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Plane to See

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

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Four Points on a Cube

What is the surface area of the tetrahedron with one vertex at O the vertex of a unit cube and the other vertices at the centres of the faces of the cube not containing O?

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Instant Insanity

We have a set of four very innocent-looking cubes - each face coloured red, blue, green or white - and they have to be arranged in a row so that all of the four colours appear on each of the four long sides of the resulting cuboid.

Cheese Cutting

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This is a difficult visualisation problem to tackle, not least because is it very difficult to create a physical model with which it is possible to experiment with the slicing. Visualisation is thus crucially required.


The problem can be attempted at various levels, from having an educated guess at the answer to providing a proof of the maximum possible number of pieces.

If solvers are unable to prove the maximum number of pieces they should certainly be encouraged to describe their best effort at a slicing as clearly and convincingly as possible. Try to establish the existence of an upper bound on the number of pieces.

In practice, the solver may be able to spot the answer quite quickly, but providing an explanation of the answer may be very hard. Solvers should be encouraged to try to explain their answer as clearly as possible in words if a sound mathematical argument cannot intially be provided.