ambiguous rectangle | fault-free tiling

A dissection of a rectangle into smaller rectangles such that the original rectangle is not divided into two subrectangles. Rectangle dissections into 3, 4, or 6 pieces cannot be fault-free but, as illustrated above, a dissection into five or more pieces may be fault-free. More precisely, a complete existence criterion for fault-free rectangles with congruent tiles is given by the following theorem due to Graham. A rectangle with integer sides p and q admits a (nontrivial) fault-free tiling by a×b tiles (where a and b are relatively prime integers) if and only if all the following conditions are fulfilled: 1. Each of a and b divides one of p and q.

Blanche's dissection | Mrs. Perkins's quilt | nowhere-neat dissection | perfect square dissection | rectangle