Combinatorial Game Theory
who wins, and by how much
Concept

Orthodox accounting — where it appears

The Go endgame bookkeeping that plays the hottest region first and counts what each exchange was worth. It is legitimate because a sum is almost always as hot as its hottest part, which is a measurement rather than an assumption.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The endgame, accounted for. Several independent regions, each a fight with a settled value and a size. The account plays them hottest first: add up what each is worth on average, then add the largest amount at stake, subtract the next, and so on down. The exact value of the whole position is computed beside it, and the figure prints both.

The endgame, accounted for

Add up what each region is worth, then add the biggest thing at stake, subtract the next, and so on down. On a board of simple fights the result is exact — and the moment one region has a fight inside it, the account is out by a point.

temperature · Thermograph Is {{5 | 3} | {2 | −4}} double sente?. One local fight with a follow-up on each side, played out inside a sum with a switch whose temperature rises. The two middle columns are what optimal play does when each player opens the fight. Whether the opponent has to answer is settled by the ambient temperature and not by the shape, so the same position is double sente, sente for one player, and gote for both, at three different ambients.

Double sente is not a property of the position

A fight with a threat on each side is called double sente, as though the phrase named a shape. Swept against a rising ambient temperature, one such fight — { {5 | 3} | {2 | −4}} — is double sente up to an ambient of 1, sente for one player only from 3/2 to 3, and gote for both from 7/2, and the two band edges are the temperatures of the two follow-ups.

temperature · Sente The endgame, accounted for. Several independent regions, each a fight with a settled value and a size. The account plays them hottest first: add up what each is worth on average, then add the largest amount at stake, subtract the next, and so on down. The exact value of the whole position is computed beside it, and the figure prints both.

The first time it told somebody something

A theory earns its keep when it produces an answer nobody had. Temperature did that for Go endgames — the orthodox account gives a move order that is provably right and is not the one experience offers, and the position it is right about is small enough to check here completely.

history · Temperature Two operators that undo the same tax. Heating leaves every number alone; the warming operator leaves every number alone except an integer, which comes back with a star on it. That single clause is the whole difference between them, and it is what the Go endgame literature needs, because a chilled integer is usually a fight that has been frozen. The rows shown are the ones whose four entries fit in sixteen characters — a warmed day-three value runs to fifty-two, and the clause is legible only in the short ones.

The operator that puts the star back

Chilling is not invertible: it freezes, and 400 values born by day three collapse onto 29. Both heating and Norton's warming operator are exact right inverses of it — each lands back where it started, on all 400 — and they pick different preimages, differing on 396 of them and differing by exactly a star on 335. The clause that separates them is one line long and it is about the integers.

temperature · Chilling

Named alongside it

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

TemperatureThermographCoolingExhaustive searchGo endgameMean valueSenteSwitchAll-smallAlternating sumAmbient temperatureApproximation

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