Tiling a Rectangle with L-tetrominoes

Solutions

Find a necessary and sufficient condition that an a×b rectangle can be exactly covered (completely, and without overlaps) with L-tetrominoes.

Solution

Since 4 divides ab, a may be taken even. Let a/2 alternate rows of b squares each be colored black in the rectangle.

Then every L-tetromino in the covering must cover three squares of one color and one square of the other. If m L-tetrominoes cover three black squares and n L-tetrominoes cover three white squares, then 3m + n = ab/2 = 3n + m; hence m = n. This means that the covering must use an even number of L-tetrominoes, and hence that 8 divides ab. Except for the 1×8k rectangle, every rectangle of area 8k can be partitioned into exhaustive, disjoint rectangles of dimensions 2×4 and/or 3×8, but both the 2×4 and 3×8 rectangle can be packed with L-tetrominoes in an obvious way. Hence, the necessary and sufficient conditions are that a and b be greater than 1 and ab = 8k.