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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Wythoff Nim canonical | 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
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Wythoff Nim Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.