Indistinguishability — where it appears
Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.
What survives misère play
Misère play destroys the value theory, and something much smaller grows back. Fix one game, look only at sums of its own positions, and the classes that behave alike form a monoid — computed here, and larger than the normal-play answer every time.
Misère play has no negatives
Put a position beside its own mirror image and answer every move with the mirror move. Under normal play the answerer wins and the sum is worth zero. Under misère the answerer still has every reply and loses because of it — so there is no zero, no subtraction, and no comparison, which is why the misère theory had to be rebuilt rather than adjusted.
Equal in every company
Two games are equal when no third game can tell them apart — a quantifier over every position there is, discharged by one finite test. A search over 184 contexts separates all 5,790 unequal pairs it is handed and still calls two different games the same, which is exactly why G − H = 0 is a theorem and an exhaustive search is not.
A pass is not a move
Put a single pass token on a Nim board and one clause decides everything. If it may be taken at any time — including as the move that ends the game — the value of the whole is the nim-sum with a one added, in all 120 positions swept: the pass is a heap of one. Forbid it as the final move and the value stops being a function of the nim-sum at all, and 3 and 1 + 2 come apart.
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.
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.
A quotient that identifies nothing
The dead-ending class is famous for quotients rather than comparisons, so the matched pair was asked the question its own subject is about. Neither quotient identifies a single pair of positions, and both are separated by exactly five addends — because a quotient is small when its universe is poor, which is a choice of company and not a property of a class.
What two numbers cannot tell apart
A thermograph summarises a position in a mean and a temperature — nine characters against the brace form's twenty-two, and readable in a way the expression is not. It is also not exact: 1,454 of the 1,474 games born by day three share a pair with some other game, 291 of them share one pair, and adding a star to two of those gives different winners.
Every chance but a certainty
The coin's number lands on a grid of dyadic fractions, and which points of that grid arrive is a count rather than a guess. Over the 1,474 values born by day three it reaches every one of the fifteen interior sixteenths and neither end — no position is ever certain. The groups sharing a chance run 1, 2, 4, 8 on the small pool, which looks like doubling, and 1, 2, 4, 20 on the large one, which is not.
What a component would have to carry
For a held pass to be decided by a summary of each component, the summary must separate every pair of components some company tells apart. The Grundy value does not — Nim 1 and Kayles 8 are equal games that a held pass separates beside a single Nim heap of two. Nor does the Grundy value with the component's own held-pass value: Kayles 3 and Kayles 6 agree on both and are split by a company of two Nim heaps. Over twenty-four components, fifteen classes against fourteen pairs, and the gap widens as the pool grows.
Two heaps of testing are enough
A misère quotient is computed by testing positions against positions, and the universe used to find twelve classes of Dawson's chess was every position of up to four heaps tested against every other — 511,225 outcomes. Varied one size at a time, the count stops growing at tests of two heaps and positions of three: 12,100 outcomes find the same twelve classes. The narrower universe the earlier essay drew did not merge anything; it held fewer positions. And the corner that is enough moves: for Kayles at heap twelve, two-heap tests miss a class.
Twelve classes, seven questions
Twelve misère classes of Dawson's chess were found by testing 715 positions against 715 others. Seven of those tests are enough to tell every class from every other — a greedy choice against a floor of four, since each test is one yes-or-no question. Kayles needs nine of 715 and Nim sixteen. The seven cost almost nothing to use and cannot be found without the whole closure, and they do not carry: the tests found with heaps up to seven tell apart only seven of the twelve classes with heaps up to nine.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchEqualityMisère quotientBounded universeDisjunctive sumEquivalenceOutcome classComparisonCanonical formMisère playNimComponent