We discuss a number of related equipartition problems. Their common feature is that if the number of tiles is a prime power, the proof explores some topological facts that are no longer true for the general case.
We are mainly interested in going beyond prime powers.
(1) Nandakumar & Ramana Rao conjectured that every convex polygon P in the plane
can be partitioned into any prescribed number n of convex pieces that have equal area and equal
perimeter.
First, we discuss the proof for n=p^a (based on https://arxiv.org/pdf/1202.5504.pdf )
And then the proof for arbitrary n (https://arxiv.org/pdf/1804.03057.pdf)
(2) Equipartition (related to some evaluating function) of a segment (https://arxiv.org/pdf/2009.09862.pdf)