In a standard 3x3 tic-ta-toe there are 8 distinct possibilities to create “three in a row”- 3 horizontal, 3 vertical and 2 diagonal.

How many ways are there in a three-dimensional 5x5x5 tic-tac-toe?

How about 6x6x6 ?

Assuming my solutions are correct here's how to generalize to nxnxn with n≥4

7's are at the 2x2x2 corner cubes at each of the 8 corners (I'll divide by 2 at this step.)
7x8x8/2 = 224

11's are on the edges.  2x2x(n-4) for each of the 12 edges
2x2x(n-4)x12/2 = 264(n-4)

17's are at the face centers.  2x(n-4)² for each of the 6 faces
2x(n-4)²x6/2 = 102(n-4)²

26's are at the very center.  26x(n-4)³/2 = 13(n-4)³

For a grand total of 224 + 264(n-4) +102(n-4)² + 13(n-4)³

This formula checks for n=5 and n=6, assuming those answers were correct.

Posted by Jer on 2014-10-23 10:18:48