I was interested in this problem in GCJ Finals and thought of its general case (= the board size is infinite).
You are given two (or more) type of polyominoes. Find a shape satisfies the condition "can be filled completely with some number of polyominoes of the same type in no overlaps" for each type of polyomino.
A shape with fewer cells is considered better but it's not necessary to minimize.
Extra: Are these possible or not? Proof that.
I welcome your solutions or new problems!