The gap number of the TT-tetromino

A famous result of D. Walkup (1965) states that the only rectangles that may be tiled by the TT-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 TT-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 GG such that if both sides of the rectangle are at least 12, then no more than GG gaps will be required. We prove that GG is either 5, 6, 7 or 9.