In 2022, Jacques Ferroul sent some notes on a remarkable exploration in polyominoes to Kate Jones, who shared them with George Sicherman, who in turn forwarded them to me. I quickly saw that there was quite a lot of potential there, and exchanged a few emails with Ferroul, where we shared ideas riffing off of his original notes. And then I let the matter go, since it would seem unkind to scoop his discoveries in a blog post when he still hadn’t written about them for public consumption. Recently, I came back to thinking about them in the context of a notation system for polyform tile sets that I had been noodling upon. And when I looked to see what he had been up to lately, I found a note on Kadon’s page for a puzzle he designed, stating that he died in December 2023.
Well, crap.
So I guess I can write this post now. A fuzzy pentomino, in Ferroul’s conception, is an equivalence class of tetrominoes connected to weighted adjacent cells where the weights sum to one. All of the figures below are the same fuzzy pentomino:
Equivalence classes of polyominoes shouldn’t be wholly unfamiliar. We use them for aspects of the same polyomino with different symmetries applied. Ferroul was inspired by fuzzy logic, where truth values can take on any value between 0 and 1. (I can also see an analogy to the “cloud of probabilities” model of electrons in an atom.) A simpler version, where the added cell is constrained to have a weight of 1, Ferroul calls “boolean polyominoes”.
When we use fuzzy polyominoes in a tiling, we allow cells from different polyominoes to overlap as long as their weights sum to one. Now it’s reasonable to ask: does this lead to interesting tiling problems? And I think the answer is, not directly! Restricting ourselves to the “boolean” case, a tiling with these would be equivalent to making a tiling with an extra monomino next to each tile. And tiling generally gets pretty easy when you can throw in a bunch of extra monominoes! Ferroul was interested in finding a tiling that required non-boolean polyominoes to realize. I’m pretty sure this is impossible, but I don’t know how to prove it.
We can however make problems where we put some additional constraints on the extra cells. For example, let us fuzzily join each tetromino with two half-weight cells, and for each piece put one copy of the same color in each of two 5×5 squares. The number of extra cells is 10, exactly the same as the number of pairs of tetrominoes. Then a “fuzzy” tiling can be turned into a puzzle using regular tetrominoes and unit tiles matching every color pair:
The generalization of tiling to weighted cells where weights must sum to one may also be used without the equivalence rule. Here are all of the ways to join a dihex or trihex to a weight-½ monohex.
And here’s a tiling where the monohexes overlap:
Problem 62: Find a tiling where the monohex positions have some symmetry. Bilateral or threefold rotation symmetry seem likely to work. Dihedral threefold symmetry seems less likely, but would be cool.
I have a couple more problems I’d like to share in the fuzzy polyform vein, but this is a good place to stop for now. It’s also worth mentioning that some of the previously produced polyomino piling problems can be modeled as “subtractive fuzzy polyominoes”, where for each piece we take an equivalence class of pieces where one of the cells has been reduced to a fractional weight, and we are again making a weight-1 generalized tiling. I mentioned before that working on a notation system for polyform sets was what brought me back to this subject matter. In a future post, I intend to elaborate on some of what I’ve come up with so far. But for a small spoiler, check the tags on this post.
