Every square can be tiled with T-tetrominos and no more than 5 monominos

TL;DR

This paper proves that any square with odd side length can be tiled with T-tetrominos and no more than five monominos, resolving a longstanding question in tiling theory.

Contribution

The paper introduces a new construction that tiles all odd-sided squares with exactly five monominos, improving previous bounds and settling the minimal number needed.

Findings

All odd-sided squares can be tiled with T-tetrominos and five monominos.

Previous bounds on the number of monominos needed are improved.

The construction applies to all odd n, confirming the minimal number.

Abstract

If n is a multiple of 4, then a square of side n can be tiled with T-tetrominos, using a well-known construction. If n is even but not a multiple of four, then there exists an equally well-known construction for tiling a square of side n with T-tetrominos and exactly 4 monominos. On the other hand, it was shown by Walkup that it is not possible to tile the square using only T-tetrominos. Now consider the remaining cases, where n is odd. It was shown by Zhan that it is not possible to tile such a square using only one monomino. Hochberg showed that no more than 9 monominos are ever needed. We give a construction for all odd n which uses exactly 5 monominos, thereby resolving this question.