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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?

Cubic Covering

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2
We first imagine the cross with the blue cubes only:

We now glue the yellow faces to the up and bottom faces of the blue cross. We require 5 yellow cubes for wrapping each of the two blue cubes. (We used transparent yellow cubes)

Now we need one yellow cube for each of the four corners:
Finally, we have 4 faces to cover, so we need 4 more squares.

Therefore, we used $5+5+4+4 = 18$ yellow cubes.

This problem is taken from the UKMT Mathematical Challenges.