In a standard 3x3 tictatoe 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 threedimensional 5x5x5 tictactoe?
How about 6x6x6 ?
Having originally thought that on a 5x5x5, for example, 5 in a row would be needed. But I see, what's really wanted is still any 3 in a row.
Just to get started, then, is the solution where the number in a row matches the size of one edge of the cube:
5 x 5 x 5:
Orthogonal (changing only one of x, y or z coordinates):
3*5*5 = 75
Diagonal along slice parallel to a face:
3*2*5 = 30
Body Diagonal:
4 (starting at any top corner)
Total: 109
6 x 6 x 6:
Orthogonal (changing only one of x, y or z coordinates):
3*6*6 = 108
Diagonal along slice parallel to a face:
3*2*6 = 306
Body Diagonal:
4 (starting at any top corner)
Total: 418
General:
Orthogonal (changing only one of x, y or z coordinates):
3*n^2 = 108
Diagonal along slice parallel to a face:
6*n = 306
Body Diagonal:
4 (starting at any top corner)
Total: 3*n^2 + 6*n + 4

Posted by Charlie
on 20141023 09:38:17 