Wythoff Nim
E653413
Wythoff Nim is a classic impartial combinatorial game involving two piles of tokens, whose optimal play is characterized by positions related to the golden ratio.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf | two-pile take-and-remove game ⓘ |
| hasAlternativeName |
Wythoff game
NERFINISHED
ⓘ
Wythoff’s game NERFINISHED ⓘ |
| hasAnalysisMethod | Sprague–Grundy theory NERFINISHED ⓘ |
| hasApplication | illustration of number-theoretic structure in impartial games ⓘ |
| hasBeattySequenceProperty | P-positions correspond to complementary Beatty sequences for φ and φ² ⓘ |
| hasBoardRepresentation | two-dimensional lattice of pile sizes ⓘ |
| hasComplexityProperty | optimal play is computable in constant time from pile sizes using golden ratio formulas ⓘ |
| hasGameType |
finite impartial game
ⓘ
normal-play impartial game ⓘ |
| hasGeneralization |
Wythoff variants with restricted diagonal moves
ⓘ
multi-pile Wythoff-type games ⓘ |
| hasGoldenRatioValue | (1 + √5) / 2 ⓘ |
| hasGrundyValueProperty | P-positions have Grundy value 0 ⓘ |
| hasInventor | Willem Abraham Wythoff NERFINISHED ⓘ |
| hasMathematicalCharacterization | P-positions are described using the golden ratio ⓘ |
| hasMathematicalTool | Beatty sequences NERFINISHED ⓘ |
| hasMoveRule |
a move consists of removing any positive number of tokens from exactly one pile
ⓘ
a move consists of removing the same positive number of tokens from both piles ⓘ |
| hasOptimalPlayProperty | from any N-position there exists a move to a unique P-position ⓘ |
| hasOriginalLanguage | Dutch ⓘ |
| hasOriginalPaperTitle | A modification of the game of Nim ⓘ |
| hasPositionNotation | ordered pair of nonnegative integers (a, b) ⓘ |
| hasPositionType | normal play (last move wins) ⓘ |
| hasPPositionDefinition |
P-positions are pairs (a_k, b_k) with b_k − a_k = k and a_k = ⌊kφ⌋
ⓘ
P-positions are pairs (⌊kφ⌋, ⌊kφ²⌋) for k ≥ 0 ⓘ |
| hasPPositionExample |
(1, 2)
ⓘ
(3, 5) ⓘ |
| hasPPositionExample |
(4, 7)
ⓘ
(6, 10) ⓘ |
| hasPublicationVenue | Nieuw Archief voor Wiskunde NERFINISHED ⓘ |
| hasRelatedConcept |
Beatty pair
NERFINISHED
ⓘ
cold game ⓘ octal games ⓘ |
| hasRelatedGame |
Euclid’s game
NERFINISHED
ⓘ
Nim NERFINISHED ⓘ |
| hasResearchArea | combinatorial game theory ⓘ |
| hasStartingPosition | two nonnegative integer pile sizes ⓘ |
| hasSymbolForGoldenRatio | φ ⓘ |
| hasSymmetryProperty | P-positions are symmetric under exchanging the two piles ⓘ |
| hasTerminalPosition | both piles empty ⓘ |
| hasWinningCondition | player making the last legal move wins ⓘ |
| hasWinningStrategyDescription | move to the nearest P-position whenever possible ⓘ |
| hasYearOfIntroduction | 1907 ⓘ |
| hasZeroPosition | (0, 0) is a P-position ⓘ |
| usesConstant | golden ratio NERFINISHED ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.