Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface of the water make around the cube?
Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
In the game of Noughts and Crosses there are 8 distinct winning
lines. How many distinct winning lines are there in a game played
on a 3 by 3 by 3 board, with 27 cells?
As Allen (Sha Tin College) discovered, the trick to cracking this problem is to work systematically:
"My method was to start of with a small cube and work onwards from that. I started off with a $3 \times 3 \times 3$ cube and stared counting the number of lines for each Vertical, Horizontal and Diagonal"
Doing this he discovered that there are 13 winning lines that go through middle-middle-middle.
The winning lines in general seperate into three types:
"Lines" of three cubes can be made from cubes joined face to face
"Diagonals" are cubes joined edge to edge in a line - so the diagonal of a face for example
"Long Diagonals"are lines of three cubes joined vertex to vertex, going through the middle from a vertex to one diagonally opposite.
Counting these gives 27 lines, 18 diagonals and 4 long diagonals for the $3 \times 3 \times 3$ cube. In general, for an $n \times n \times n$ cube:
Long Diagonals: 4