Recently, I came across a unique problem for which I couldn't find a complete solution.
I want to determine if a given rectangle is fully "compatible" (for the lack of a better word, please suggest if there is any) with all members of a given Polyomino set.
A rectangle is said to be "compatible" for a given member of a Polyomino set if it can be formed by using at least one of that given polyominoes and the rest can be any combination (with repetition) of all members of Polyomino set.
I am sure I have confused you by now, so here is the example, Consider a 4x4 rectangle with Tetrominoes. There exists at least one solution for all (I, O, Z, T, L) Tetrominoes. So it is compatible. Likewise 5x8 and 4x10 are compatible as they involve ALL tetrominoes.
For Trominoes, any rectangle with area A where A > 3 and A % 3 = 0 is compatible.
Please note that I am only interested in finding out if it is "Compatible", the actual solutions are not required.
I could find that there is no solution for all N-Ominoes where N >= 7, as they all involve at least one omino shape with a hole which is impossible to fill up without causing overlaps.
I am having trouble finding solution for 4,5 and 6-Ominoes.
I am sure someone out there would have worked this out.