Recently I came upon the following result:
Theorem (*): Let $n$ be a positive integer not equal to $1,3,5,7,9$. Then it is possible to partition a rectangle into exactly $n$ congruent non-rectangular parts.
For the $n$ among $3,5,7,9$ it is not know whether this is possible. What is amazing to me is that even $n=3$ is unknown??? It seems that this case is easy to visualize and moreover it seems intuitive/obvious that the answer is no. However as the history of mathematics has shown, an intuitive/obvious result can turn to be false. Is it that proving rigorously that a partition can't exist is hard? Or maybe I'm missing some key aspect of the problem, and actually the answer is not intuitive or obvious? This is the most simple looking open problem I've ever seen.
Can someone also direct me to a proof of (*)?
Very likely not. This is in the realm of rectifiable shapes. You should mention what the source for your theorem is.
It is tricky to divide a rectangle into an odd number of congruent non-rectangular shapes. The minimum known solution is for the P-hexomino.
There are some order-15 solutions known for pentominoes. I don't know of any order-13 solutions.