Combinatorial Game Theory
who wins, and by how much
Concept

Wall — where it appears

One side of a thermograph, tracing what a player gets moving first at each level of tax on moving. A bend in one is where a follow-up stops mattering, and two positions whose walls bend at different heights cannot cancel each other.

Named by 12 essays across 3 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.

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 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 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 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 The switch formulas, off the hypothesis they were stated for. Values born by day three with exactly one option a side, split by whether both options are numbers. On the twenty-one that satisfy the textbook hypothesis the midpoint and half-gap formulas are exact; on the 146 that do not, the same formulas read off the two stops instead hold about three quarters of the time.

A fight with no midpoint

The mean of {a | b} is the midpoint and the temperature is half the gap — on the twenty-one values of day three where a and b are numbers. One hundred and forty-six others have the same shape and not the hypothesis, and the repair that suggests itself, reading the two stops instead of the two options, holds on about three quarters of them and no more.

values · Switches Where sente stops, by what the answer costs. The largest ambient temperature at which a local move is still answered, sorted by how deep the fight below the answer runs. When the answer ends the fight the crossover is the follow-up's temperature; when the answer starts another fight it is exactly half of it, and a third level does not halve it again.

The answer that starts another fight

A local move is answered while the ambient temperature stays below the follow-up's — and that rule, which this site has carried since the anchor opened, is exact only when the answer ends the fight. When the answer starts another one the crossover is exactly half the follow-up's temperature, on every position tested, and a third level of fight does not halve it again.

temperature · Sente Fifty-two errors, put to four instruments. The fifty-two discrepancies the lattice identity leaves on day two, counted by what distinguishes them. As values no two are the same; as pairs of stops there are seven; as means three and as temperatures three. Not one of them is a number, and only three are values born by day two.

Fifty-two errors and seven sizes

Day two is a lattice and a group and not a lattice-ordered group, and the fifty-two incomparable pairs it fails on leave fifty-two different error terms. Measured rather than listed, the fifty-two collapse: seven pairs of stops, three means, three temperatures, and a rule that predicts the temperature from the pair on forty-four of them.

values · Lattice Two feet, and the one that halves the crossover. The thermographs of two follow-ups, drawn to the same scale. The first has a right wall that leans in from the axis and its move is answered up to the follow-up's full temperature; the second has a wall rising vertically first and its move is answered only to half of it.

What the halving is a function of

A move is answered while the board is cooler than the follow-up's temperature — or half of it, depending which of two classes the position is in, and the classes were stated in terms of forms. They are a feature of one wall: whether the follow-up's right wall rises straight up before it leans. Fifty-five positions, no exception, and the crossover becomes something a reader can see.

temperature · Sente The wall as an envelope. A thermograph with each option's contribution to its wall drawn over it, built from that option's two stops alone. The wall is the envelope of those contributions and it bends where the envelope has a corner.

The bend is in the stops

The rung below reduced the whole stop reading to one question — does this wall bend? — and asked whether that could be answered from the options' stops instead of from a diagram. It can, in four lines, and it gives more than the bend: on all 1,459 non-number values born by day three the options' stops determine the entire thermograph. One day deeper it breaks, and every failure is a value with a bent-walled option.

values · Switches One quantity, two currencies. Each bent value with how far its temperature falls short of its stop reading and how far its stop falls short of the translation bound. The second is exactly twice the first.

The same number in two currencies

The rung below found bent-walled values falling strictly inside the translation bound and asked how far. The shortfall is the value's own hottest follow-up's temperature — exactly, on 400 of 408 pairs, and twice it on the other eight — which makes the whole error one expression. And it is the switches ladder's constant: a half there and a whole here, because a temperature is half a stop gap.

sums · Translation Where the two diagrams part. Pairs of hot day-two values with the true thermograph of their sum against the one made by adding the components' walls. Every pair parts, and every pair parts at the lower of the two temperatures.

The second bend is the boundary

Adding two thermographs wall by wall gives a diagram that is right at the mast and wrong below it. Over every pair of hot values born by day two, the added walls sit outside the true ones at every height — an outer envelope with the truth somewhere inside — and the two pictures separate at exactly the lower of the two temperatures, on all twenty-eight pairs. Above that height both components are still fights and the addition is exact; one sixteenth below it, every pair has parted.

temperature · Thermograph

Named alongside it

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

TemperatureThermographMean valueMastSwitchHot gameStopsFollow-upCoolingCounterexampleEnumerationDisjunctive sum

All concepts