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A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

### 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.

### 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?

# Cheese Cutting

##### Age 16 to 18 Challenge Level:

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.