Consider a polyomino
\(S\). We want to divide \(S\) into several (may be one) smaller polyominoes. The division is called a Fillomino solution
if no two polyominoes of equal size share a side. (They may touch at a point.) In other words, the division is a valid solution of a Fillomino puzzle.
Determine the number of distinct Fillomino solutions of the \(3 \times 3\) square.
Try here for an easier version, or here for a much harder version.