2018/07/18 by Jack Grahl, Grahl, Jack
Biochemistry, Genetics and Molecular Biology · Engineering · Materials Science · #Combinatorics (math.CO) #DNA and Biological Computing #FOS: Mathematics #Quasicrystal Structures and Properties #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1807.09201
openalex publication_date 2018/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.