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 Extending tic-tac-toe (Posted on 2014-10-23)
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?

 No Solution Yet Submitted by Ady TZIDON No Rating

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 10-min p&p based on a misinterpretation -- not a solution | Comment 1 of 6
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 2014-10-23 09:38:17

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