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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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It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of another cube is 8cms. What is the side length of this cube? Another cube has an edge length of 12cm. At each vertex a tetrahedron with three mutually perpendicular edges of length 4cm is sliced away. What is the surface area and volume of the remaining solid?

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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?

Nine Colours

Stage: 3 and 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Each colour must appear on all six faces of the larger cube.

Small cubes can be placed
  • at the corners of the large cube,
  • on the edges of the large cube,
  • in the middle of the faces of the large cube,
  • or at the very centre of the large cube.
How many faces of the small cube will be visible in each of these different positions?

A small cube will need to go in at the very centre of the larger cube.
Where will the other two small cubes of the same colour go?