Combinatorial Game Theory
who wins, and by how much
Concept

Tax on moving — where it appears

A charge levied on every move, which is what cooling does and what turns a fight into a number. It is the same operation as cooling, seen as something charged rather than as something read off a diagram.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

Cooling {5 | 1}, one degree at a time. The same position under a rising tax on moving. Each bar is what Left gets moving first and what Right gets moving first, once every move costs the tax. The bars close as the tax rises, and at the temperature they meet — and from there on the position is worth its mean value and neither player wants to touch it.

Cooling

Charge a tax on every move and a fight becomes a number. The height of tax at which that happens is the temperature — so cooling is not a technique for finding the temperature, it is what the temperature is.

temperature · Temperature The thermograph of {6 | {5 | {4 | 0}}}. Temperature runs up the page and value across it. Each wall is where a player is willing to move once a tax of that much is charged per move; above the temperature at which they meet, neither wants to move and the position is worth its mean value. The height of the meeting point is what is at stake.

A thermograph is built from its options'

The diagram is not measured, it is computed — each wall is an extremum over the options' opposite walls, shifted by the tax. That construction is why a hot follow-up lowers the temperature instead of raising it, and why two fights with the same swing can differ by a factor of two in what is at stake.

temperature · Thermograph Which of the two operators distributes over a sum. Cooling and heating, each asked whether applying it to a sum is the same as applying it to the parts and adding. The pools are the values born by day two and a set of deliberately hot positions; the counts are of ordered pairs.

Cooling adds and heating does not

The two operators are presented as a pair, and they are not one. Cooling a sum is the same as cooling the parts and adding, on every one of the 1,768 pairs tried, at two taxes and on two pools. Heating fails on 263 — and not for the obvious reason: in every failure neither part and not the sum is a number, so the clause exempting numbers never fires at the top. It fires two levels down, where an option of a sum is one part's option plus the whole of the other.

temperature · Heating The whole Domineering catalogue, chilled. Every Domineering board this site evaluates, with its value, its temperature, and what cooling by exactly one does to it. Thirteen of the fifteen become cold — a number, or a number plus a star — and the two that do not are the two whose temperature was above the tax.

The operator chosen for one game

Chilling is cooling by exactly one, and the one is not derived from anything. It is chosen because Domineering mostly runs at that temperature — and measured against this site's whole Domineering catalogue it turns thirteen of fifteen boards into numbers or numbers plus a star, and warms thirteen of the fifteen back exactly. They are not the same thirteen: eleven boards do both, two freeze too far to be recovered, and two stay hot and come back on the nose.

temperature · Chilling 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 Adding two thermographs. Every pair drawn from the 15 values born by day two that are not numbers — 120 sums — with the two walls added pointwise and compared against the true diagram of the sum. The added walls are always an outer bound and the means always add; the whole diagram is right for 92 of the 120, and the 28 it is wrong for are exactly the pairs in which both components are hot.

When two thermographs can be added

The temperature of a sum is not the sum of the temperatures, and the natural repair is to add the whole diagrams instead. Over every pair of hot values born by day two the added walls always bound the true ones and the mast always comes out right — and the whole diagram is right exactly when at most one of the two components is hot, which is precisely the case a reader has no use for.

temperature · Thermograph How hot a day gets. The hottest value born by each of the first three days, with every temperature that occurs on it. Day one tops out at nought, day two at one, day three at two — a day buys exactly one degree — and the value attaining the maximum is unique each time. Each temperature was computed as the height at which that value's two thermograph walls meet. The temperatures of day three are exactly the half-gaps between the numbers born by day two, which is what puts a hole in the scale at 7/4.

How hot a day gets

A day of construction buys exactly one degree of temperature — nought, then one, then two — and the value attaining the maximum is unique on every day: ∗, then {1 | −1}, then {2 | −2}. The distribution underneath is not tidy at all: it peaks at a half, leans to the right of the peak, and has a hole in it at one and three quarters where nothing is born.

temperature · Cold

Named alongside it

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

TemperatureCoolingMean valueThermographExhaustive searchInfinitesimalFreezing pointHeatingStar (∗)SwitchAll-smallHot game

All concepts