Combinatorial Game Theory
who wins, and by how much
Concept

All-small — where it appears

The class of positions in which one player has a move exactly when the other does, so that no side is ever ahead in moves. Every all-small position is an infinitesimal, so no number says anything about it, and Clobber is all-small by the shape of its rule rather than by inspection.

Named by 23 essays across 6 fields — each of them below, with the objects they name alongside it.

Smaller than every positive number, and not zero. Values that sit between zero and every positive number, each compared with zero and with 1/1024. Every relation drawn was computed by playing the difference, and one of them is confusion — neither greater, smaller nor equal. None of these is a number, and in a close game they are the entire margin.

Infinitesimals

Some positions are positive — Left wins them whoever moves first — and smaller than every positive number, including a millionth and a millionth of that. They are the values that decide close games, and the smallest of them is a single move's worth of nothing.

values · Infinitesimals How many ups, bracketed. Every position here is all-small, so no number says anything about it and the yardstick has to be ↑ instead. Each bar spans the multiples of ↑ the position lies between: the largest it is at least, and the smallest it is at most. Four of the seven are pinned to a single multiple of ↑; the rest keep a band that comparison cannot narrow, the widest being ∗ at four ups of slack.

How many ups

When every component of a position is smaller than every positive number, no number can decide it. What decides it is a count of ups — and comparison can pin that count down exactly, except when a star is present, when it cannot.

sums · Infinitesimals Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal.

A game where nobody can be ahead in moves

A blue stone beside a red one is a move for both players at once. So neither player can run out while the other still has something to do — and every value the game produces is smaller than every positive number, by the shape of the rule rather than by inspection.

positions · Clobber Tiny and miny: infinitesimals with a scale. Positions that are greater than zero and smaller than every positive number, and which are nevertheless strictly ordered among themselves — the larger the subscript, the smaller the value. Being smaller than everything positive is not one size of thing; it is a whole scale, and up sits above all of it.

Tiny, miny, and the sizes below every size

An empty two-by-four Domineering board is worth less than nothing and more than every negative number. It is not up, not down and not a fraction — it is a miny, and the minies come in sizes, strictly ordered among themselves below a floor no number reaches.

values · Infinitesimals Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal.

The class where nobody runs out first

Three stones in a row — blue, blue, red — and the position is worth exactly up. Clobber cannot produce anything else, because adjacency is symmetric — a player has a move precisely when the opponent does, and a game with that shape can never be worth a whole move to anybody.

values · Infinitesimals 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 The bracket of a sum, against the sum of the brackets. Two all-small positions, the interval of multiples of ↑ each lies between, those two intervals added coordinatewise, and the interval the sum actually lies between. The added one always contains the computed one — greater-than survives addition — so the bracket never widens under a sum. Where it narrows, the parts were each too vague to pin down and the sum is not.

When the ups add

Atomic weight brackets do not add over a sum — they bound it. Over all 120 pairs from a fifteen-game family the sum's bracket came out exactly the sum of the parts' brackets 56 times, strictly narrower 64 times, and wider never; and the rule separating the two is one line long, because every one of the 54 pairs with a pinned part is exact and only 2 of the other 66 are.

sums · Infinitesimals 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 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 Clobber: every value smaller than every number. Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal.

One row of Clobber

Every string of blue, red and empty squares up to eight long — 9,840 rows — carries one of only 111 values, and every one of them is infinitesimal. A third of the rows are worth exactly zero. Six alternating stones are worth zero and eight are worth a form that takes four lines to print, so the values do not simplify as the row grows: they explode, while the row stays trivial to describe.

positions · Clobber How much company equality needs. Each row restricts the quantifier in the definition of equality to the games named, and counts how many of the 22 values born by day two survive as distinct. The bar is the same number drawn; the jump from nine numbers to four games is the whole argument.

Equal in this company

Equality quantifies over every game there is, and the quantifier can be made smaller. Restricted to a company of nine numbers, the twenty-two values born by day two collapse to seventeen; restricted to four games — nought, one, minus one and star — they stay twenty-two, and no three of the four will do. The company that decides equality is tiny, and it has to contain a star.

limits · Universes 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 Seven infinitesimals, added to a whole day. Each row adds one infinitesimal to every value born by day three. The stops never move — that is what being smaller than every number means — the temperature moves a handful of times, and the outcome class moves in a quarter of the additions.

What an infinitesimal does to a fight

Adding a number moves both stops by exactly itself. Adding something smaller than every number moves neither — across 10,318 additions to a whole day of values, not once — and the outcome class changes anyway, 2,622 times. It changes at exactly one kind of position: the ones with a stop sitting on zero, which is where the numbers have run out of things to say.

sums · Translation Every Domineering shape up to four squares. The pieces a partly played board falls into, each with the value the recursion gives it. Left plays vertically and Right horizontally, so a tall shape is worth something positive and a wide one something negative, and the quarter turn is not a symmetry of the game.

A board that is a sum of its regions

A table of rectangles is a table about the openings. A partly played Domineering board is not a rectangle, and evaluating one means splitting it into pieces no domino can straddle, looking each piece up and adding. The catalogue of 104 shapes does it correctly on every one of the 3,227 positions of a 3×4 board it covers — and among the shapes are two worth an up and a down, which no rectangle ever is.

positions · Domineering What an approximation is worth. Every pair of Clobber rows up to six squares, judged twice: by their up-brackets and by the comparison itself. The bracket is never wrong where it speaks, and most of what it declines to answer has no answer.

When the bracket decides

A Clobber row's value is an all-small game nobody can hold in their head, so the practical answer is the up-bracket: a pair of integers between which its atomic weight must lie. As an approximation it is worth exactly what it settles — 1,585 of 7,875 pairs of rows are ordered by it, every one of those orders is right, and of the 6,290 it declines, 4,222 have no answer either.

positions · Clobber 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.

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.

values · Stops 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 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.

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.

sums · Negation 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.

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.

positions · End-Nim Which clause of the rules produces which kind of value. Every combination of pawn-file clause in range, sorted by the kind of value it produces. Files where both pawns can advance are all-small and their values are nimbers and infinitesimals. A file where one pawn is stuck behind a friendly piece gives the other side free moves and is worth an integer. A file whose middle square can be held stops the other pawn the moment somebody reaches it, and is worth a switch — a position both players want to move in. The dictionary is read off the evaluation rather than asserted.

What has to break before a pawn is worth a number

Every value the blocked-file model can hold is an infinitesimal, and the reason is one sentence about the move rule rather than anything about pawns. Break that sentence — a pawn stuck behind a friend, a square only one side can hold — and integers, switches and positions worth fighting over arrive at once.

applied · Chess

Named alongside it

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

InfinitesimalStar (∗)Exhaustive searchTemperatureAtomic weightUp (↑)Canonical formComparisonMean valueDomineeringThermographCooling

All concepts