Negation — the series · Combinatorial Game Theory

Every position has an exact opposite. A position beside its negative, which is the same game with the players exchanged, and the sum of the two. The sum is worth zero every time — a second-player win — because the second player can answer each move with its mirror image. It is the fact that makes values a group, and it is what lets one position be subtracted from another.

Turn the board through a right angle

A two-by-four Domineering board is worth something no number can express, and Right is ahead on it. Turn a second board through a right angle, put the two side by side, and the total is exactly zero. Every position has an exact opposite, and that single fact is what makes subtraction — and therefore comparison — possible at all.

part 1 · sums

The values born by day three that are their own negatives. Every game satisfies G + (−G) = 0, so a game equal to its own negative satisfies G + G = 0 — it has order two. The nimbers do, and they are not the only ones: a switch symmetric about zero is unchanged by negation, and so is anything whose Left options are the negatives of its Right options. Each row carries the value, whether it is a nimber, and its outcome.

The values that are their own negatives

Every game satisfies G + (−G) = 0, so a game equal to its own negative satisfies G + G = 0 — it has order two in a group whose elements otherwise have infinite order. The nimbers do. So does ±1, on sight. Over the 1,474 values born by day three there are 30 of them and only four are nimbers, every one of the 900 sums of two is another, and the equality test and a symmetry of the written form agree 1,474 times out of 1,474.

part 2 · sums

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.

part 3 · sums

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.

part 4 · sums

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.

part 5 · sums

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.

part 6 · sums

What identifies two subsets

Every subset of a day gives a self-negative value by mirroring it, and 1,793 subsets of day two give thirty values. The collapse happens in two stages with different characters: domination takes the 1,793 to 96 antichains and is a theorem, and the rest of the reduction takes 96 to 30 and is concentrated almost entirely on two values — nought, which has an exact description, and star, which has none.

part 7 · sums

A mex with no impartial game in it

The rung below described the zero fibre of the mirror map and left star's fourteen undescribed. Star's fibre is 'some element is at least nought, and none is at least star' — and the two rules are one rule: the mirror value is the least nimber no element of the set reaches. That is a mex, in a construction built entirely from partizan values.

part 8 · sums

Four moves, three arguments. Every first move in the sum, with what answers it and how many cases of each the census holds.

The case that was supposed to be hard

The mex rule for the mirror construction was to be proved by induction, and the step flagged as needing care was the one where an option is incomparable with the nimber. There is no induction: the argument is four lines, and incomparability is what makes two thirds of the cases go through — because a fuzzy sum is a first-player win and the first player is the opponent.

part 9 · sums

Twenty-six other values

The mex rule accounts for sixty-six of the ninety-six antichains and is silent on the other thirty. Every one of those thirty is worth a self-negative value born by day three — and the same mex, run over that family instead of over the nimbers, is exact on all ninety-six. The nimber rule is this one cut short after its fourth member.

part 10 · sums