IMC 2006 Q5
5. A solid ‘star’ shape is created by gluing a square-based pyramid, in which each edge is of length 1 unit, precisely onto each face of a cube of edge 1 unit. How many faces does this ‘star’ have?
- A 18
- B 24
- C 30
- D 36
- E 48
Intermediate Mathematical Challenge 2006, Question 5. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 24
Each pyramid has five faces, but the square base of each one is glued to the cube and therefore does not form a face of the star. So each pyramid contributes four faces to the star. As there are six such pyramids, the star has 24 faces.
(It can be shown that the height of each of the pyramids is \(1/\sqrt{2}\). If, and only if, the pyramids had been of height 1/2, however, then the angle between a triangular face of the pyramid and its square base would have been 45°. This would result in the solid having 12 faces rather than 24, as pairs of triangular faces would combine to form rhombi. The solid so formed is known as as a rhombic decahedron: a space-filling polyhedron.)
(Diagram from http://dogfeathers.com/mark/rhdodec.html ).
Solution © UK Mathematics Trust, from the IMC 2006 solutions.