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 (1981, p. 125). A rectangle with integer sides and admits a (nontrivial) fault-free tiling by tiles (where and are relatively prime integers) if and only if all the following conditions are fulfilled:
1. Each of and divides one of and .
2. Both the Diophantine equations and have at least two distinct solutions in positive integers.
3. If and , then and are not both equal to 6.
A nowhere-neat dissection is a special case of a fault-free rectangle in which no two squares have a side in common.