Imagine a $3 \times3 \times3$ cube, made up from 27 unit cubes, all
of which are made from clear plastic that can be filled with ease.
The location of a unit cube is described according to the following
positions with respect to the three axes or directions:
A marble is placed in the unit cube at left-middle-bottom.
Another is placed at middle-middle-middle.
Where should the third marble be placed to make a winning line of
How many winning lines go through middle-middle-middle?
How many different types of winning lines are there?
How many winning lines are there altogether?
How many winning lines of four are there altogether in a $4 \times
4 \times 4$ cube?
How many winning lines of $n$ are there altogether in an $n \times
n \times n$ cube?
This problem will feature in
Maths Trails - Visualising, one of the books in the Maths Trails
series written by members of the NRICH Team and published by
Cambridge University Press. Maths Trails - Visualising is due to be
published later this year, but for more details about the other
books in the series, please see our publications page .