Tiling one-deficient rectangular solids with trominoes in three and higher dimensions

Abstract

A classic theorem of Solomon Golomb’s states that if you remove a square from a chess board of size 2N × 2N then the resulting board can always be tiled by L-shaped trominoes (polyominoes of three squares). We show that if you remove a cube (hyper-cube) from a board of size K1 × ・・ ・×KN, where K1 ・ ・ ・KN ≡ 1(mod 3), for N ≥ 3, and at least three of the Ki > 1, then the remaining board can always be tiled by solid L-shaped trominoes. This extends 2D results of Chu and Johnsonbaugh from the 80s and results of Starr’s on 3D cubical boards from 2008. We also study the analogous problem for straight trominoes, showing that the same types of boards are never generically tilable (i.e., tilable regardless of square/cube/hypercube removed) using straight trominoes.