Chapter 5: Two Dimensions and Beyond

Section 6: Multiway Systems

Game systems

One can think of positions or configurations in a game as corresponding to nodes in a large network, and the possible moves in the game as corresponding to connections between nodes. Most games have rules which imply that if certain states are reached one player can be forced in the end to lose, regardless of what specific moves they make. And even though the underlying rules in the game may be simple, the pattern of such winning positions is often quite complex. Most games have huge networks whose structure is difficult to visualize (even the network for tic-tac-toe, for example, has 5478 nodes). One example that allows easy visualization is a simplification of several common games known as nim. This has k piles of objects, and on alternate steps each of two players takes as many objects as they want from any one of the piles. The winner is the player who manages to take the very last object. With just two piles one player can force the other to lose by arranging that after each of their moves the two piles have equal heights. With more than two piles it was discovered in 1901 that one player can in general force the other to lose by arranging that after each of their moves Apply[BitXor, h] 0, where h is the list of heights. For k > 1 this yields a nested pattern, analogous to those shown on page 871. If one allows only specific numbers of objects to be taken at each step a nested pattern is again obtained. With more general rules it seems almost inevitable that much more complicated patterns will occur.

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From Stephen Wolfram: A New Kind of Science [citation]