Combinatorial Game Theory
who wins, and by how much
Generator

Two numbers from the same tree

Two numbers from the same tree
Two numbers from the same tree. Four subtraction games, each with its Grundy sequence and its remoteness sequence. The Grundy value decides a disjunctive sum and the remoteness decides a conjunctive one; the only thing they always agree about is which heaps are losses for the player to move.

Four subtraction games, each with its Grundy sequence and its remoteness sequence. The Grundy value decides a disjunctive sum and the remoteness decides a conjunctive one; the only thing they always agree about is which heaps are losses for the player to move.

3 essays call remoteness-table. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.

Two numbers from the same tree. Four subtraction games, each with its Grundy sequence and its remoteness sequence. The Grundy value decides a disjunctive sum and the remoteness decides a conjunctive one; the only thing they always agree about is which heaps are losses for the player to move. Sums and comparison

How long it lasts

Move in every component at once and the game ends the moment any one of them does. Grundy values say nothing about that game; what decides it is the remoteness, a second number computed from the same tree that measures how long a component can be made to last. Over 2,268 positions the rule is right every time, and the two numbers determine each other in neither direction.

 The number nobody needs. The shortened selective compound — move in any non-empty set of components, and the game stops as soon as any one component stops — solved directly on 1,176 three-heap positions across four subtraction sets, with four predictions beside it. The suspense number was introduced for this compound and it is right; so are three cheaper things, and the shortening leaves the winner unchanged. Sums and comparison

The number nobody needs

The compound theory carries a third quantity — the suspense number — computed by the remoteness recursion with both preferences reversed, for the compound that stops as soon as any component stops. It governs that compound correctly. So does remoteness, so does the plain Grundy value, and the shortening does not change the winner on any of 1,176 positions.

 Four rules, asked of compounds made of two different games. Compounds whose two components come from different subtraction games, solved in full and compared with what each rule predicts. The three rules the compound theory supplies are exact on every position; the shortcut a reader carries instead is not. Sums and comparison

A compound of two different games

Every rule the compound theory has survives mixing exactly — the minimum-remoteness rule is right on all 5,184 mixed pairs and all 7,560 triples — and the reason is not that the rules are strong. It is that each of them reads one number per component, and a number does not remember which ruleset produced it. The thing mixing damages is the shortcut a reader carries instead.

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