Combinatorial Game Theory
who wins, and by how much
Generator

The thermograph of {5 | 1}

The thermograph of {5 | 1}
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.

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.

34 essays call thermograph. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

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. Temperature

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.

 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. Temperature

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.

 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. Temperature

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.

 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. Values

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.

 How much changes hands, against how much is at stake. Two ways of choosing where to move, run against optimal play over every board from a pool of three components. Biggest-first takes the component where the most changes hands, which is the count in every endgame book; hottest-first takes the one with the highest temperature. They disagree on most of these boards, the count costs points more often, and — the difference that matters — the count sometimes loses more than the largest temperature on the board, which is the bound the theory's rule is guaranteed to keep. Temperature

Big is not the same as hot

A player sizes a move by how much changes hands when it is played, which is the number in every endgame book. The theory sizes it by temperature. On a plain switch the two agree exactly, so nothing shows; on a move with a follow-up they come apart, and the count gives up more than the guarantee the theory's rule carries.

 {5 | {4 | 0}} beside one other fight. A local position and a single switch, played out together at each of several ambient temperatures. The middle columns are what optimal play does: whether it opens the local fight, and whether it answers when the opponent opens it. The answer stops being forced at a temperature the local position alone does not name. Temperature

Sente is a fact about the rest of the board

A move that must be answered is called sente, and the word is used as though it described the local position. It does not. The same fight is answered while the rest of the board is quiet and ignored once it is busy, and the crossover — measured by solving the whole board at every temperature — sits at the follow-up's own 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. Temperature

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.

 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. Temperature

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.

 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. How it was found

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.

 {4 | 0} played out in a stack of 5 coupons. An idealised environment: coupons worth a fixed step less each, which either player may take instead of moving in the game. The rows are the line optimal play takes over the whole board, in order. What the game turned out to be worth is set beside its mean value, and the coupon the players stopped at beside its temperature — two quantities measured from the play, and two computed from the thermograph. Temperature

An environment made of coupons

Beside the game sits a stack of coupons worth 4, 3, 2, 1, 0, and a player may always take the top one instead of moving. Play the whole thing out and two quantities the theory computes are measured instead: {4 | 0} comes out worth exactly 2, its mean value, and the coupons stop at 2, its temperature. For {10 | {9 | 1}}, whose temperature is 1, they stop at 7/2 — because what the stopping coupon measures is the hottest temperature anywhere in the tree.

 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. Particular games

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.

 the temperature that bounds the loss. 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 it costs

A rule that is never right and cannot be far wrong

Playing the hottest component is not optimal — over 440 measured lines it costs something on 17 of them. What makes it worth having is that the cost is bounded by the temperature, provably, and that the same test run with the ordering reversed breaks the bound on 54.

 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. Temperature

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.

 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. The two marks on the base line are the stops — what each player gets by moving first and playing the fight out with no tax charged at all. Values

Where the fight stops

Keep taking the biggest thing on offer until somebody is left facing a number, and the number reached is a stop. Every position has two of them, one for each player moving first, and between them they say how a fight ends — except for the part of it no number can reach.

 Small Domineering boards and what they are worth. Every value here was computed from the moves rather than looked up. Even on boards this small the values are switches and infinitesimals rather than numbers, which is the ordinary situation for a partizan game and the reason the theory needs more than arithmetic. Particular games

The values of every small board

Thirty Domineering rectangles, every value computed from the moves rather than looked up. The 1×n row obeys a formula and the 2×n row does not: its outcomes run L N N R three times over and then 2×13 comes out worth exactly 0, and its temperatures climb to 19/16 and fall back without settling.

 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. Temperature

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.

 Amazons on one line. A one-dimensional Amazons board: an amazon slides along the row and shoots along the row, and the square the arrow lands on is burnt for the rest of the game. The whole board fits in a sentence, and the values it produces are already of several different kinds. Particular games

Amazons on one line

A board one square high is small enough to evaluate completely: every strip from two to ten squares with one amazon a side is 37,886 positions taking 81 distinct values, and every one of them is an integer, a switch, a number plus a star, or a bare star. Not one is a fraction — and forcing the arrow onto the square just vacated, which takes a freedom away rather than adding one, produces 1,196 that are.

 Every day-three value, moved by every quarter. Four claims counted over 3,000 translations — 120 values, each moved by every quarter from −3 to 3: that adding a number leaves the temperature alone, that it shifts both stops by exactly itself, that the interval between the negated stops is where the two players are confused, and that the only failures of the last are on its endpoints. Sums and comparison

What a number does to a fight

Adding a number to a position moves everything and changes nothing: over three thousand translations the temperature never once shifted and both stops moved by exactly the number added, every time. What the number decides is whether the fight is worth having — and the interval where the two players are confused is exactly the open interval between the negated stops, right in all 2,890 cases away from its endpoints and wrong in 110 that are all on them.

 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. Values

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.

 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. Values

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.

 The same row, cut and toppled. Rows of blue and red drawn once and evaluated twice: as a Hackenbush string, where a player cuts an edge of their own colour and everything above it falls, and as Toppling Dominoes, where a player knocks one over and everything on the chosen side falls. Both values are computed by the same recursion from the two rulesets. Particular games

Topple it from either end

A row of blue and red is the picture this site opens with, and under Hackenbush's rules it is always a number. Knock the pieces over instead of cutting them — everything on the chosen side falls — and 480 of the 510 rows up to eight pieces stop being numbers. The two games agree on sixteen rows, every one of a single colour, and the temperature of the hottest row climbs by exactly a half for each domino added.

 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. Temperature

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.

 The numbers each position is confused with. Each row is a position. The bar runs from its right stop to its left stop; the filled part is the set of numbers the position is genuinely confused with, computed one comparison at a time. The two coincide except at the ends, and a position whose stops meet is confused with nothing at all even when it is not a number. Values

The numbers it is confused with

A position is confused with a number when neither is at least as good as the other, and the set of such numbers is an interval. It is exactly the open interval between the two stops: over 36,850 comparisons the rule is wrong nowhere it speaks, and the 2,596 comparisons it declines are precisely the ones at an endpoint, where the position and the number differ by an infinitesimal.

 Which end of the interval is open. Every value born by day 3 compared with each of its own two stops — 2,948 comparisons, each one a search over the difference. The two rows are mirror images because the day is closed under negation, and the small number in each row is the exception class: the 352 values whose two stops coincide, for which the left stop is the right stop and the law has nothing to bite on. Values

Which end of the interval is open

The confusion interval is open at both ends, and the two ends are not the same kind of open. At its own left stop a position can be below the number, confused with it or equal to it, and — over 2,948 comparisons — above it exactly thirty-three times, every one of them a value whose two stops are the same number and whose left end is therefore also its right one.

 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. Temperature

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.

 How hot a background has to be. Every pair of values born by day two that share a reduced canonical form, added to backgrounds of seven temperatures and three means — 609 comparisons in all — with the count of pairs whose outcome the swap changes. Safety is not monotone in the background's temperature, so the threshold the question asks for does not exist; every one of the 48 changes is at a position with a stop exactly on nought. Sums and comparison

How hot a background has to be

The reduced canonical form throws away infinitesimals, and the rung below asked for a bound: how hot must the rest of the board be for the discarded part not to matter? There is no such bound. Safety is not monotone in the background's temperature — an eighth is safe, a quarter is not, two is safe again — and the quantity that does decide it is not a temperature but a stop.

 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. Values

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.

 Where the thirty sit on the scale. How many of the values equal to their own negatives carry each temperature. Fifteen sit at nought, fourteen are hot, and one is a number — so the subgroup runs the whole length of the scale rather than living at the cold end of it. Sums and comparison

The thirty that cancel themselves

Thirty values born by day three are equal to their own negatives, and every one of them has a mean of exactly nought and two stops that are exact opposites. Neither property comes close to picking them out — 496 values of the day have a mean of nought — and half of the thirty are hot, one of them the hottest value the day produces.

 Two thousand values and no temperature. Every End-Nim row of up to six heaps, with its temperature. Not one of the 5,460 is hot, every one of them is worth an infinitesimal, and the 361 worth numbers are 361 rows worth nought. Particular games

A game with nothing at stake

The rung below explained a coincidence with a claim it did not compute: that End-Nim carries no hot self-negative values. The census says something stronger. Not one of 10,919 rows across three shapes of board is hot, every one of them is worth an infinitesimal, and the 361 rows the earlier census called numbers are 361 rows worth nought. The reason is one line of the rule, and 2,693 distinct values sit underneath it.

 The bend decides it. Whether the stop reading gives the mean and temperature, against whether either wall bends below the meeting point. Both off-diagonal cells are empty on all 138 values. Values

The bend is the condition

The rung below offered a description of the class its stop reading is exact on — neither wall bends below the meeting point — and a route to proving it: that a bend happens precisely when some option is neither a number nor an infinitesimal. The first is exact on all 138 values, both directions, no exception. The second is half right: every bent value has such an option and 49 unbent ones do too. And the eight apparent exceptions to the first turn out to be a bookkeeping convention.

 The same temperature, and four different departures. Positions with a temperature of one whose follow-ups are worth different amounts, with the coupon at which the players leave the environment. The departure tracks the follow-up. Temperature

How big the answer is

The rung below found every early departure from a coupon stack caused by a position with a follow-up, and could not say more: its follow-ups were all of a similar size, so the class it measured was one bit. A pool graded by follow-up size answers it. With the position's own temperature held at one, the departure runs from coupon 1 to coupon 3.5 as the follow-up's temperature runs from 1 to 4 — and over the whole grid the players leave at the larger of the two temperatures.

 The error is half the follow-up. Each bent value's true temperature, the temperature the stop reading gives it, the difference, and half the temperature of its hot option. The last two columns agree on every value. Values

Half a follow-up out

The rung below settled which values the stop reading is wrong about — the ones whose walls bend — and left the size of the error unmeasured. It is not bounded by anything readable off the diagram; it equals something readable off the diagram. On all thirty-two, the mean and the temperature are each out by exactly half the follow-up's temperature, and the temperature is always read too low.

 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. Values

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.

 Two numbers instead of an expression. The mean and the temperature of a position, set against its brace expression. The pair is readable in a way the expression is not and is half its length, and it is not exact: most of the games born by day three share a pair with some other game. The cost is not abstract — two games written the same way here can be separated by adding an ordinary small position to each, which is exactly the test a table of outcomes fails one level down. How it was found

What two numbers cannot tell apart

A thermograph summarises a position in a mean and a temperature — nine characters against the brace form's twenty-two, and readable in a way the expression is not. It is also not exact: 1,454 of the 1,474 games born by day three share a pair with some other game, 291 of them share one pair, and adding a star to two of those gives different winners.

The whole library · The position index · The figures that play back