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All in the Mind

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?

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Painting Cubes

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?

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Tic Tac Toe

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?

Noughts and Crosses

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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:
Lines: $3n^2$
Diagonals: $6n$
Long Diagonals: 4