This is related to the Banach-Tarski paradox. One of the sites mentioned in that thread has a beautiful "experiment" on equidcomposability: Let A be a unit circle, and B be a unit circle with one point missing. It turns out that you can divide B into two subsets, rotate one of them, and then put the pieces back in such a way that you obtain a full circle.
I just couldn't resist posting this.Consider set B and let U be the subset consisting of all points that are a positive integer number of radians clockwise from X along the circle. This is a countably infinite set (the irrationality of Pi prevents two such points from coinciding). Let set V be everything else.
If you pick set U up and rotate it counterclockwise by one radian, something very interesting happens. The deleted hole at X gets filled by the point 1 radian away, and the point at the (n-1)-th radian gets filled by the point at the n-th radian. Every point vacated gets filled, and in addition, the empty point at X gets filled too!
Thus, B may be decomposed into sets U and V, which after this reassembling, form set A, a complete circle!