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?

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?

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?

Cubist Cuts

Stage: 3 Challenge Level:

A $3 \times 3 \times 3$ cube may be reduced to unit cubes ($1 \times1 \times1$ cubes) in six saw cuts if you go straight at it.

If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer? Answer the same question with a $4 \times 4 \times 4$ cube:

What about a cube of any size (an $n \times n \times n$ cube)?

This problem is taken from "Sums for Smart Kids" by Laurie Buxton, published by BEAM Education. To obtain a copy call the BEAM orderline on 020 7684 3330 quoting product code SMAR. (Price: £13.50 plus handling and delivery.)