Analysis Of Tic Tac Toe

Introduction
Tic Tac Toe is a classic game that everyone must have layed once in your life time. Tic Tac Toe, also known as noughts and crosses or Xs and Os. It is a paper-and-pencil game for two players, X and O, who take turns marking the spaces in a 3×3 grid. The player who succeeds in placing three of their marks in a horizontal, vertical, or diagonal row wins the game.
Tic Tac Toe is a game that needs stragegy, observation and tactics. Therfore players soon discovered that the best players will always lead to a draw. Hence Tic Tac Toe is often played among children.
Theory
The way to solve Tic Tac Toe is to use combinatorial Game Theory – which is a branch of mathematics that allows us to analyses all different outcomes of an event. This
By looking only for distinct positions, positions that are isometric under refection and rotation we can.
When considering only the state of the board, and after taking into account board symmetries (i.e. rotations and reflections), there are only 138 terminal board positions. Assuming that X makes the first move every time:
91 distinct positions are won by (X)
44 distinct positions are won by (O
3 distinct positions are drawn (also known as a cat's game)
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. While it is true that the number of moves is very large, there are really only a few basic patterns (because of symmetry). In fact, we will see that there are only 12 essentially different games.
But even though there are nine different spaces for your first move, there are essentially only three different places that you could put your opening move (as you will see below). In the language of Game Theory, Tic Tac Toe is a two-person contest that is finite (comes to an end), has no element of chance, and is played with “perfect information” (all moves being known to both players).
One variation uses 6 counters (one player uses three circles, another uses three triangles). Players take turns placing a counter on the board until all 6 counters are down. If neither player has won by getting 3 in a row, they continue to play by moving on each turn a single counter to any adjacent square (one variation allows only moves horizontally or vertically, while another allows one step in any direction, while still another permits any piece to be moved to any vacant cell). Try some of these. Many variations of moving counter Tic Tac Toe have been applied to 4x4 boards, each player using 4 counters and striving to get 4 in a row. There is an interesting 5x5 version called “Teeko.” There are eight markers in a Teeko game, four black and four red. One player, "Black" plays the black markers, and the other, "Red", plays the red. Black moves first, and places one marker on any space on the board. Red then places a marker on any unoccupied space; black does the same; and so on until all eight markers are on the board. The object of the game is for either player to win by having his markers situated in a straight line (vertical, horizontal, or diagonal) or square of four adjacent spaces. If neither player has won after the "drop" (when the eight pieces are put on the board), then it's attempted via the following method: The players alternate moving pieces one at a time, with Black playing