Combinatorial Game Theory
who wins, and by how much
Series

Subtraction — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Grundy values for subtraction of 1, 3, 4. The Grundy value of every heap size for a take-away game, computed by the mex rule. A period, if the figure marks one, was found by searching the computed sequence rather than assumed — and where no period is marked, none was found in the range drawn, which is not the same as there being none.

    Take one, three or four

    A heap and a list of legal takes. It is the smallest interesting impartial game there is, and the only family in the subject where eventual periodicity is not observed, not conjectured, but guaranteed — with a bound on when it must appear.

    part 1 · impartial
  2. Grundy values for subtraction of 2, 5, 7. The Grundy value of every heap size for a take-away game, computed by the mex rule. A period, if the figure marks one, was found by searching the computed sequence rather than assumed — and where no period is marked, none was found in the range drawn, which is not the same as there being none.

    The period is small and the proof does not say so

    Every subtraction game repeats eventually — that is a theorem, and its proof gives a bound of sixteen thousand for a three-move set. Over 112 sets the longest period measured is twenty-two. The proof and the fact are four orders of magnitude apart, and the rule of thumb that closes the gap is broken by one set in the sweep.

    part 2 · impartial

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