Combinatorial Game Theory
who wins, and by how much
Generator

Every impartial position is a Nim heap

Every impartial position is a Nim heap
Every impartial position is a Nim heap. A heap in a subtraction game, its Grundy value, and the Nim heap it is equivalent to. The equivalence is exact: the two positions have the same options up to value, so they behave identically in any sum, which is the Sprague–Grundy theorem.

A heap in a subtraction game, its Grundy value, and the Nim heap it is equivalent to. The equivalence is exact: the two positions have the same options up to value, so they behave identically in any sum, which is the Sprague–Grundy theorem.

2 essays call sprague-grundy. 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.

The positions it draws

22 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
heap 0 0 P Every impartial game is a Nim heap
heap 1 ∗1 N Every impartial game is a Nim heap
heap 2 ∗2 N Every impartial game is a Nim heap
heap 3 ∗3 N Every impartial game is a Nim heap
heap 4 ∗4 N Every impartial game is a Nim heap
heap 5 ∗5 N Every impartial game is a Nim heap
heap 6 ∗6 N Every impartial game is a Nim heap
heap 7 ∗7 N Every impartial game is a Nim heap
heap 8 ∗8 N Every impartial game is a Nim heap
heap 0 0 P Every impartial game is a Nim heap
heap 1 ∗1 N Every impartial game is a Nim heap
heap 10 ∗1 N Every impartial game is a Nim heap
heap 11 ∗2 N Every impartial game is a Nim heap
heap 12 ∗3 N Every impartial game is a Nim heap
heap 2 0 P Every impartial game is a Nim heap
heap 3 ∗1 N Every impartial game is a Nim heap
heap 4 ∗2 N Every impartial game is a Nim heap
heap 5 ∗3 N Every impartial game is a Nim heap
heap 6 ∗2 N Every impartial game is a Nim heap
heap 7 0 P Every impartial game is a Nim heap
heap 8 ∗1 N Every impartial game is a Nim heap
heap 9 0 P Every impartial game is a Nim heap

Where it is called

Changing this generator changes every one of these figures.

Every impartial position is a Nim heap. A heap in a subtraction game, its Grundy value, and the Nim heap it is equivalent to. The equivalence is exact: the two positions have the same options up to value, so they behave identically in any sum, which is the Sprague–Grundy theorem. Impartial games

Every impartial game is a Nim heap

Sprague and Grundy proved, independently and four years apart, that any position in any impartial game is equivalent to a single heap of counters. Not similar to one — equal to one, interchangeable with it inside any larger game.

 Grundy values for subtraction of 1, 3, 4. The Grundy value of every heap size for a take-away game, computed by the mex rule. A period, if the figure marks one, was found by searching the computed sequence rather than assumed — and where no period is marked, none was found in the range drawn, which is not the same as there being none. Impartial games

Take one, three or four

A heap and a list of legal takes. It is the smallest interesting impartial game there is, and the only family in the subject where eventual periodicity is not observed, not conjectured, but guaranteed — with a bound on when it must appear.

The whole library · The position index · The figures that play back