Combinatorial Game Theory
who wins, and by how much
Concept

Codes — where it appears

The compact notation for an impartial take-and-break game, one digit per number of counters removed, its bits saying how many heaps a move may leave. It is what makes a whole family sweepable — a code is a game, and enumerating codes enumerates games.

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

Two counters, not one. The four periods of the odd-saltus class against the two base-three counters, with which each follows.

Two counters, and one displaced term

The rung below found four Grundy sequences in the odd-saltus class and asked which term each displaces and whether the digits predict it. They do — but there are two base-three counters and not one, chosen by whether a heap of one can be taken away. And there are three sequences rather than four: the fourth is the third with three isolated values, and was counted separately because its period had not settled.

impartial · Hexadecimal Not rare at all. How many hexadecimal codes whose sequence settles have a stretch of heaps before the pattern begins.

A pattern that has not started yet

A pre-period was supposed to be rarer in this family than a defect. Two hexadecimal codes in five have one, 321 have a pre-period longer than their own period, and the code the rung below found slow takes fifty-four heaps to settle rather than two blocks — which is also the account of three defects the rung below recorded and could not explain.

impartial · Hexadecimal

Named alongside it

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

EnumerationGrundy sequencesHexadecimalImpartialOctal gamePeriodicitySaltusApproximationClosed formCounterexampleEventual periodicityGrundy value

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