References:
Question 1: Given a number $N$, can we construct a convex planar region that can be cut into $N$ mutually congruent, connected, convex pieces but not into any other number of connected, mutually congruent convex pieces?
Partial Answer (guess): For prime $N$, there seems to be a simple way. Take a regular $N$-gon and mark from it $N$ mutually congruent quadrilaterals by drawing lines from center to mid points of the N faces. Now in each quadrilateral, replace the two 'outward' edges by copies of a polyline with say $p$ edges and with angles that are irrational fractions of $\pi$ (see ref 3 for some justification for 'irrational') in such a way that the $N$-gon becomes a convex $Np$-gon. This $Np$-gon seems to allow partition into $N$ and only $N$ pieces that are mutually congruent, convex and connected.
Remark: I have no answer for $N$ non-prime even when the pieces are allowed to be non-convex.
Question 2: Are there convex planar regions that allow partition into mutually congruent and connected pieces only when the number of pieces is one of exactly $2$ specified values — for example is there a convex region that can only be cut into $3$ connected congruent pieces or $5$ congruent pieces but not into any other number of congruent pieces?
Remark: Answer to question 1 can be slightly modified to yield planar regions that seem to allow partition into only $N$ mutually congruent pieces or $kN$ mutually congruent pieces where $N$ and $k$ are primes.
