Combinatorial Game Theory
who wins, and by how much
Concept

Cold game — where it appears

A position neither player gains by moving in, whose value is a number and which is left until last. Its value is a number, so an infinitesimal added to it changes nothing and the numbers alone decide who wins.

Named by 19 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 Col and Snort on a path of four. One graph, two games, and one word of difference between the rules. Col forbids painting next to your own colour, which makes every move a small self-harm and drives the values towards numbers. Snort forbids painting next to your opponent's, which makes every move a land grab and drives them towards fights. Both values are computed from the same recursion.

One board, two rules

Col forbids painting next to your own colour. Snort forbids painting next to your opponent's. One word differs, the boards are identical, and the values that come out are not the same kind of object.

positions · Colouring Cutcake: every value an integer. The value of an m by n cake, for every small m and n. Left cuts down, Right cuts across, and neither player ever gains by moving — so nothing is ever at stake, every value is a whole number, and the number says exactly how many spare moves one player has.

Cutcake, where every value is a whole number

A partizan game in which no position is ever worth a fraction, a star or a fight. Every value is an integer, the integer is a count of spare moves, and the pattern it follows is decided by binary digits.

positions · Cutcake {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 Every position of both games, and where the fights are. Every colouring of every graph in the library, under both rules, with the temperature of each computed. A position with a positive temperature is one both players want to move in. Col has none anywhere; Snort has them on every graph, and the hottest grows with the graph.

One rule makes it cold, the other hot

Col forbids painting next to your own colour and Snort forbids painting next to your opponent's. Over all 540 colourings of six graphs, not one Col position has anything at stake and 61 Snort positions do — the difference between a game that is a count and a game that is a series of fights, produced by one word.

positions · Colouring Maundy Cake: the pieces must be equal. The same cake as Cutcake, cut by the same two players, with one extra rule: a cut must divide the cake into equal pieces, and every piece stays in play. The values are still whole numbers, but the arithmetic that decides them is not Cutcake's — it counts prime factors rather than binary digits.

Maundy Cake

Cutcake with one word added: a cut must divide the piece into equal parts. The values are still whole numbers, and the rule this site has been repeating about them is false — over all 1,296 cakes to 36×36 the largest-odd-divisor account has 946 counterexamples. What survives is a count of prime factors, and it says who wins without saying by how much.

positions · Cutcake 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 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 The fight never runs backwards. Each of the 22 values born by day two, drawn from its right stop to its left stop — what Right gets moving first, and what Left gets moving first, once the fight has been played out to a number. Every bar runs the same way. The left stop is never below the right one, which is what "both players are trying to improve their own position" amounts to, and the cold rows, where the two coincide, are drawn as a single point.

The fight never runs backwards

Left's stop is never below Right's — in every one of 1,780 distinct values, computed twice by two independently written routes, with nothing that disagreed anywhere. The inequality is what makes a mean value well defined and a fight a fight; and where it collapses to equality, 433 of the 460 cold positions turn out not to be numbers at all.

values · Stops Shove strips, and what each is worth. A shelf of positions with the value the recursion returns beside each. Every one is a number: Shove has no hot positions at all, which is unusual for a partizan game and is the first of the essay's three claims.

Nothing worth fighting over

Shove is a strip of coins beside a cliff, and both players have completely different moves. Every one of its 728 positions is worth a number, so nobody ever wants to move; the winner is the owner of the coin furthest from the cliff, in all 728; and the number the board is worth is not the sum of its coins — that reading is exact on 126 strips and wrong on 588 of the other 602.

positions · Shove 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 What the reduction collapses. Each reduced form with the values that reduce to it. The largest class is the one that reduces to zero and it holds every infinitesimal on the list, which is exactly what the reduction is for — against a hot background, none of them is distinguishable from nothing.

What is left when the small change is thrown away

Canonical form answers a demanding question: which positions are interchangeable inside every sum whatever. A player with a hot board does not have every sum — an infinitesimal difference cannot decide anything against a genuine fight — so there is a coarser question with an exact answer. The reduced canonical form takes the 1,474 values born by day three to 61, with 292 of them collapsing to zero, and it is a homomorphism on all 8,100 pairs tested only when a second pass is made.

sums · Reduced form 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 What each move is worth to the player making it. For each position: every incentive, whether they are all strictly negative, whether the position is a number, and its temperature. The middle two columns are two different computations of the same fact.

Nobody wants to move here

A position is a number exactly when every move loses ground for the player making it. The test never mentions numbers, it disagrees with the ordinary one on none of the 1,474 values born by day three — and the reason a position fails it is not that somebody wants to move. It is that somebody cannot afford to wait.

values · Incentives Push and Shove over every strip up to 6 squares. The same strips under both rules. A cliff lets coins fall off and a wall does not, and the census says what that one clause is worth: both games are entirely made of numbers, they never agree on a value, and the obvious board-reading is right far more often under the wall than under the cliff.

The other way to move a row

Shove has a cliff and Push has a wall, and that is the whole of the difference. Both games make every one of the 728 strips up to six squares a number, so neither ever has anything worth fighting over — and the two rules do not agree on the value of a single position. The obvious board-reading is exact on 446 strips under the wall and on 140 under the cliff, and 486 strips contain a coin its owner cannot move at all.

positions · Push 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 The same population counted twice. The temperature scale over the positions this site has enumerated, once with every position counted and once with every distinct value counted. The two disagree about how much of the subject is hot, about what the commonest hot temperature is, and about whether a number is the usual thing for a position to be worth.

How hot a real position is

Counted one value at a time, a tenth of the subject is hot. Counted one position at a time — every board this site has enumerated, all 11,397 of them — it is a twentieth, two thirds of the positions are worth numbers outright, and ten of the seventeen rulesets never produce a hot position at all.

temperature · Cold What a Domineering region is worth, by size. Every connected shape of at most six free squares, sorted by whether its value is a number, an infinitesimal distance from a number, or hot. Hot shapes do not appear at all until four squares, and the hottest shape of six is the two-by-three rectangle.

Which shapes are worth fighting over

Forty-four of the 104 Domineering regions of at most six squares are worth numbers and the rest are not, and the rung below said no visible property of a shape predicts which. Half of that is wrong: a region only one orientation fits in is a whole number, on all eleven of them, for a reason a reader can supply in a sentence. The other half stands, and thirty-three shapes are what makes it stand.

positions · Domineering

Named alongside it

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

TemperatureHot gameExhaustive searchThermographMean valueSwitchNumbersStar (∗)InfinitesimalCoolingCanonical formPartizan

All concepts