General question: Given two integers m and n, to find a convex polygonal region that can be cut into sets of m mutually congruent convex pieces in exactly n ways - the shape of pieces in each set should be unique. For many values of m and n, there might not be such a polygonal region.
Some simple cases:
- m=2 and n=1. A thin isosceles triangle can be cut into exactly one set of 2 convex and mutually congruent pieces (by the bisector of its apex).
- m=3 and n=2. A non-square rectangle can be cut into exactly 2 different sets of 3 mutually congruent convex pieces (by cutlines parallel to length or width).
- Any centrally symmetric planar region other than circular disk can be cut into infinitely many different sets of 2 mutually congruent convex pieces (different lines through its center give differently shaped pairs of congruent pieces) - the case m=2, n=$\infty$.
- An equilateral triangle answers m=3, n=$\infty$.
Special cases: Find a polygon that can be cut into sets of 3 convex congruent pieces in exactly three different ways (m=3, n=3). Or (m=4, n=4)…
Guess: For some m and n, if the pieces are allowed to be non-convex there could be interesting possibilities.
