Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece? 
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).
 A: A new paper was posted to the arXiv on related questions: 
"Infinite families of monohedral disk tilings," by Joel Haddley and Stephen Worsley
(arXiv abs.).
Here are tilings of the disk into congruent pieces where at least one piece
does not touch the center (a result mentioned by Anton Geraschenko):



They conjecture (with extensive support) that,


"for any monohedral tiling of the disk, the centre may only intersect a tile at a vertex."

This would answer the original posed question in the negative.
A: Given that this is an open problem, I figured I may as well make these comments an "answer".
Here's a related problem that I don't know how to answer:

1. Is any dissection of the disk into a finite number of congruent pieces rotationally symmetric?

It seems likely (to me) that any example of a dissection into a finite number of congruent pieces so that the center is in the interior of one of them is going to fail to be rotationally symmetric.

Actually, there's an easier problem that I don't know the answer to:

2. Is every dissection of the disk into a finite number of congruent pieces one of the following?
  
  
*
  
*Slices of pizza. (i.e. every two pieces are congruent via a rotation around the center of the circle)
  
*This one:


