A fault-free rectangle is a dissection of a rectangle into smaller rectangles such that no uninterrupted horizontal or vertical cut along tile boundaries divides the original rectangle into two smaller rectangles. Dissections into three or four pieces cannot be fault-free, but, as illustrated above, a dissection into five pieces can be fault-free. Dividing the central piece of the five-piece spiral dissection gives a six-piece fault-free dissection.
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) iff
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.
