SOME MODELS OF COMBINATORIAL GAMES
Hassan Mohamed Nasr Aly Mostafa;
Abstract
Combinatorial games CGT are two-player game with perfect information and no chance. For example, the child’s play TIC-TAC-TOE. In combinatorial game theory, we make analysis of the combinatorial game, we describe the game and we try to predict the winner for any game. Studying the winning strategy for the combinatorial games is the most important part of combinatorial game theory. In this thesis, we construct some models of combinatorial games.
CGT arise in relation to the theory of impartial games, in which any play available to one player must be available to the other as well. One such game is Nim, which can be solved completely. Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses. In the 1930s, the Sprague–Grundy theorem [1] showed that all impartial games are equivalent to heaps in nim, this implies that major unifications are possible in games considered at a combinatorial level, in which detailed strategies matter, not just pay-offs.
In the 1960s, Elwyn R. Berlekamp, John H. Conway [2] jointly introduced the theory of a partisan game, in which the requirement that a play available to one player is available to both is relaxed. Their results were published in [2]. However, the first work published on the subject was Conway's book [3] , in which as been introduced the concept of surreal numbers and the generalization to games.
Combinatorial games are generally, by convention, put into a form where one player wins when the other has no moves remaining. It is easy to convert any finite game with only two possible results into an equivalent one where this convention applies. One of the most important concepts in the theory of combinatorial games is that of the sum of two games, which is a game where each player may choose to move either in one of the two games any point in the game, and a player wins when his opponent has no move in either game. This way of combining games leads to a rich and powerful mathematical structure.
the thesis consists of four chapters which are organized as follows:
CGT arise in relation to the theory of impartial games, in which any play available to one player must be available to the other as well. One such game is Nim, which can be solved completely. Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses. In the 1930s, the Sprague–Grundy theorem [1] showed that all impartial games are equivalent to heaps in nim, this implies that major unifications are possible in games considered at a combinatorial level, in which detailed strategies matter, not just pay-offs.
In the 1960s, Elwyn R. Berlekamp, John H. Conway [2] jointly introduced the theory of a partisan game, in which the requirement that a play available to one player is available to both is relaxed. Their results were published in [2]. However, the first work published on the subject was Conway's book [3] , in which as been introduced the concept of surreal numbers and the generalization to games.
Combinatorial games are generally, by convention, put into a form where one player wins when the other has no moves remaining. It is easy to convert any finite game with only two possible results into an equivalent one where this convention applies. One of the most important concepts in the theory of combinatorial games is that of the sum of two games, which is a game where each player may choose to move either in one of the two games any point in the game, and a player wins when his opponent has no move in either game. This way of combining games leads to a rich and powerful mathematical structure.
the thesis consists of four chapters which are organized as follows:
Other data
| Title | SOME MODELS OF COMBINATORIAL GAMES | Other Titles | بعض نماذج الألعاب التآلفية | Authors | Hassan Mohamed Nasr Aly Mostafa | Issue Date | 2020 |
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