Three and four-dimensional noughts and crosses

In ordinary two-dimensional noughts and crosses, there are eight winning lines.

How many in three-dimensional noughts and crosses, where the “board” is a cube made up of 27 smaller cubes, and taking a turn is putting an X or a O in a small cube?

There are various different ways to count the winning lines, as Aniqa explained to us on 23 March 2018.

The answer is 49.

The reason why this is an interesting exercise is that it develops our skills at counting possibilities – a branch of maths called combinatorics, which has become vastly more active in recent decades because it is highly useful for computer science and for encryption.

It also starts to take us into the branch of maths which discusses symmetries of things. In mathematics, the symmetries of a thing are the transformations you can do to it which leave it looking the same as when you started. Every thing has at least one symmetry, namely, leave it as it is. Many have more. The study of systems of symmetry is a fundamental part of group theory, which in turn is fundamental to modern algebra and to large parts of physics.

If you go on to four-dimensional noughts and crosses, then you start also to develop skills at thinking in four (and higher) dimensions, which again are central in modern physics.

The number of winning lines is 272 in 4-dimensional noughts and crosses, and 1441 in five-dimensional.

The general formula for the number of winning lines in k-dimensional noughts and crosses (thanks for this to Marcel Armour) is

\sum_0^{k-1} 3^j \cdot \, ^k C_j \cdot 2^{k-1-j}