In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap...

no piles or a Hackenbush diagram with nothing drawn on it. The Sprague–Grundy theorem applies to impartial games (in which each move may be played by...

last object or to take the last object. Nim is fundamental to the Sprague–Grundy theorem, which essentially says that every impartial game is equivalent...

theory to assign nim-values to impartial games. According to the Sprague–Grundy theorem, the nim-value of a game position is the minimum excluded value...

from ordinal addition and ordinal multiplication. Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap...

successful landing and must be careful not to block themself. Hales–Jewett theorem m,n,k-game Number Scrabble Garcia, Dan. "GamesCrafters: Tic-Tac-Toe". gamescrafters...

has no moves left and loses The game can be analysed using the Sprague–Grundy theorem. This requires the heap sizes in the game to be mapped onto equivalent...

games are more difficult to analyze than impartial games, as the Sprague–Grundy theorem does not apply. However, the application of combinatorial game theory...

Roland Percival Sprague (11 July 1894, Unterliederbach – 1 August 1967) was a German mathematician, known for the Sprague–Grundy theorem and for being the...

the Sprague–Grundy theorem indicates that in the normal version any Cram position is equivalent to a nim-heap of a given size, also called the Grundy value...

through a formal application of Bayes' theorem; among them books by Gill and Henze. Use of the odds form of Bayes' theorem, often called Bayes' rule, makes...

which means that a player who cannot move loses. In the 1930s, the Sprague–Grundy theorem showed that all impartial games are equivalent to heaps in Nim,...

theory) Sperner's theorem (combinatorics) Sphere theorem (Riemannian geometry) Spin–statistics theorem (physics) Sprague–Grundy theorem (combinatorial game...

co-discoverers of the Sprague–Grundy function and its application to the analysis of a wide class of combinatorial games. Grundy received his secondary...

they rely on chance. Impartial games can be analyzed using the Sprague–Grundy theorem, stating that every impartial game under the normal play convention...

quotient is a commutative monoid that generalizes and localizes the Sprague–Grundy theorem for a specific game's rule set. In the specific case of misere-play...

snort. The development includes their scoring, a review of the Sprague–Grundy theorem, and the inter-relationships to numbers, including their relationship...

Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem Key figures Albert W. Tucker Amos Tversky Antoine Augustin...

Sprague–Grundy theorem, every position in a poset game has a Grundy value, a number describing an equivalent position in the game of Nim. The Grundy value...

comparatively straightforward to give a complete analysis using the Sprague–Grundy theorem. Thus the versions of Hackenbush of interest in combinatorial game...

Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem Key figures Albert W. Tucker Amos Tversky Antoine Augustin...

for the Sprague–Grundy theorem Paul Grundy (engineer) (1935–2013), Australian engineer Rebecca Grundy (born 1990), English cricketer Reg Grundy (1923–2016)...

information games, making it also analyzable by Nim because of the Sprague–Grundy theorem. For any rectangular starting position, other than 1×1, the first...

Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem Key figures Albert W. Tucker Amos Tversky Antoine Augustin...

piles whose nim-sum is zero, and this strategy is central to the Sprague–Grundy theorem of optimal play in impartial games. However, when playing only with...

games per turn. It is the fundamental operation that is used in the Sprague–Grundy theorem for impartial games and which led to the field of combinatorial...

results in a completely different game that can be analyzed with the Sprague–Grundy theorem. Blockbusting (game) A combinatorial game whose analysis has been...

n {\displaystyle K_{n}} ; it is a nimber, not a number. By the Sprague–Grundy theorem, K n {\displaystyle K_{n}} is the mex over all possible moves of...

This fact—that all closed games are determined—is called the Gale–Stewart theorem. Note that by symmetry, all open games are determined as well. (A game...

publication of the Sprague–Grundy theorem, the basis for much of combinatorial game theory, later independently rediscovered by P. M. Grundy. Weiszfeld, E...