Combinatorial Game Theory
who wins, and by how much
Concept

Cooling — where it appears

Charging a tax on every move, which shrinks a hot position toward its mean and freezes it at its own temperature. Cool far enough and any position becomes a number; the height at which that happens is what how much is at stake means.

Named by 26 essays across 4 fields — each of them below, with the objects they name alongside it.

The thermograph of {5 | 1}. 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.

What is at stake

Some positions both players are desperate to move in, and some neither player wants to touch. The difference is a number — how much the move is worth — and it turns out to be the most useful single quantity for deciding where to play.

temperature · Temperature The thermograph of {5 | 1}. 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.

Reading a thermograph

A thermograph is two walls rising from a number line, closing in as the tax on moving increases, and meeting where the position stops being worth fighting over. Everything about a position's hotness is in the shape.

temperature · Thermograph 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 {2 | 0} + {2 | 0} — where the temperature goes. Three thermographs on one frame: two positions and their sum. The mean of the sum is the sum of the means, every time. The temperature is not: it is bounded by the hottest of the parts and is often far below it, so the number that says how much is at stake in a whole board cannot be got by adding up the parts.

Two hot fights that add to a cold number

The mean of a sum is the sum of the means, every time. The temperature is not — it is bounded by the hottest part and is often far below it. Two positions each worth fighting over can add to a plain number that neither player wants to touch.

temperature · Thermograph {2 | 0}, added to itself. The value of n copies of one position, for each n, beside n times its mean and the smallest distance between the two. The mean value theorem says that distance stays bounded however many copies are piled up — and the bound is the position's temperature, which is what makes the temperature a second genuine measurement rather than a diagram-reading convenience.

The same fight, eight times over

The mean is not roughly what a position is worth. It is the number that eight copies of the position stay close to — and the theorem is that the closeness does not decay as the copies pile up. The gap stops at the temperature, and stays there for ever.

temperature · Temperature A switch, its mean and its temperature. Positions of the form {a | b} with a above b: both players want to move there, so neither is settled. The bar spans the two options, the marked point is the mean the position is worth once the fighting is over, and the temperature is half the gap — which is exactly what moving first is worth.

Worth nothing, and worth fighting for

A switch is a position both players want to move in. Its average value can be zero while the difference between getting there first and second is enormous, and that gap is a second number every position carries.

values · Switches 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 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 The thermograph of {{5 | {3 | 1}} | 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. The marks partway up the walls are the bends: the heights at which the option holding a wall up stops holding it up. Each stands at the temperature of a follow-up. Every one of them is below the temperature of the position itself, which is where the two walls meet.

A thermograph with two bends

A wall bends where the option holding it up stops holding it up, and most drawn thermographs bend once. { {5 | {3 | 1}} | 0} bends twice, at 1 and 3/2, and neither height is its temperature of 7/4 — every bend lies strictly below where the walls meet. Over a census of 17,255 hot positions two levels deep, a second bend in one wall never happened once.

temperature · Thermograph Cooling by 1, and heating back. Each row is a position, its temperature, what it becomes when every move is taxed, and what comes back when the tax is refunded. The refund is not an inverse: a position whose temperature was below the tax has already frozen into a number, and heating a number does nothing at all.

Cooling by exactly one

Cooling is usually met as a way of reading a thermograph: a tax, and two numbers at the height of the tax. Applied as an operator it returns a position instead — and then the obvious question is whether heating gives the position back. Over the 1,474 values born by day three, 27 survive the round trip and 15 of those are numbers that never moved. Cooling throws away almost everything it touches.

temperature · Chilling A hot position, split into its mean and what is left. Each row is a position cooled by exactly its own temperature — the tax at which it stops being worth moving in. The result is the mean value with something small still attached, and the last column is that something, obtained by subtracting the mean from the cooled game rather than by inspection.

A number and a fight

Charge a position exactly what it is worth fighting over and the fight disappears, leaving the mean value — with something still attached to it. Over all 1,122 hot values born by day three the residue is smaller than every positive number, it is a star in 942 of them, and it is never nothing. So a hot game is its mean plus a fight plus a remainder that no number reports, and the remainder is what decides close games.

temperature · Dissociation Where a switch stops being a switch. The same Left option with the Right one raised past it. While the Left option is above the Right one both players want to move, the bar spans a real fight and the temperature is half the gap. Where the two meet the position is the number plus a star — no longer a switch, and not a number either. Above that the simplicity rule takes over: the value is the simplest number strictly between the options, and there is nothing to fight about.

When a switch is not a switch

A position {a | b} with numbers on both sides is a fight only while a is above b. Sweeping the boundary with a fixed at 2 and b climbing from −2 to 3 turns up three regimes rather than the two the definition suggests: eight fights whose temperature is exactly half the gap, one position at a = b that is 2∗ and is not a number, and beyond that numbers chosen by the simplicity rule — which on 8 of 12 sampled cases is not the midpoint.

values · Switches 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 Where 1,474 values sit on the scale. The temperature of every value in the pool, counted. The floor is −1 and only numbers are on it; the next rung up is 0, and everything there is a number with an infinitesimal added. Above that the scale is continuous and the counts thin out.

Below zero

Temperature is described as urgency and urgency has no obvious bottom, but the scale stops at −1 and only the numbers are on it. The rung above is exactly zero, and the 337 values born by day three that sit there are the ones a number cannot be told from — flat thermograph, nothing at stake, and an infinitesimal that no number can see. Two independent computations agree on the classification for all 1,474.

temperature · Cold 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 Where a sum's temperature actually lands. Every pair drawn from 45 hot positions, with the temperature of the sum set against the larger of the two temperatures. The bound is never broken and it is almost never used: 864 of the 1035 sums sit exactly at the maximum and 160 are frozen.

How cold a sum of hot games can be

The temperature of a sum is at most the largest temperature in it, and the bound leaves the whole interval below it open. Over 1,035 pairs the sums do not use that interval: 864 sit exactly at the maximum, 160 are frozen outright, and eleven land anywhere in between — every one of them with a component whose wall bends.

temperature · Thermograph 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 Five premises, and the step. The claims an induction would need, with what checks each. The last row is the step and nothing here checks it.

The premises an induction would need

The rung below settled by a grouping test that a position's crossover depends on its own temperature and its answer's and on nothing below them, and asked for the induction. The four paragraphs are not written here; the checking they would rest on is. The law holds at five levels, survives translation, heating and cooling — and none of that is the step.

temperature · Sente The counts, beside what happened next. The hottest Domineering region of each size with the number of shapes attaining it, and whether the next size was hotter.

A description, and not a detector

The rung below noticed that the count of shapes attaining the hottest temperature grew across a plateau and collapsed at the step, and proposed it as a way to read a plateau off a single size. The growth is exact — five plateaus, no exception — and the rule is impossible: five orbits precede a rise at seven squares and no rise at eight.

temperature · Cold The bound survives and the rule does not. The census against the widened sweep, with the bound and the two-line rule scored separately on each.

One expression proved, and one withdrawn

The census closed with three expressions exact on 1,440 pairs, and the rung above asked for derivations. The cold one has a four-line proof. The straight one has a threshold the census cannot determine — any constant between 4/3 and 3/2 fits it — and eight more addends of the same family break it on 38 pairs while leaving the bound above it untouched.

sums · Translation Two claims read as one. The proposed geometry separated into the claim the rung below established and the claim it needs but did not.

The four paragraphs prove something else

Three rungs earned the right to write the crossover law's proof as two straight walls meeting where the law says. The walls are straight — one of them everywhere, for a trivial reason, and the other only above the answer's own temperature. The crossover sits below that height, so the geometry holds nowhere the law is about, and where it does hold it proves the temperature instead.

temperature · Sente The residue of a sum is the hotter part's. Every hot day-two value added to every hot day-three value, 7,854 sums in all, sorted by what happens to the residues: when the parts differ in temperature, on all 4,712 such pairs, the sum's residue is the hotter part's alone; when they are equally hot the residues add on 880, the sum grows colder on 2,237 and cancels to a number on 25.

The hotter residue survives

Charge a hot position its own temperature and subtract its mean, and an infinitesimal is left — its residue. Add two hot positions and the means add, the temperatures do not, and the residues do something stranger than either: on every one of 4,712 pairs whose parts differ in temperature, the residue of the sum is the hotter part's residue and nothing of the colder's. Cooling is additive, and a part cooled past its own temperature is a bare number. Residues add only when two parts are exactly as hot, and even then the fights usually cancel into something colder. The summary forgets; the game does not.

temperature · Dissociation Colder exactly when the residues cancel. Equally hot pairs of hot values, 3142 from days two and three and 3000 sampled from day three, sorted by whether the sum stays as hot, grows colder or is a number, and by whether the two residues cancel. Every sum that stays as hot has residues that do not cancel, and every sum that grows colder or is a number has residues that do.

Colder exactly when the residues cancel

Add two fights of one temperature and the sum either stays that hot or grows colder, and which one happens is decided by the smallest thing either fight carries. The sum grows colder exactly when the two residues are negatives of each other — on all 3,142 equally hot pairs from days two and three and 3,000 sampled from day three, and by a two-line argument from cooling's additivity. What survives the cancellation is another matter. Group the colder sums by their parts' thermographs and the survivor differs in every group that holds more than one pair: {1 | 0} and {1 | 0, ∗} draw the same diagram and leave different infinitesimals behind. The diagrams fix only the survivor's temperature: nought, or exactly the lowest bend of either part.

temperature · Dissociation

Why cooling has to add

Cooling came through a sweep of 1,768 sums untouched while heating failed 263 times, and the sweep's own account was cautious: cooling is not additive in general, and a richer pool would break it. Aimed exactly where that account said to look — infinitesimals hidden under low temperatures, at taxes of a quarter and a half — 107,816 more pairs break nothing, because there is nothing to break. Cooling distributes over every sum of short games. Its freezing clause fires on one side of the equation and not the other in 6,032 of those pairs, and costs nothing every time, because what it hands back is the mean, and means add.

temperature · Heating

The bonus an integer is owed

Heating fails to distribute over a sum where cooling never does, and the repair proposed for it was to change what its exception returns: heat a number into a switch instead of leaving it alone. Run on the same 1,768 pairs, that repair fails on every one of them, beginning with nought plus nought. What adds is a heating that changes which numbers it exempts and pays the bonus on the moves an integer holds in reserve — Norton's product, and heating with no exception at all, which at a whole-number bonus is the same operator. Given heating's exception, cooling stops adding too.

temperature · Heating

Named alongside it

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

TemperatureMean valueThermographExhaustive searchSwitchHot gameDisjunctive sumStar (∗)InfinitesimalHeatingStopsAdditivity

All concepts