Alex, Ed, and anyone else who enjoys recreational mathematics: A _golyhedron_ is a simply-connected lattice polyhedron whose face areas form the sequence {1, 2, 3, ..., n} for some n. Joseph O'Rourke conjectured on MathOverflow that no such golyhedra exist. After some deliberation, however, I was able to find a 32-faced counter-example, given below: http://cp4space.files.wordpress.com/2014/04/labelled-golyhedron.png Here's an article introducing golyhedra and explaining how I discovered this elusive beast: http://cp4space.wordpress.com/2014/04/30/golygons-and-golyhedra/ My golyhedron has 32 faces. I include a proof that no golyhedron can have fewer than 11 faces, so the challenge is to decrease the (rather large!) gap between 11 and 32 by either providing smaller examples or stronger proofs. I have really no intuition as to what an optimal golyhedron would resemble. I have a feeling that you can probably reduce 32 quite considerably by using a similar approach with more work (I made no attempt to optimise my golyhedron), but you may need an entirely different approach to get down to 20 or fewer faces. Sincerely, Adam P. Goucher