Taming the wild in impartial combinatorial games

Abstract

We introduce a misere quotient semigroup construction in impartial combinatorial game theory, and argue that it is the long-sought natural generalization of the normal-play Sprague-Grundy theory to misere play. Along the way, we illustrate how to use the theory to describe complete analyses of two wild taking and breaking games.

35 pages. Accepted by INTEGERS: The Electronic Journal of Combinatorial Number Theory. See also http://www.plambeck.org/oldhtml/mathematics/games/misere

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• Compound Node-Kayles on Paths
• Absolute Combinatorial Game Theory
• Misère canonical forms of partizan games
• Dead Ends in Misere Play: The Misere Monoid of Canonical Numbers
• Misere quotients for impartial games: Supplementary material
• Algorithms for lattice games
• An Investigation of Partizan Misere Games
• Misère Games and Misère Quotients
• Analysis of misere Sprouts game with reduced canonical trees
• Binary dicots, a core of dicot games
• Restricted developments in partizan misere game theory
• Misere quotients for impartial games
• The strange algebra of combinatorial games
• The structure and classification of misère quotients
• Simultaneous Combinatorial Game Theory
• The Sprague-Grundy Functions of Saturations of Misere Nim
• Equality Classes of Nim Positions under Misère Play
• Affine stratifications from finite misère quotients
• Invertibility modulo dead-ending no-P-universes
• Lattice point methods for combinatorial games
• Invertible Elements of the Misere Dicotic Universe