A no-touch dissection is a nowhere-neat dissection into squares in which squares of the same size do not share a line segment of positive length. Contact at a single point, including a corner, is allowed.
For an integer 
, Scherer and Hamlyn (Friedman 2005) showed that an 
square
has a no-touch dissection into smaller squares with integer side lengths iff 
, 18, or 
.
The figure above shows Scherer's 
example (Friedman 2005). It uses four squares
of each side length 1 through 5 and one of side length
6, for 21 squares altogether, and has quarter-turn rotational symmetry.
