Impartial achievement and avoidance games for generating finite groups
نویسندگان
چکیده
منابع مشابه
Impartial achievement games for generating generalized dihedral groups
We study an impartial game introduced by Anderson and Harary. This game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nimnumbers of this game for generalized dihedral groups, which are of the form Dih(A) = Z2 n A for a finite abelian group A.
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A b s t r a c t : For any finite group G, the DO GENERATE game is played by two players Alpha and Beta as follows. Alpha moves first and chooses x 1 C G. The k-th play consists o f a choice of x g E G S k _ 1 where S n = ~x 1 . . . . . X n } . Let G n = . The game ends when G n = G. The player who moves x n wins. In the corresponding avoidance game, DON'T GENERATE, the last player to move l...
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We announce misère-play solutions to several previously-unsolved combinatorial games. The solutions are described in terms of misère quotients—commutative monoids that encode the additive structure of specific misère-play games. We also introduce several advances in the structure theory of misère quotients, including a connection between the combinatorial structure of normal and misère play.
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We define the family of locally path-bounded digraphs, which is a class of infinite digraphs, and show that on this class it is relatively easy to compute an optimal strategy (winning or nonlosing); and realize a win, when possible, in a finite number of moves. This is done by proving that the Generalized Sprague-Grundy function exists uniquely and has finite values on this class.
متن کاملThree-player impartial games
Past efforts to classify impartial three-player combinatorial games (the theories of Li [3] and Straffin [4]) have made various restrictive assumptions about the rationality of one’s opponents and the formation and behavior of coalitions. One may instead adopt an agnostic attitude towards such issues, and seek only to understand in what circumstances one player has a winning strategy against th...
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ژورنال
عنوان ژورنال: International Journal of Game Theory
سال: 2017
ISSN: 0020-7276,1432-1270
DOI: 10.1007/s00182-017-0602-x