Combinatorial Game Theory
who wins, and by how much
The collection

Every essay — page 22

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Sums and comparison

Real games break into independent parts. Adding them up is the whole method, and comparing them is how it is checked.

Cancellation, by exhaustion. The law checked on every triple of values born by day two, and then put to work: two Domineering regions compared directly and compared again inside a larger board. The comparison never changes, which is the licence every decomposition on this site is drawn under.

What can be struck out

From G + X = H + X it follows that G = H, in one line, by adding −X to both sides. It is the shortest theorem here and the most used: it is what makes comparing two boards region by region legitimate. Over 10,648 triples the hypothesis fires 484 times and the conclusion holds 484 times — and the licence expires in three separate directions, each of which loses the same axiom in a different way.

7 figures · Negation Two numbers from the same tree. Four subtraction games, each with its Grundy sequence and its remoteness sequence. The Grundy value decides a disjunctive sum and the remoteness decides a conjunctive one; the only thing they always agree about is which heaps are losses for the player to move.

How long it lasts

Move in every component at once and the game ends the moment any one of them does. Grundy values say nothing about that game; what decides it is the remoteness, a second number computed from the same tree that measures how long a component can be made to last. Over 2,268 positions the rule is right every time, and the two numbers determine each other in neither direction.

6 figures · Remoteness 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.

7 figures · Translation The number nobody needs. The shortened selective compound — move in any non-empty set of components, and the game stops as soon as any one component stops — solved directly on 1,176 three-heap positions across four subtraction sets, with four predictions beside it. The suspense number was introduced for this compound and it is right; so are three cheaper things, and the shortening leaves the winner unchanged.

The number nobody needs

The compound theory carries a third quantity — the suspense number — computed by the remoteness recursion with both preferences reversed, for the compound that stops as soon as any component stops. It governs that compound correctly. So does remoteness, so does the plain Grundy value, and the shortening does not change the winner on any of 1,176 positions.

8 figures · Remoteness 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.

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.

7 figures · Reduced form Four rules, asked of compounds made of two different games. Compounds whose two components come from different subtraction games, solved in full and compared with what each rule predicts. The three rules the compound theory supplies are exact on every position; the shortcut a reader carries instead is not.

A compound of two different games

Every rule the compound theory has survives mixing exactly — the minimum-remoteness rule is right on all 5,184 mixed pairs and all 7,560 triples — and the reason is not that the rules are strong. It is that each of them reads one number per component, and a number does not remember which ruleset produced it. The thing mixing damages is the shortcut a reader carries instead.

6 figures · Remoteness 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.

7 figures · Negation How much of the value the colon respects. Every form whose option lists are antichains of day-two values, grouped by the value it reduces to, and each group asked whether all its forms give the same ordinal sum with star. On 636 of the 640 groups they do.

What the colon respects

The ordinal sum reads the form of its base rather than its value, which is why the colon principle is stated for positions and not for values. Built over 9,604 forms it turns out to read the value on 636 of the 640 values that have more than one form, and the four it can tell apart are zero, one, minus one and star — the values born by day one, and no others.

7 figures · Ordinal sum Four candidate bounds, and the one that holds. Each candidate bound tested against every failing cut. One domino and the height of the cut both fail on six; twice the height holds on all twenty-two; the whole board's temperature fails on eighteen.

How wrong a nearly-independent split is

Treating a connected board as a sum of two halves is a claim, and the rung below counted how often it fails. This one prices it: over every vertical cut of every small Domineering rectangle the error is a game rather than a number, it is never in Right's favour, and it is bounded below by twice the height of the cut — a bound the height alone does not supply.

6 figures · Disjunctive sum A self-negative value costs a day. The values born by day three, by temperature, with the earliest self-negative one at each. The earliest is always the day after the temperature's own birthday, and the four temperatures with none are the four whose birthday is three.

A self-negative value costs a day

The rung below placed the thirty values equal to their own negatives on the temperature scale and asked whether being self-negative forces anything about when a value can be born. It does, exactly: the earliest self-negative value of temperature t is born the day after t itself, which accounts for the four temperatures that carry one and the four that carry none. The guess it offered — that the first value of each temperature is a self-negative one — holds at four temperatures out of five and is not the shape of the answer.

6 figures · Negation The stops stop working. Adding a switch to each of 120 day-three values, and asking whether the two stops of the sum are the sums of the two stops. They are on 330 of 720.

What a fight does to a fight

A number added to a position shifts both its stops by itself; an infinitesimal moves neither. A hot game does neither: over 720 sums the two stops add on 330 and are wrong on the rest. What survives is the mean, which adds on every one of the 720 — and the failure has a bound, since no stop is ever out by more than twice the smaller of the two temperatures, a bound 222 of the sums attain exactly.

6 figures · Translation No fifth value. Forms of day-three values built by adding day-three gift horses, and the ordinal sums they give. Over eighteen thousand forms and four followers, no value's forms disagree.

No fifth value

The colon reads a form rather than a value, and the rung below found the forms of a value disagreeing at exactly four of them — the values born by day one. It could only check forms whose options came from day two. Built one day deeper, by adding day-three gift horses to day-three values, eighteen thousand forms give no disagreement at all, while the same treatment still splits nought four ways. The class is about the width of the base's form and not the depth of its options.

6 figures · Ordinal sum The bound that needed no third number. The rung below's bound and the conjectured replacement, scored over 1,440 sums whose addend has a follow-up. The two-number bound holds everywhere and is attained; the three-number one fails.

A bound with one number too many

The rung below bounded how far a hot addend can drag a stop — twice the smaller of the two temperatures — over sums whose addends were all plain switches, and conjectured that an addend with a follow-up would need twice the smaller of three numbers. Over 1,440 sums with bent addends the two-number bound holds everywhere and is attained 358 times, and the three-number version fails on 66.

6 figures · Translation Self-negative values, day by day. How many values each day of the construction supplies and how many of them are their own negatives. Day four cannot be counted; a corner of it supplies 571.

At least five hundred and seventy-one

The rung below dated the self-negative values — the earliest of temperature t is born the day after t — and left the count to a day-four census nobody can run. The construction settles it instead: a value is its own negative exactly when its form is a mirror, so the subgroup can be built from subsets of the day below rather than sifted out of the day above. Day four supplies at least 571 against day three's 26, and the share of a day that is self-negative keeps falling.

6 figures · Negation Add, then reduce again. The arithmetic the homomorphism promises, measured: summing two reduced forms gives a reduced form on 88 per cent of pairs and needs a second reduction on the rest.

Add, then reduce again

The homomorphism promises that a sum's reduced form can be computed from its parts', and says nothing about what the operation is. It is addition followed by a second reduction — needed on 431 of 3,600 pairs of day-three values, and on not one of the 1,751 pairs with a cold part. What the second pass removes is an option that only becomes dominated once the two fights are side by side.

6 figures · Reduced form

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