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Wrapping Gifts

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?


Age 16 to 18
Challenge Level

Once you have solved a problem it's instructive to try to think of another (perhaps better?) method of solution. On one famous occasion I did not expect different methods but two school students sent in not 2 but 8 different solutions of the same problem. Not content with that these two students then generalised the problem.

See Eight Methods for Three by One and also Why stop at Three by One ?

Can you beat that? I expect their teacher at Madras College will remember it. Perhaps that school can score again, but what about other schools? This is a challenge calling for the combined efforts of a group of students or a whole class.