Combinatorial Game Theory
who wins, and by how much
Series

Grundy sequences — 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, 2, 3. 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.

    Grundy sequences, and where they stop being predictable

    Computing one Grundy value is a mex. Computing all of them produces a sequence, and the sequences do something nobody has fully explained — most of them eventually repeat, some of them take thousands of terms to start, and for a few nobody knows whether they ever do.

    part 2 · impartial
  2. The octal game ·137, read out. An octal code is a rule table. The kth digit says what a player may do after taking k tokens from one heap: end that heap, leave one heap, or split the rest into two. Three bits, one digit, and the whole family of take-away games becomes something that can be listed and swept.

    Naming a game with a number

    An octal code is a rule table compressed into an integer. It turns "which game" into something that can be counted through, which is how the family was swept — and how the games nobody can solve were found.

    part 3 · impartial

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