Checker Boards and Polyominoes

Tiling Deficient Rectangles with Trominoes

“…Trominoes were introduced by Golomb [3], who proved that deficient squares whose side length is a power of two can be tiled. Chu and Johnsonbaugh first extended Golomb's work to the general cases of deficient squares [1].…”

“…We will indicate decompositions into nonoverlapping subrectangles by means of an additive notation. For example, a 3i × 2 j rectangle can be decomposed into i j 3 × 2 subrectangles and we write this fact as R(3i, 2 j) = i µ=1 j ν=1 R(3, 2) = i j R (3,2). It follows from this and the tiling in FIGURE 2 that any 3i × 2 j or 2i × 3 j rectangle can be tiled.…”

“…But in cases B and C, the tiling must tile the leftmost 3 × 2 subrectangle of Q, so that the original tiling is also a tiling of the third column of Q, which is an R (3,1). This is impossible.…”

“…This is a special case of a well-known and beautiful example of mathematical induction [3], [4, page 4], [6, page 45…”

“…− is tiled by the ChuJohnsonbaugh Theorem and reference [3]. Finally if m = 11, n must be congruent to either 8 − ; the first terms are tiled by the Chu-Johnsonbaugh Theorem and the tiling of the last term can be found in reference [1].…”