Self-negative — where it appears
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
The colour swap pays through a mirror
A Clobber piece with its colours swapped is the same piece with the players exchanged, so a solver could store one and read the other off by negation. On its own that folds the piece table by about a tenth, because a piece's negative is rarely in the table at all. Beside symmetry it nearly doubles the fold on three boards of four, because on a board with an even side a mirror carries every position to the negative of another. And the opening of every such board is worth its own negative — a theorem from the mirror, not a search.
Named alongside it
The objects these essays reach for when they reach for this one.
NegationCanonical formEnumerationNimberAll-smallBirthdayBorn on dayClobberCounterexampleDecompositionDisjunctive sumDominated option