  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.
