When Can You Tile a Box With Translates of Two Given Rectangular Bricks?

When can a d-dimensional rectangular box R be tiled by translates of two given d-dimensional rectangular bricks B1 and B2? We prove that R can be tiled by translates of B1 and B2 if and only if R can be partitioned by a hyperplane into two sub-boxes R1 and R2 such that Ri can be tiled by translates of the brick Bi alone (i = 1, 2). Thus an obvious sufficient condition for a tiling is also a necessary condition. (However, there may be tilings that do not give rise to a bipartition of R.) There is an equivalent formulation in terms of the (not necessarily integer) edge lengths of R, B1, and B2. Let R be of size z1

• Type: Journal
• Year: 2004
• Where: COMBINATORICS
• Authors: Richard J. Bower, T. S. Michael
• Keywords: COMBINATORICS 2004, D-dimensional Rectangular Box, D-dimensional Rectangular Bricks, Obvious Sufficient Condition