Combinatorial Game Theory
who wins, and by how much
Concept

Self-negative — where it appears

A position equal to its own negative, so that whose turn it is stops mattering and the sum of two copies is nought. Every one is a second-player win or fuzzy with nought, and the values with this property are where a mex turns up in a partizan setting.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

sums · Negation 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.

sums · Negation

Named alongside it

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

Canonical formEnumerationNegationNimberBirthdayBorn on dayCounterexampleDisjunctive sumDominated optionGroupIncomparableInvariant

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