God numbers for Graphs, Games and Groups

Abstract

We describe and axiomatize finite solitaire puzzles and zero sum sequential games graph theoretically. Zermelo's theorem telling that there is a win for one of the players or a draw follows from the definitions. The god number is a geometric quantity that quantifies the number of moves necessary to solve the puzzle. In the solitaire case, the god number is the minimal distance from the initial state $v$ to the solution space $A$. If $v$ and $A$ are not specified, the god number is the graph diameter. God number computations are related to combinatorial sorting problems and is a NP-complete problem in general even when restricted to concrete sliding problems. In the two-player case, the god number is a minimax critical value: it minimizes the maximal game event length over the set of all strategies. A strategy is a sub-graph of the game graph that contains the initial vertex. The definition is done so that a ``mate in k" chess problem has god number k. As for examples: in the solitaire case, we look at group games like Rubik type problems, transposition games related to sorting, at sliding puzzles like the 15 puzzle or rainbow ball, or the tower of Hanoi. For two-player games, we illustrate the story using examples of small chess games, a small card game or tic-tac-toe type problems.