Winning possiblities in 3D tic-tac-toe

Question

How many ways to get 3 in a row are there in a $3 \times 3 \times 3$ tic-tac-toe game? I arrived at 49, but I'm not sure if there are more of them.

Answer 1

In each of the three horizontal planes, there are eight combinations. There are also nine vertical columns, and eight vertical planes (four outer and four inner ones) in which two diagonals can be drawn. Indeed, the number of combinations $n$ equals:

$$n = 24 + 9 + 8 \cdot 2 = 49$$

A second way to solve this is by distinguishing four different types of points:

1. A vertex, which contributes to 7 line segments;

2. A corner on an edge, which contributes to 4 line segments;

3. The middle of a facet, which contributes to 5 line segments;

4. The center of the cube, which contributes to 13 line segments.

Since we counted each line segment thrice, we find:

$$n = \frac{8 \cdot 7 + 12 \cdot 4 + 6 \cdot 5 + 13}{3} = \frac{147}{3} = 49$$

Answer 2

• $3 \times 3 \times 3$ simple rows or columns parallel to edges
• $3 \times 3 \times 2$ diagonals on planes parallel to faces
• $4$ diagonals between pairs of opposite vertices

making the same $49$ in total as jvdhooft found