Convex Planar Regions Cut into Congruent Pieces – Metric and Discrete Geometry

Question

References:

1. https://math.stackexchange.com/questions/1838617/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-parts

2. On congruent partitions of planar regions

3. https://research.ibm.com/haifa/ponderthis/challenges/December2003.html

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.

Answer 1

This construction (the picture has the case $N=6$) seems to work. It is obtained in 2 steps:

-Begin with a convex set formed by $N$ equal pieces, each having a boundary formed by two segments and a piece of circumference.

-Change the orientation of one piece.