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All in the Mind

Imagine you are suspending a cube from one vertex and allowing it to hang freely. 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?

Changing Areas, Changing Volumes

Age 11 to 14 Challenge Level:
Try working on Changing Areas, Changing Perimeters first, as it is a two-dimensional version of this three-dimensional problem.
 
If I know two rectangles have the same area, how can I decide, just by looking at their dimensions, which has the greater perimeter?
If I know two cuboids have the same volume, how can I decide, just by looking at their dimensions, which has the greater surface area?