Tic-tac-toe and Gomoku (five-in-a-row) are common games that have fairly mathematical rules. Players alternately choose points from some subset of a lattice and try to form a line segment of a certain length.
The Hales–Jewett theorem is a result from Ramsey theory that essentially says that however long the lines must be, a draw is not possible in a sufficiently large dimension.
Gomoku has been solved, constructively. (The first player wins.)
The game of Connect Four adds the additional element of "gravity". It has also been solved. (The first player wins on the standard board size, but not on some boards of slightly different size.)
