Tic-Tac-Toe

The usual game of tic-tac-toe (also called Ticktacktoe) is 3-in-a-row on a $3\times 3$ board. However, a generalized n-in-a-Row on an $u\times v$ board can also be considered. For $n=1$ and 2 the first player can always win. If the board is at least $3\times 4$, the first player can win for $n=3$.

However, for Tic-Tac-Toe which uses a $3\times 3$ board, a draw can always be obtained. If the board is at least $4\times 30$, the first player can win for $n=4$. For $n=5$, a draw can always be obtained on a $5\times 5$ board, but the first player can win if the board is at least $15\times 15$. The cases $n=6$ and 7 have not yet been fully analyzed for an $n\times n$ board, although draws can always be forced for $n=8$ and 9. On an $\infty\times\infty$ board, the first player can win for $n=1$, 2, 3, and 4, but a tie can always be forced for $n\geq 8$. For $3\times 3\times 3$ and $4\times 4\times 4$, the first player can always win (Gardner 1979).

See also Pong Hau K'i

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 103-104, 1987.

de Fouquières, B. Ch. 18 in Les Jeux des Anciens, 2nd ed.. Paris, 1873.

Gardner, M. ``Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another.'' Sci. Amer. 240, 18-28, Jan. 1979a.

Gardner, M. ``Ticktacktoe Games.'' Ch. 9 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, 1983.

Stewart, I. ``A Shepherd Takes A Sheep Shot.'' Sci. Amer. 269, 154-156, 1993.