Combinatorial Game Theory
who wins, and by how much
Concept

State — where it appears

What a position has to carry beyond the pieces on the board, such as the size of the previous move or which component was played in last. A game whose components need one is a game the disjunctive sum cannot be applied to unaltered.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

What a component has to carry. Four impartial games, one of which is Nim. In the other three a component cannot say what its own legal moves are without knowing something about the past or about the rest of the board, so the Sprague–Grundy recipe does not apply — and the table says by how much. Every outcome was obtained by solving the sum outright rather than by any formula.

What a component has to carry

Three impartial games on this site break the sum, and they break it for the same reason: a component cannot say what its own legal moves are. Measured with one instrument — one number per part, exclusive-ored — the failure rate runs from a quarter to nearly half, against a control where the same recipe is a theorem and is never wrong.

limits · Memory One number per heap, and one number per state. Sums of Fibonacci Nim components solved in full, against two predictions. Giving each component the number its heap size suggests gets a quarter of the pairs wrong; giving it the Grundy value of its state — the pair of heap size and cap — gets every pair and every triple right.

What restores the theorem

Fibonacci Nim breaks the recipe every impartial game is supposed to obey: one number per heap, exclusive-ored, gets a quarter of two-heap sums wrong. Index the recursion on the pair of heap size and cap instead and the recipe is exact on every pair and every triple — and the number a heap of nine carries turns out to be five rather than one.

limits · Memory The Fibonacci numbers are one row of a table. The losing heaps of Fibonacci Nim with the factor two replaced by one, three, four and up to eight. Each factor gives a different integer sequence: the powers of two, the Fibonacci numbers, and four more with no common name.

The family the Fibonacci numbers belong to

Fibonacci Nim lets a player take at most twice what the last one took, and the heaps the opener loses are the Fibonacci numbers. Two is an arbitrary number. At one the losing heaps are the powers of two, at three and four and five they are four more sequences, each with a linear recurrence whose lag is twice one less than the factor — until the factor is six, where the pattern stops.

impartial · Fibonacci nim The gap is the factor. The separation condition of each factor's greedy numeral system: the smallest gap between the indices of two terms. It is one at factor one, two at factor two — Zeckendorf's non-adjacency — and the factor itself at every factor swept.

What the numerals knew

Every factor in the Fibonacci Nim family gives a numeral system, and the rung below predicted its separation condition would be the lag of the recurrence the losing heaps satisfy. It is not. The gap is the factor — one at c = 1, Zeckendorf's two at c = 2, and c at every factor to eight — while the lag goes 1, 2, 4, 6, 8, 11, 14, 17 and leaves its own pattern at six. The numerals then solve every one of 194,480 states, cap and all.

impartial · Fibonacci nim A gap that widens without bound. Both savings as the number of components grows, enumerated where possible and given by the closed forms beyond.

One half multiplies, the other adds

The rung below priced the two halves of a substitution licence on sums of two Cram boards and predicted that the first half's saving would grow with the number of components while the second's would not. It is right, and both halves have closed forms: the component licence saves s^(k−1)/k and the subposition licence k·s over a shape count that never moves.

limits · Universes Two clauses, and what each is about. Four rulesets against the two clauses of the condition. A ruleset passes both or the one-number-per-component recipe fails on it, and the two clauses fail for different reasons: locality is about the state proposed, isolation is about the rule.

Two clauses and a third question

A component can carry its own rule when two things hold: its moves are a function of what it carries, and a move in it leaves every other component alone. Two rulesets built to fail one clause each are both caught on a named witness. The four real games sort exactly — every one the recipe gets right fails no clause, every one it gets wrong fails one — and the two clauses still miss something, because Fibonacci Nim and a held pass fail the same clause and only one of them can be repaired.

limits · Memory

Named alongside it

The objects these essays reach for when they reach for this one.

ImpartialDecompositionEnumerationExhaustive searchInvariantComponentCounterexampleFibonacci nimSprague–GrundyZeckendorf representationContextFibonacci

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