This post is based on the answer to this question: A claim on partitioning a convex planar region into congruent pieces
A perfect congruent partition of a planar region is a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent. The answer from above post shows how to construct a convex polygon with odd number of edges that can be perfect congruent partitioned into 2 non-convex pieces but not into 2 convex pieces.
Question: Given an n-vertex convex polygon P, how does one algorithmically decide if it has a perfect congruent partition into 2 pieces - and find a dividing line/polyline if one exists? Can an algorithm linear in n be found? Is the convexity of P important?
If n is odd, one end of the partitioning line/polyline has to be at a vertex of P. Not sure about the n even case. The number of straight portions in the 'simplest' dividing polyline (if it exists) appears limited in some way by the number of edges of P. One suspects that increasing the number of mutually congruent pieces even from 2 to 3 could make the problem very difficult.
An old, related blog post: https://nandacumar.blogspot.com/2009/05/identical-partitions-of-polygons.html
