Combinatorial Game Theory
who wins, and by how much
Generator

The misère quotient of Nim, heaps up to 2

The misère quotient of Nim, heaps up to 2
The misère quotient of Nim, heaps up to 2. Each row and column is a class of positions that no sum in this universe can tell apart, and each entry is the class their sum falls into. The shaded classes are the ones a player wants to hand over. Under normal play the same positions need only the Nim values; the extra classes here are what misère play costs.

Each row and column is a class of positions that no sum in this universe can tell apart, and each entry is the class their sum falls into. The shaded classes are the ones a player wants to hand over. Under normal play the same positions need only the Nim values; the extra classes here are what misère play costs.

9 essays call misere-quotient. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

The misère quotient of Nim, heaps up to 2. Each row and column is a class of positions that no sum in this universe can tell apart, and each entry is the class their sum falls into. The shaded classes are the ones a player wants to hand over. Under normal play the same positions need only the Nim values; the extra classes here are what misère play costs. Where it stops

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.

 The mirror strategy, and the ending that punishes it. A position beside its negative and the sum of the two, with the outcome under both endings. Under normal play the sum is worth zero every time, because the second player answers every move with its mirror image. Under misère the same answers are available and the same player runs out last, so every one of these sums is a first-player win — there is no zero, and no subtraction. Where it stops

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.

 Classes needed, as the heaps get bigger — Dawson's chess ·137. How many kinds of position there are, against how large a heap the universe allows. Under normal play the answer stops growing as soon as the Grundy values stop growing. Under misère play it does not stop, and every new class is a pair of positions that behave identically under normal play and differently under misère. How it was found

"Hopeless" was a claim about a method

Misère analysis was declared intractable in the 1970s, and the verdict was correct about what was being attempted. Quotients did not refute it thirty years later — they changed the question from a value per position to a monoid per universe, and the computed sizes show why the first question has no good answer.

 Kayles ·77: what each heap may be replaced by. Each heap with its genus, the Nim position carrying that genus, and the Nim heap a reader would substitute from the normal-play value alone. The two columns agree except where the genus belongs to no single heap — and there the second one is wrong, in sums, by exactly the amount the census counts. Where it stops

What a tame heap may be replaced by

Calling a heap tame is only worth anything because a tame heap can be swapped for a Nim position with the same genus in any misère sum. The swap is not always a single heap: Kayles' heap of eight is worth ∗ under normal play and carries the genus of 2 + 3, and substituting ∗ instead gets three of the twenty-eight Kayles pairs wrong.

 Classes needed, as the heaps get bigger — Dawson's chess ·137. How many kinds of position there are, against how large a heap the universe allows. Under normal play the answer stops growing as soon as the Grundy values stop growing. Under misère play it does not stop, and every new class is a pair of positions that behave identically under normal play and differently under misère. What it costs

The cost is in the closure, not in the positions

Under normal play, Dawson's chess needs four classes for every heap up to twelve, because its Grundy values stay at three or below there. Under misère play the same game needs six, then twelve, and the number rises with the universe rather than with the position — which is a different kind of expense entirely.

 How much company equality needs. Each row restricts the quantifier in the definition of equality to the games named, and counts how many of the 22 values born by day two survive as distinct. The bar is the same number drawn; the jump from nine numbers to four games is the whole argument. Where it stops

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.

 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. Sums and comparison

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.

 The genus of a sum. Every pair of heaps up to 9 counters, from nine impartial games, filed by the genus symbols of its two parts. The claim under test is that the file determines the answer; it does, and neither half of the symbol determines it alone. Where it stops

The genus of a sum

A genus symbol is meant to be carried one per heap, so that a solver never has to look at the heap again. That is a claim that the pair of symbols determines the sum's, and across nine games and 405 pairs it holds without exception — while the bases alone determine it in only 38 of 50 cases and the superscripts alone in 70 of 74. Both halves of the symbol are load-bearing, and two wild heaps can add to a tame sum.

 One misère outcome, searched. The number of positions a misère search visits to decide the outcome of a sum of k heaps of Dawson's chess, each heap at most 9, on a logarithmic scale: the average over the sums and the worst single sum, for k from one to eight. Normal play decides the same sums from 20 stored values. What it costs

A misère sum is searched, not added

Under normal play the outcome of a sum of heaps is a nim-sum of numbers already known: twenty stored values decide every sum of Dawson's chess with heaps up to nine, however many heaps it has. Under misère play each sum is a new position to search. One outcome costs six positions for a single heap, two hundred for four heaps and over five thousand for eight, and a table of every eight-heap outcome costs a hundred thousand. The misère quotient is the only thing that brings the price back down.

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