Cube factors

Abstract

The puzzle "Soma" is a 3 × 3 × 3 cube consisting of 27 unit cubes, which is to be constructed from 7 different pieces, each containing 3 or 4 unit cubes. Variants of the puzzle include a 4 × 4 × 4 cube and a 3 × 4 × 5 brick. In this paper, we show how to construct all k × l × m bricks with small sets of different pieces. A defective cube is a cube with one of its unit cubes missing. We show that every 2^k × 2^k × 2^k defective cube can be constructed from the 2 × 2 × 2 defective cube. This follows from a more general result involving factors of products of graphs.

This research was supported by N.S.E.R.C. grant no. A3013.