We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
A red square and a blue square overlap. Is the area of the overlap always the same?
A 10×10×10 cube is made from 27 2×2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.
An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?
There are 27 small cubes in a 3 × 3 × 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you can colour every face of all of the smaller cubes?
