@mbonsma I saw the initial claim, disagreed, came here, this is the key point.

You have some nice, orderly definition of jigsaw pieces which you haven't explicated in this thread which does not allow for topologically complicated pieces. For example, what if many piece have thin "tendrils" which are as long and as wide as the whole puzzle? (I won't even go into what might be possible with jigsaw pieces that aren't measurable sets, see the Banach-Tarski paradox!)

@stillmoms @mbonsma

"Topologically complicated" might be simple as a bunch of long, thin right-angle pieces.

The proof seems to assume that puzzle pieces are neatly bounded by small circles, which is very specific, you can think of all sorts of interesting puzzles where this would not be the case.

If you're going to claim to have a proof, the conditions need to rule any edge cases where it doesn't work, or it isn't a proof.

@TomSwirly @mbonsma I'm no scientist, nor mathematician, but my impression is that this is not being offered as some sort of rigorous, theoretical mathematical proof applicable to all real or imagined puzzles—it's a study of "typical" jigsaw puzzles in the real world, meant to predict “realistic assembly scenarios”. The word "proof" appears nowhere.

Grousing that it doesn't address unrealistic, imaginary puzzles is like complaining that a Klein bottle is a bad vessel for transporting liquids.

@stillmoms @mbonsma

Oops, I'm sorry, you're right - "proof" didn't appear in there, I thought it did. Yes, my standards are lower for non-proofs. 😃

I don't see a puzzle with long or thin shapes in it as being "unrealistic" though.

One of the simplest non-trivial tiling puzzles are pentominos https://en.wikipedia.org/wiki/Pentomino#Constructing_rectangular_dimensions which have both long and right angle pieces.