Wythoff Nim

E653413

two-pile take-and-remove game

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.

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All labels observed (3)

Label	Occurrences
Wythoff Nim canonical	2
Wythoff game	1
Wythoff’s game	1

How this entity was disambiguated

This entity first appeared as the object of triple T7278066 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Wythoff Nim
Context triple: [Sprague–Grundy theorem, relatedTo, Wythoff Nim]

A. Sprague–Grundy theorem
The Sprague–Grundy theorem is a fundamental result in combinatorial game theory that assigns each impartial game position a nonnegative integer (its Grundy value), allowing such games to be analyzed and combined via nim-like addition.
B. Hackenbush
Hackenbush is a combinatorial game played on colored line-graphs, famous in recreational mathematics for illustrating concepts in game theory and surreal numbers.
C. Kayles
Kayles is a classic impartial combinatorial game in which players alternately remove one or two adjacent pins from a row, with the goal of making the last move.
D. Winning Ways for your Mathematical Plays
Winning Ways for your Mathematical Plays is a multi-volume book on combinatorial game theory that popularizes and systematically explores mathematical games and their underlying structures.
E. Conway’s soldiers
Conway’s soldiers is a mathematical puzzle and thought experiment in combinatorial game theory that explores how far checkers-like pieces can advance on an infinite grid under specific movement rules.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Wythoff Nim
Target entity description: 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.

A. Sprague–Grundy theorem
The Sprague–Grundy theorem is a fundamental result in combinatorial game theory that assigns each impartial game position a nonnegative integer (its Grundy value), allowing such games to be analyzed and combined via nim-like addition.
B. Hackenbush
Hackenbush is a combinatorial game played on colored line-graphs, famous in recreational mathematics for illustrating concepts in game theory and surreal numbers.
C. Kayles
Kayles is a classic impartial combinatorial game in which players alternately remove one or two adjacent pins from a row, with the goal of making the last move.
D. Winning Ways for your Mathematical Plays
Winning Ways for your Mathematical Plays is a multi-volume book on combinatorial game theory that popularizes and systematically explores mathematical games and their underlying structures.
E. Conway’s soldiers
Conway’s soldiers is a mathematical puzzle and thought experiment in combinatorial game theory that explores how far checkers-like pieces can advance on an infinite grid under specific movement rules.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	two-pile take-and-remove game ⓘ
hasAlternativeName	Wythoff game ⓘ linked to: Wythoff Nim Wythoff’s game ⓘ linked to: Wythoff Nim
hasAnalysisMethod	Sprague–Grundy theory ⓘ linked to: Sprague–Grundy theorem
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 ⓘ
hasMathematicalCharacterization	P-positions are described using the golden ratio ⓘ
hasMathematicalTool	Beatty sequences ⓘ
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 ⓘ
hasRelatedConcept	Beatty pair ⓘ cold game ⓘ octal games ⓘ
hasRelatedGame	Euclid’s game ⓘ Nim ⓘ
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 ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Sprague–Grundy theorem → relatedTo → Wythoff Nim ⓘ

Nim → hasVariant → Wythoff Nim ⓘ

Wythoff Nim → hasAlternativeName → Wythoff’s game ⓘ

linked to: Wythoff Nim

Wythoff Nim → hasAlternativeName → Wythoff game ⓘ