Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Pairing Strategy for Tic-Tac-Toe on the Integer Lattice with Numerous Directions. | A Pairing Strategy for Tic-Tac-Toe on the Integer Lattice with Numerous Directions Klay Kruczek Mathematics Department Western Oregon University Monmouth OR 97361 kruczekk@ Eric Sundberg Mathematics Department Occidental College Los Angeles CA 90041 sundberg@ Submitted Jan 16 2008 Accepted Nov 10 2008 Published Nov 24 2008 Mathematics Subject Classification 91A46 Abstract We consider a tic-tac-toe game played on the d-dimensional integer lattice. The game that we investigate is a Maker-Breaker version of tic-tac-toe. In a MakerBreaker game the first player Maker only tries to occupy a winning line and the second player Breaker only tries to stop Maker from occupying a winning line. We consider the bounded number of directions game in which we designate a finite set of direction-vectors Sc Zd which determine the set of winning lines. We show by a simple pairing strategy that Breaker can win this game if the length of each winning line is at least 3 S . It should be noted that Breaker s winning strategy can be used as a drawing strategy for Player 2 in the strong version of this game. 1 Introduction The traditional game of 3 X 3 tic-tac-toe is a type of positional game. In particular 3 X 3 tic-tac-toe is an example of what we call a strong positional game. In general a positional game 1 is a two-person game with complete information played on a hypergraph V H where V is an arbitrary set called the board of the game and H is a family of subsets of V called the winning sets. The two players Player 1 and Player 2 alternately occupy previously unoccupied elements of V. In a strong positional game the first player to occupy all points of some winning set wins. We say that Player 1 has a winning strategy if no matter what Player 2 does Player 1 can follow that strategy to win the game. If neither player has a winning strategy we say that the game is a draw. The traditional game of 3 X 3 tic-tac-toe is an example of a strong positional game where the nine .
