Combinatorial Game Theory
who wins, and by how much
Concept

Number avoidance — where it appears

The theorem that moving inside a numeric component is never right while any other component has a move. It is a rare thing in this subject: a theorem that names a move rather than merely a winner.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Why nobody moves in the number. A hot position added to a number. Left wins the sum whoever moves — but only by moving in the fight. Spending the move on the number instead hands the position back as a first-player win, with Right to move, which throws the win away. The theorem says this is always so, and here it is happening.

Numbers avoid numbers

In a position with a number in it and anything else, the number is never the right move. That is a theorem rather than a heuristic, and it is the closest this subject comes to advice a player can carry into a real game.

values · Numbers Comparing two positions means playing a third. Pairs of positions with the relation between them, and the game whose solution decided it. There is no way to compare two games by looking at them: the question “is G at least H?” is answered by playing G − H and asking who wins, which is a search, and its cost is counted here beside each answer.

What a move is worth to the player making it

The gain from a move is the option minus the position it was played from — and that is a game rather than a number, so two moves can be incomparable instead of one of them being best. Temperature is what happens when the largest of those games is asked for a single number.

values · Incentives Every day-three value, moved by every quarter. Four claims counted over 3,000 translations — 120 values, each moved by every quarter from −3 to 3: that adding a number leaves the temperature alone, that it shifts both stops by exactly itself, that the interval between the negated stops is where the two players are confused, and that the only failures of the last are on its endpoints.

What a number does to a fight

Adding a number to a position moves everything and changes nothing: over three thousand translations the temperature never once shifted and both stops moved by exactly the number added, every time. What the number decides is whether the fight is worth having — and the interval where the two players are confused is exactly the open interval between the negated stops, right in all 2,890 cases away from its endpoints and wrong in 110 that are all on them.

sums · Translation

Named alongside it

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

Hot gameTemperatureComparisonMean valueMove selectionNumbersSwitchThermographBorn on dayCold positionConfusionDifference

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