
York College (CUNY)


  1. Introduction
  2. Hex
  3. Hackenbush
  4. Counting and sets
  5. Surreal numbers
  6. Surreal numbers and games
  7. References