Sprague–Grundy theorem
E163078
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.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Grundy numbers | 1 |
| Grundy theorem | 1 |
| Grundy value | 1 |
| Sprague–Grundy function theorem | 1 |
| Sprague–Grundy theorem canonical | 1 |
| Sprague–Grundy theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1422430 — 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: Sprague–Grundy theorem Context triple: [Winning Ways for your Mathematical Plays, coversConcept, Sprague–Grundy theorem]
-
A.
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.
-
B.
Conway’s Game of Sprouts
Conway’s Game of Sprouts is a pencil-and-paper topological game in which players alternately connect dots with lines under simple rules, leading to rich combinatorial and mathematical analysis.
-
C.
On Numbers and Games
On Numbers and Games is a mathematical book by John H. Conway that introduces surreal numbers and explores combinatorial game theory in a rigorous yet playful style.
-
D.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
-
E.
Surreal numbers
Surreal numbers are a class of numbers introduced by John H. Conway that form an extensive ordered field encompassing the real numbers, infinite quantities, and infinitesimals within a unified framework.
- 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: Sprague–Grundy theorem Target entity description: 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.
-
A.
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.
-
B.
Conway’s Game of Sprouts
Conway’s Game of Sprouts is a pencil-and-paper topological game in which players alternately connect dots with lines under simple rules, leading to rich combinatorial and mathematical analysis.
-
C.
On Numbers and Games
On Numbers and Games is a mathematical book by John H. Conway that introduces surreal numbers and explores combinatorial game theory in a rigorous yet playful style.
-
D.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
-
E.
Surreal numbers
Surreal numbers are a class of numbers introduced by John H. Conway that form an extensive ordered field encompassing the real numbers, infinite quantities, and infinitesimals within a unified framework.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in combinatorial game theory ⓘ |
| alsoKnownAs |
Sprague–Grundy theorem
ⓘ
surface form:
Grundy theorem
Sprague–Grundy theorem ⓘ
surface form:
Sprague–Grundy function theorem
Sprague–Grundy theorem ⓘ
surface form:
Sprague–Grundy theory
|
| appliesTo |
finite impartial games under normal play
ⓘ
impartial combinatorial games ⓘ |
| associates | each game position with a nonnegative integer ⓘ |
| assumes |
finite game positions
ⓘ
no chance moves ⓘ normal play rule where last move wins ⓘ perfect information ⓘ two-player play ⓘ |
| characterizes | positions of impartial games by nonnegative integers ⓘ |
| classification | P-positions and N-positions via Grundy values ⓘ |
| defines |
Sprague–Grundy theorem
self-linksurface differs
ⓘ
surface form:
Grundy value
nim-value ⓘ |
| determines |
a position is losing iff its Grundy value is 0
ⓘ
a position is winning iff its Grundy value is nonzero ⓘ |
| field |
combinatorial game theory
ⓘ
combinatorics ⓘ |
| formalizes | equivalence of impartial games to Nim heaps ⓘ |
| foundationFor |
nim-heap decomposition of games
ⓘ
systematic analysis of impartial games ⓘ |
| historicalPublication |
Patrick Michael Grundy's work in the 1930s
ⓘ
Roland Sprague's work in the 1930s ⓘ |
| implies |
every impartial game under normal play is equivalent to a Nim heap
ⓘ
outcome of a sum of impartial games can be determined by xor of Grundy values ⓘ |
| namedAfter |
Patrick Michael Grundy
ⓘ
Roland Sprague ⓘ |
| relatedTo |
Sprague–Grundy theorem
self-linksurface differs
ⓘ
surface form:
Grundy numbers
Hackenbush (impartial variants) ⓘ Kayles ⓘ Nim ⓘ Wythoff Nim ⓘ impartial game theory ⓘ octal games ⓘ |
| statesThat |
each impartial game position has a unique nonnegative integer Grundy value
ⓘ
every position in a finite impartial game under normal play is equivalent to a heap of Nim of some size ⓘ the Grundy value of a disjunctive sum of impartial games is the bitwise xor of the Grundy values of the components ⓘ |
| usedFor |
analyzing impartial games
ⓘ
combining impartial games via nim-sum ⓘ deciding winning and losing positions ⓘ |
| usesConcept |
disjunctive sum of games
ⓘ
minimum excluded value (mex) ⓘ normal play convention ⓘ |
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
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Input
Subject: Sprague–Grundy theorem Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Sprague–Grundy function theorem
this entity surface form:
Grundy theorem
this entity surface form:
Sprague–Grundy theory
this entity surface form:
Grundy value
this entity surface form:
Grundy numbers