Win Count - Futility Closet

Imagine a game of tic-tac-toe (noughts and crosses) played in three dimensions in an 8×8×8 cube. A player wins by marking some straight line of eight cells through the large cube. How many such winning lines are there?

	Select Click for Answer
This problem was proposed in the American Mathematical Monthly by A.L. Rubinoff of the University of Toronto in 1948 (Problem E773). It’s possible simply to count up the lines, but Leo Moser of the University of Manitoba hit on a simpler solution. Encase the 8×8×8 cube in a “shell” of unit thickness, and extend each winning line so that it pierces the shell at either end. Each winning line pierces two cells in the shell, and each cell in the shell is pierced by only one winning line. Thus each winning line corresponds to a unique pair of cells in the shell, and the number of winning lines is just half the number of cells in the shell: $\displaystyle \frac{10^{3} - 8^{3}}{2} = \frac{1000 - 512}{2} = 244.$ This solving strategy can be generalized to cubes of any size. (Via Ross Honsberger, Mathematical Morsels, 1978.)

Click for Answer