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Cut Cube

Find the shape and symmetries of the two pieces of this cut cube.

Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

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.

Napoleon's Hat

Age 16 to 18 Challenge Level:
One possibility is to use coordinates. You need to use the coordinatesof the centroids of the triangles.