Tag: pilings

Notation Notions: Actions on Fractions

~ munizao ~ Leave a comment

Previously, we looked at this set of dihexes and trihexes with half-weight hexes affixed, which we used in a generalized type of tiling where fractional cells may overlap if the weights on those cells sum to one.

This notion of generalized tiling turns out to be powerful enough to describe all kinds of problems involving tilings with overlap, or “pilings” as I have called them.Because of this, I would like to extend my notation system for polyform sets to encompass the kinds of sets that could appear in such tilings.

Conveniently, I’m not already using fractions, so I can use them unambiguously to denote a fractional weight cell. The set used in the tiling from that previous post can then be notated as (2⊕3)+½⬣. We may also choose to use a fractional weight cell as a base cell of a polyform. There are 12 5·½▲. Here’s a piling:

I’m representing the half-weight cells here with a pinwheel of smaller triangles. Note that when agglomerating half-weight cells, we may stack them in place to make a full-weight cell in addition to placing them in adjacent positions.

Pilings of entire polyforms of fractional weight are possible. The ⅓·5■ can tile a 4×5 rectangle, as shown here. (If the pentominoes could tile three copies of a 4×5 rectangle, you could use this as the basis for such a piling, but they cannot.)

Since we have two different kinds of multiplication with the same notation, we’ll want to be careful about the details of how they work. It makes sense for fractional weight multiplication and agglomeration multiplication to both be right associative. Writing out the implicit multiplication and parentheses for the above examples, 5·½▲ = 5·(½·■), and ⅓·5■ = ⅓·(5·■). Consecutive instances of fractional weight multiplication work like arithmetic multiplication; the ½·½·5■ should be identical to the ¼·5■.

As a final example, here is a set using the “with” (colon) operator. These are the trominoes with two half-weight monominoes, or 3:(2⊙½)■. There are 22 of these. Here the overlaps form a single loop.

There are certainly more things we can do with fractional weights, but some of them will have to wait until we have other operations to work with, subtraction in particular. You can look forward to more posts on my polyform set notation system, but don’t hold your breath; I don’t appear to be a very prolific blogger these days.

Fuzzyominoes: Weighty equivalence

~ munizao ~ Leave a comment

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.

Piling Polyominoes

~ munizao ~ Leave a comment

In my previous post I offhandedly tossed off a taxonomy of polyform positioning problems:

No OverlapOverlap
No holesTilingPiling
HolesPackingTacking

The vast majority of the problems you will find in the wild are tiling problems, with an occasional sprinkling of packings. The other side of the matrix is rare enough that it didn’t already have established terms. Piling in particular is a topic that I haven’t focused on since before the blog, so it was overdue for another look.

The earliest appearance of pentomino piling that I’m aware of is a set of problems that appeared in Puzzle Fun #7. Ariel Arbiser filled a 6×9 rectangle with 6 pairs of pentominoes that overlap in one cell, and asked if the positions of overlaps could be made symmetric. Pieter Torbijn’s solution was printed in Puzzle Fun #9:

If pentominoes overlap two different other pentominoes, you get a chain. I explored this type of problem before the blog, and wrote up some results. One problem that I set at that time was making such a chain in a 7×7 square such that every overlap cell was a knight’s move from the next. (The X and I pentominoes are the only ones that do not contain two cells a knight’s move apart, so they must be at the ends of the chain.) Recently, Bryce Herdt solved it:

Remarkably, Herdt reports that the only other solution is the one made by flipping the F so that it overlaps with the I in a different cell.

There are 12 different tetrominoes with a single marked cell. It was these that inspired me to look at overlap problems again, since, (with three T’s) they have a parity problem that makes it impossible for them to tile a rectangle. It seemed natural to ask if they could pile a rectangle with the overlaps occurring at the marked cells. Herdt found a symmetrical solution:

Herdt noted that this is the only type of symmetry that a solution can have. If a piling had vertical reflection symmetry or rotation symmetry, it would have unworkable parity.

There are 20 tetrominoes with two marked cells. These are the tetrominoes in Kate Jones’s Fill-Agree puzzle. They could make chains, and should not have any parity issues. (Edit: There is, in fact a parity issue. There is an odd number of pieces where the marks have opposite parity. Since the chain must end on the same checkerboard color it begins on, we can’t close the loop. Thanks to Bryce for pointing this out.)

Problem 56: Pile a 6×10 rectangle 5×12 torus with the tetrominoes with 2 marked cells, such that overlaps only occur on the marked cells, and the overlaps form a single circuit containing all of the pieces.

Is there more that we can do with polyomino piling? Will tacking be useful for future puzzles? Will I stop trying to make “piling” and “tacking” happen? Stay tuned for the answers to these questions and more! (Well, more silly questions, at any rate.)