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The ideas in this problem have been expanded and made into another, more detailed problem. See Marbles in a Box.
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 three marbles?
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 .