Combinatorial Game Theory
who wins, and by how much
Series

Welter — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Welter positions and what they are worth. Coins on a strip, with the Grundy value the recursion returns and the nim-sum the squares would have if they were independent heaps. The two columns are the essay: they hardly ever agree.

    No two heaps alike

    Welter's game is Nim with one extra clause — no two heaps may be the same size — and the clause is fatal to the nim-sum, which gives the right answer in none of the 120 three-coin positions. What replaces it is a function of pairs: ⟨a | b⟩ = (a ⊕ b) − 1, exact on all 55 two-coin positions, and nim-added over every pair it is exact on the whole board provided the number of coins is even.

    part 1 · impartial
  2. Four readings, one game. The closed form, the strict mating, the cancelling matching and the parity term, with how much of the game each accounts for.

    The pairing the formula hides

    Welter's closed form sums a function over every pair of coins and needs an extra term when the count is odd, which the rung below called a surprise. Read as a matching it is not: an odd number of coins cannot be paired, the left-over coin contributes its own square, and some matching gives the value on every position measured.

    part 2 · impartial

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