The Gap Number of the T-Tetromino

A famous result of D. Walkup states that the only rectangles that may be tiled by the T-tetromino are those in which both sides are a multiple of four. In this paper we examine the rest of the rectangles, asking how many T-tetrominos may be placed into those rectangles without overlap, or, equivalently, what is the least number of gaps that need to be present. We introduce a new technique for exploring such tilings, enabling us to answer this question for all rectangles, up to a small additive constant. We also show that there is some number G such that if both sides of the rectangle are at least 12, then no more than G gaps will be required. We prove that G is either 5, 6, 7 or 9.Comment: 17 pages, 17 figure

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

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