Similarly, 31 3, a player cannot satisfy the winning condition without violating the rules in a 1 tic tac toe. Take turns marking your space, and try to get three in a row across, down, or diagonally. Part of the lecture notes in computer science book series lncs, volume 8591. Tic tac toe 2 different games to play total 15 one player is odd numbers 1, 3, 5, 7, 9. Some of the systems are simple and some are rather complex. Lastly, the boards fields and squares are counted from the topleft to the bottomright, reading left to right, then top to bottom, like a book.
Skim though them and play with the ones that make sense then, try going back to the ones that seem a little more. Thus the game of fifteen and the game of tic tac toe are equivalent. Lastly, the boards fields and squares are counted from the topleft to the bottom right, reading left to right, then top to bottom, like a book. Math lottery most systems at least all of the ones in this book only require rudimentary mathematics. Analysis of a contemporary 2x2 super tictactoe board uci math. Play jungle vines throw 1 to practice multiples of numbers up to 6. Thus, if we are playing tic tac toe on this board, the winning tic tac toe lines correspond precisely to the triples of numbers that sum to 15, and hence the winning sets of the game of fifteen.
Pdf theoretical work on generalized tictactoe relevant to computer science and. A positional game is a game in which players alternate in taking possession. Although it has been shown that no one can ever win at tic tac toe unless a player commits an error, the game still seems to have a universal appeal. The first person to complete a line that adds up to 15 is the winner. Number the cells of a tictactoe board with the integers from 1 to 9 arranged in the form of a magic. The mathematics of playing tic tac toe david pleacher. We study the analogue of tictactoe played on a kdimensional hypercube of side n. More careful power series analysis shows that the difference.
It also includes an extensive analysis of tictactoelike symmetric lineforming. Thompson, k retrograde analysis of certain endgames. Hypercube tictactoe is a twoperson game played on an nk board i. In general, i started asking questions about the nature of the game. In ordinary tic tac toe, there are three winning patterns. Later, when we study the extensive form of a game, we will see that many other types of games may be modeled and described in strategic form. Above all you learn in this book, remember advance seasoned pick 3 players have one thing in common. Ill probably follow up with an analysis of that game, but for starters, id like to analyze the simple game of tictactoe. Evolution of noloss strategies for the game of tictactoe. On the other hand, the number of games of tic tac toe is rather small, so that it is possible to study all strategies and. Tictactoe american english, noughts and crosses british english, or xs and os is a. Ticktacktoe by martin gardner in scientific american book of mathematical puzzles.
The player who succeeds in placing three of their marks in a horizontal, vertical, or diagonal row is the winner. Recently, after playing a game involving dice, i asked myself the question. March, 1957, included in the scientific american book of mathematical. There is no nice mathematical theory into which we can place ttt.
It was published in 2008 by the cambridge university press as volume 114 of their encyclopedia of mathematics and its applications book series isbn 9780521461009 topics. Play tic tac toe against another player or the computer. On the other hand, the number of games of tictactoe is rather small, so that it is possible to study all strategies and. While it is true that the number of moves is very large, there are really only a few basic patterns because of symmetry. The mathematics of playing tic tac toe by david pleacher although it has been shown that no one can ever win at tic tac toe unless a player commits an error, the game still seems to have a universal appeal. To find all winning patterns in torus tic tac toe, all you have to do is try shifting each of the ordinary.