Problem: is it possible to dissect the interior of a circle into a finite number of congruent pieces (mirror images are fine) such that some neighbourhood of the origin is contained in just one of the pieces?
It may be conceivable that there is some dissection into immeasurable sets that does this. So a possible additional constraint would be that the pieces are connected, or at least the union of connected spaces.
A weaker statement, also unresolved : is it possible to dissect a circle into congruent pieces such that a union of some of the pieces is a connected neighbourhood of the origin that contains no points of the boundary of the circle?
This is doing the rounds amongst the grads in my department. So far no one has had anything particularly enlightening to say - a proof/counterexample of any of these statements, or any other partial result in the right direction would be much obliged!
Edit: Kevin Buzzard points out in the comments that this is listed as an open problem in Croft, Falconer, and Guy's Unsolved Problems in Geometry (see the bottom of page 87).