I understand the missing square puzzle with the triangles (https://en.wikipedia.org/wiki/Missing_square_puzzle) but I'd be interested to see how Freer's puzzle works.

I'm having trouble finding Peter Tappan's "FuTILE Subtraction" from the citation

Look at the video after he's taken the pieces out of the frame and reassembled them on the table—say, at 1m20s (http://youtu.be/_LY7mf49l5k?t=1m20s or screen shot here: http://imgur.com/GgxONLV). He's kept the two reassembled halves separate.

Look at the diagonal edges of the halves. They don't match up! I've put them together in Photoshop to make it clear: http://imgur.com/gsP6AxG When you put the two halves together, you don't get full cells. You get a crazy mishmash, because the height of one of the halves has been increased by 3/7ths of a cell-height. He replaced some pieces when he flipped the frame over.

You can also see mismatches on the egg one. There's enough extra egg tacked on to each egg in the "hole" picture to allow another egg to be created.

It must be that, once you re-arrange the pieces, you transform the rectangle with a hole into a shape that looks like a rectangle with no hole but is slightly caved in.