Combinatorial Game Theory
who wins, and by how much
Generator

Three partizan positions against every nimber, and not one match

Three partizan positions against every nimber, and not one match
Three partizan positions against every nimber, and not one match. Sprague and Grundy give every impartial position a single number that is complete: two positions with the same value are interchangeable everywhere. The three positions here are partizan — the two players have different moves — and each is compared against every nimber up to eight. Nothing is equal to anything. The magenta cells are worse than inequality: a position confused with a nimber is not above it or below it either, so no ordering could rescue the substitution.

Sprague and Grundy give every impartial position a single number that is complete: two positions with the same value are interchangeable everywhere. The three positions here are partizan — the two players have different moves — and each is compared against every nimber up to eight. Nothing is equal to anything. The magenta cells are worse than inequality: a position confused with a nimber is not above it or below it either, so no ordering could rescue the substitution.

2 essays call no-nimber-fits. 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

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

PositionWorth OutcomeDrawn in
a heap of 4, taking 1, 3, 4 ∗2 N Where the impartial theory stops
a heap of 5, taking 1, 3, 4 ∗3 N Where the impartial theory stops
a heap of 7, taking 1, 3, 4 0 P Where the impartial theory stops
2 × 3 2 | −1/2 N Where the impartial theory stops
2 × 4 {{2 | 0} | 0} R Where the impartial theory stops
3 × 3, 4 squares taken 1∗ L Where the impartial theory stops
a switch: Left to 1, Right to −1 1 | −1 N Confused is not the same as unknown · Where the impartial theory stops
a switch: Left to 2, Right to −2 2 | −2 N Where the impartial theory stops
Left to 0, Right to 1 1/2 L Confused is not the same as unknown · Where the impartial theory stops
Left to 2, Right to 0 2 | 0 N Where the impartial theory stops
star ∗ N Where the impartial theory stops
star two ∗2 N Where the impartial theory stops
that switch plus a star {1∗ | −1∗} N Where the impartial theory stops
the asymmetry at the root ↑∗ N Where the impartial theory stops
the asymmetry one level down {0, ↑∗ | 0, ∗} N Where the impartial theory stops
the asymmetry two levels down {0, {0, ↑∗ | 0, ∗}, ∗ | 0, ∗, ∗2} N Where the impartial theory stops
up ↑ L Confused is not the same as unknown · Where the impartial theory stops

Where it is called

Changing this generator changes every one of these figures.

Three partizan positions against every nimber, and not one match. Sprague and Grundy give every impartial position a single number that is complete: two positions with the same value are interchangeable everywhere. The three positions here are partizan — the two players have different moves — and each is compared against every nimber up to eight. Nothing is equal to anything. The magenta cells are worse than inequality: a position confused with a nimber is not above it or below it either, so no ordering could rescue the substitution. Sums and comparison

Where the impartial theory stops

Sprague–Grundy gives every impartial position one number, and the number is complete. The moment the two players have different moves no number works at all — not a harder one to compute, none — and three positions here are compared against every nimber to show it.

 Comparing two positions is playing their difference. To decide whether one position is worth at least another, subtract and see who wins moving second. It is the only definition of comparison the subject has, and it produces a partial order — some pairs come out confused, which no comparison of numbers ever does. Sums and comparison

Confused is not the same as unknown

Two positions can be neither greater, nor smaller, nor equal. That is a fourth relation with its own symbol, it is a fact about the pair rather than a limit of the method, and it is what makes a game worth playing — a position is a first-player win exactly when it is confused with zero.

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