TITLE: Cutting of a regular polygon into congruent pieces
QUESTION [20 upvotes]: Question. For which $N$ it is possible to cut a regular $N$-gon into congruent pieces such that the center of the regular polygon lies strictly inside one of the pieces? For $N=3,4$ there are trivial examples. For $N=6$ there is a nontrivial example by Peter Mueller (based on idea of Matt F.) below.

Remark. There is also similar open problem about a disc. Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?
It seems that the answer is NO, but there is no proof.

REPLY [20 votes]: Using the tiles suggested in Matt F.'s answer, there is a solution for the regular $6$-gon: