Combinatorial Game Theory
who wins, and by how much
Concept

Genus — where it appears

A Grundy value with a tail of misère values attached, which classifies the games behaving like Nim and says little about the rest. Its two halves are both load-bearing: neither the base nor the tail determines the genus of a sum on its own.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

What reversing the ending destroys. Everything that makes normal play tractable is a theorem about who moves last, and misère play contradicts every one of them. The positions are unchanged; the means of evaluating them is gone, and what replaces it is far heavier.

Misère play

Change one word — the player who cannot move wins — and the games are identical, the strategies are not, and almost every theorem of the normal-play theory stops being true. It is the cheapest possible modification and the most expensive.

limits · Misère play The genus of Kayles ·77, heap by heap. One row per heap: the genus symbol, the misère outcome it implies, and whether the symbol is one a Nim heap has. A game all of whose positions are tame is played in a misère sum exactly as Nim is; a single wild heap ends that, and the normal-play Grundy value gives no warning of which heaps those will be.

Tame and wild

The genus is a Grundy value with a tail — the misère values of the position with 0, 1, 2, … heaps of ∗2 added — and a game is tame when its symbols are the ones Nim heaps have. Computed here for seven games over heaps 1 to 14: Kayles goes wild at heap 5, Dawson's chess at heap 9, the octal game ·6 at heap 7, and heaps 3 and 11 of Dawson's chess are both worth ∗2 under normal play with only one of them tame.

limits · Genus What the two outcome classes of the parts settle. For each pair of outcome classes, the set of outcomes the sums actually took. A cell with one letter is a pair of classes that decided the answer; a shaded cell with several is a pair that did not. Both conventions have ambiguous cells — the difference is that normal play repairs them with values and misère play has nothing to repair them with.

Two misère outcomes are not enough

Knowing who wins each part does not say who wins the sum. Over 676 sums built from a pool of twenty-six positions, nine of the sixteen pairs of outcome classes settle the answer under normal play and not one of the sixteen settles it under misère — and the nine that work are theorems about a value being zero, which is exactly the thing misère play does not have.

limits · Misère play 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.

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.

limits · Genus Two ways to be certain and ignorant at once. Ten positions from two games that both terminate for reasons no bound comes out of. Sylver Coinage's proof counts something that goes down and can be counted; the hydra's counts an ordinal, which cannot, and the last column shows what that difference is worth.

Two ways to end with no bound

Sylver Coinage and the hydra are both guaranteed to finish and neither will say when. The difference is that one of them carries its own bound: every move in Sylver removes at least one gap, the gaps can be counted in a moment, and over ten openings the longest play uses every single one. The hydra has no decreasing quantity a solver can hold — three hydras of five nodes each take seven chops, twenty-one, and a number past two hundred and seventy-nine that this machine never reaches.

limits · Termination 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.

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.

limits · Genus How two genus symbols make a third. The composition rule for genus symbols, stated with its cases and checked on every pair of heaps of nine games. The base exclusive-ors, the sum is fickle only when every component is, and the symbol follows from those two.

The rule the symbols follow

Two genus symbols make a third by three lines and no lookup table: the base exclusive-ors, the sum is fickle only when every component is, and the symbol follows. Checked on 252 pairs across nine games it is right on 238 — and the fourteen failures are exactly the fourteen pairs with a wild heap in them, which is the boundary the genus is defined up to arriving as a measurement.

limits · Genus A function on the wild side too. Every pair of heaps filed by the pair of genus symbols it is made from. No file holds two different sums, including the sixteen with a wild symbol in them.

A function with no formula

The rung below's composition rule is exact on tame pairs and wrong on all fourteen wild ones, which looked like an exact boundary. Two heaps further it is wrong on 34 of 35 and right on one — Kayles' five and nine — so the boundary was a boundary of the pool. What survives is stronger and stranger: the pair of symbols still determines the sum on the wild side, and no rule of that shape describes it.

limits · Genus Not closed, and not nearly. Where the table's answers live. None is a symbol a wild heap carries; some are symbols tame heaps carry; the rest are symbols nothing in the sweep carries.

The wild side does not close

The rung below asked for the wild composition table and for two things about it: whether the wild genus symbols form a small closed set, and whether that set is a misère quotient in disguise. Building the table needed a wider sweep — nine counters a heap gives a diagonal rather than a table — and both answers are no. Not one of the twelve entries is a symbol any wild heap carries, and two wild heaps added together are tame two thirds of the time.

limits · Genus The genus of Kayles ·77, heap by heap. One row per heap: the genus symbol, the misère outcome it implies, and whether the symbol is one a Nim heap has. A game all of whose positions are tame is played in a misère sum exactly as Nim is; a single wild heap ends that, and the normal-play Grundy value gives no warning of which heaps those will be.

Closing the wild side

The twenty-two wild genus symbols are not closed under addition, and the rung below offered two answers: a monoid nobody had guessed, or no algebra at any size. Neither. Five of the six games with wild heaps close at three or four heaps, with closures of two to five symbols, and the sixth is still growing.

limits · Genus The same rules under the convention they were posed in. Dawson's chess under misère play, which is how Dawson posed it. Under normal play every position of the game collapses onto one of a handful of nimbers however large the heaps are allowed to get. Under misère play the positions that behave alike form classes whose number grows with the heap limit, and a heap carries a genus rather than a value. The first wild heap is where the two accounts stop resembling each other, and the classification doubles at exactly the limit that admits it.

The convention Dawson actually used

Dawson published his puzzle as a problem where running out of moves loses you the game, and every compact result about ·137 is about the other convention. Under his own, nine values become a classification that doubles the moment a wild heap enters the range, and a heap stops carrying a number at all.

history · Dawson A staircase, not a slope. The number of misère classes of Dawson's chess positions as the largest heap allowed rises from three to 16, computed with positions of at most three heaps and tests of at most two. The count stays flat for several heaps at a time and then jumps.

A staircase, not a slope

With the misère closure cut forty-fold, Dawson's chess can be classified at heaps far beyond nine. The count of classes is a staircase: six from heap three to eight, twelve from nine to twelve, seventeen from thirteen to sixteen. Normal play steps once in that range, from four to eight at heap thirteen, where a Grundy value of four first appears. Misère play steps there too, and once more at heap nine, where normal play does not move at all — the first wild heap. Heaps eleven, fifteen and sixteen are also wild and move nothing.

complexity · Misere cost

Named alongside it

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

Grundy valueMisère quotientMisère playOctal gameExhaustive searchImpartialNimOutcome classDawsonDisjunctive sumNormal playTame

All concepts