Combinatorial Game Theory
who wins, and by how much
Ladder

Grundy sequences — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 001230412308123012123016123020123024heap size, and the value of a heap that bigperiod 4 from heap 0, holding through all 2001 values computeda heap of size n is worth ∗g(n) — and the whole game is the nim-sum of its heaps

    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.

    rung 2 · impartial
  2. ·137— one digit for each number of tokens that may be takentakedigit1 · end it2 · leave one4 · splitwhat that permits11take the whole heap23take the whole heap · take k, leave one heap37take the whole heap · take k, leave one heap · take k, split the rest in twoa filled circle is a move the code permits; an empty one is a move it forbidsa code is finite, so the game it names is a take-away game with a largest legal take

    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.

    rung 3 · impartial

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