The puzzle is made up of two metal pieces, one aluminum and one steel. The steel inner part is about three thousandths of an inch larger than the lip of the outer aluminum part (at room temperature). Here is a cross-section view of the puzzle (modeled in OpenSCAD).
Steel and aluminum have different coefficients of thermal expansion. Aluminum expands more as it is heated. A solution is to place the puzzle in the freezer for 20 minutes or so. Place the puzzle on a counter with the aluminum side up and direct a blow dryer at it. In a few seconds, the aluminum will expand enough so that the steel inner part will drop out.
Another solution is to place the puzzle in the oven. At some temperature (roughly 250 F), the aluminum will have expanded by three thousandths more than the steel.