• DOI:10.1109/PROC.1978.11005
• Corpus ID: 5166062

On numbers and games

@article{Guy1978OnNA,
  title={On numbers and games},
  author={Richard K. Guy},
  journal={Proceedings of the IEEE},
  year={1978},
  volume={66},
  pages={716-717}
}

• R. Guy
• Published 1 June 1978
• Mathematics
• Proceedings of the IEEE

Some readers know to play the game of nim well, fewer play a perfect annihilation game, and nobody knows whether there exists an opening move in chess that will guarantee a win for white. These games and many more, belong to the family of combinatorial games, by which we mean the set of all two-player perfect-information games without chance moves and with outcomes lose or win (and sometimes: dynamic tie). The motivation for ONAG may have been, and perhaps was-and I would like to think that it was-the attempt to bridge the theory gap between nim-like and chess-like games. Why is there a gap? Every combinatorial game can be described as a directed graph called game-graph, whose vertices are the game positions, and (u, v) is a directed edge if and only if there is a move from position u to position v. Denote by N the set of all positions from which the Next (first) player can force a win; by P the set of all positions from which the Previous (second) player can force a win; and by T the set of all (dynamic) Tie positions, which are positions from which no player can force a win and therefore both can avoid losing. In an acyclic game-graph there cannot be any tie positions. The N, P, T classification of any game graph R = (V, E) can be determined in 0(\V\ + \E\) steps [8]. For both nim and chess, a finite game-graph can be constructed and the N, P, T classification can be determined. So both games are solvable in principle. If we play nim with n piles, each pile containing at most k tokens, then the game-graph contains (k + \) vertices. Suppose that in (generalized) chess played on an « X « board there are k different pieces. If k is about n/2, then the game-graph of chess contains O (2") vertices. So both game-graphs have exponentially many vertices, and thus both games appear intractable in the usual sense of computational complexity [1, Chapter 10], [14, Chapter 9], namely a computation appears to be required which is asymptotically exponential. From a computational efficiency standpoint, the essential difference between nim and chess is that nim can be viewed as a disjunctive compound (sum) of independent games, namely the individual piles. A disjunctive