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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?

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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?

Classic Cube

Age 16 to 18 Challenge Level:

Ignore any tabs.

You may like to use a dynamic geometry package or a spreadsheet program to help you. But how will you then convince me that the volume you have found is greatest?