Tromino Tiling Deficient Cubes of Any Side Length

TL;DR

This paper demonstrates that three-dimensional cubes of any size can be tiled with trominoes, using singletons when necessary, extending known tiling results to all cube sizes.

Contribution

It generalizes tromino tiling results to all cube sizes, including those not divisible by three, with specific configurations of singletons and trominoes.

Findings

Cubes with side length multiple of 3 can be tiled with trominoes.

Cubes with side length ≡ 1 mod 3 can be tiled with trominoes and one singleton.

Cubes with side length ≡ 2 mod 3 can be tiled with trominoes and two singletons.

Abstract

We show that three dimensional cubes of any size can be tiled with trominoes and, when necessary, one or two singletons in any positions. Cubes of side length a multiple of three can always be tiled with trominoes (known), cubes of side length congruent to 1 mod 3 can always be tiled with an arbitrary single cube and trominoes, and cubes of side length congruent to 2 mod 3 can always be tiled with two single cubes in arbitrary locations and trominoes.