Using your Head is Permitted

May 2012 riddle

UPDATE (5 May): To clarify the bonus question: we are looking for two polyhedra with vertices in Z3 that have the same number of vertices, the same number of Z3 edge points, the same number of Z3 surface points, the same number of Z3 interior points, but two distinct volumes.

The following is a classic problem. Please submit elegant solutions only, and only if you have no prior familiarity with this type of question.

Let P and Q be two simple polygons with vertices in Z2. Let AP and AQ be their respective areas, let IP and IQ be the number of Z2 points in their respective interiors, and let BP and BQ be the number of Z2 points on their respective boundaries (their vertices, and any point on their edges).

The question: do there exist two such polygons with IP=IQ and BP=BQ that do not have the same area?

Either provide an example or prove nonexistence.

For a bonus mention (an asterisk next to your name), answer the same question for two polyhedra in Z3.

List of solvers:

Djinn Lu (*) (2 May 15:44)
Oded Margalit (*) (4 May 00:25)
Itsik Horovitz (*) (13 May 03:19)

Elegant and original solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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