**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:* As per the answers below, one can upgrade above attempt to work for all values of $N$, not only primes.

**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.

*Note:* One can widen question 2 and ask if given a set S of numbers relatively prime to one another, one can construct a planar region that allows partition only into sets of congruent pieces with cardinalities equal to each element in set S and no other number. One can also consider less constrained versions - eg. allow the mutually congruent pieces and the input region to be non-convex.