Tag: polyhexes

Stripe Club

~ munizao ~ Leave a comment

I posted last year about a path puzzle using polyiamond tiles. Those tiles were marked with a complete set of paths between cell edges on the perimeters of diamonds and triamonds. Recently I’ve been exploring a variation on tiles with marked paths. In these tile sets, the paths are constrained to straight lines aligned with the grid and connecting the midpoints of opposite cell edges. By this scheme, there are 16 ways to stripe the tetrominoes. I wasn’t able to come up with any elegant tiling using just these pieces, but with a set of unstriped trominoes, they can make a rectangle with four stripes. We follow the typical rule of path puzzles: the stripes must connect between pieces.

There are nine distinct ways to stripe the three trihexes. There is an arrangement of parallel stripes on the figure below that looks like it could have a solution, but it proved to have none when I checked it with a solver. Luckily, non-parallel stripe lines are perfectly acceptable — as long as their intersections occur outside the tiling!

Striping polyiamonds brings a new complication: the line connecting cell edge midpoints is not perpendicular to the cell edge. That means we can change the direction of paths at piece boundaries. The solution below takes advantage of this feature:

Fortuitously, the striped 2-, 3-, and 4-iamonds together contain 49 triangular cells, allowing us to tile a triangle of side length 7. The striped 4-iamonds alone contain 36 cells, but they are not able to tile the triangle of side length 6.

Where else can we go with stripe problems? Todor Tchervenkov, Roel Huisman, and Edo Timmermans looked at tetrominoes with diagonal stripes on the Puzzle Fun Facebook group. (There are 17, which makes them awkward for tiling with the full set, but there are workarounds.) We could try other stripe orientations on polyiamonds and polyhexes as well. Polytans (or polyominoes with tans added or subtracted) could have line bends at diagonal boundaries similar to what happened with the polyiamonds. Another variation I’m looking at is what can be done with multiple stripes per piece. Stay tuned for more stripe content! (Does that count as a stripe tease?)

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.

The Devil’s in the Angles

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Recently, while I was considering possible designs for a puzzle for my exchange gift for the next Gathering for Gardner, I thought about doing something with multiple layers of clear plastic, where interactions of markings on the layers define the puzzle. When you’re going to lasercut a large quantity of puzzles, keeping down the cost, and therefore the cut length, is paramount. So I wanted to be able to use the simplest possible markings on the pieces.

A straight line segment looked like a pretty good candidate, and it leads to an obvious puzzle goal: make the segments on two layers perpendicular. I still needed to choose pieces for these markings, but after a little trial and error, I landed on dominoes, with a segment centered in each square. For these, given some reasonable restriction on the allowable angles of the segments, the number of different pieces possible would land somewhere in the range of what would make for a good puzzle.

I ended up using segments that were turned either 15° or 45° off from the edges of the pieces. These admit exactly 12 different pieces, which can tile two layers of a 3×4 rectangle:


What makes this set particularly nice is that you can get two more puzzle challenges by changing the goal angle for the crossing segments. In addition to making them all perpendicular, you can make them all cross at 30° or 60°. These challenges should be easier, as there are two ways for an angle to differ from another one by 30° or 60°, but only one way to be perpendicular.

I also found a related puzzle that uses 10 dihexes. There are 13 pieces possible in this scheme, but I’ve omitted the ones with a lengthwise axis of symmetry from the puzzle:

In the end, I decided not to make either of these my exchange gift. I had a couple of prototypes made of the first puzzle, and it was clear to me that it needed to be larger than I could afford to make it and give away a few hundred copies. It also works best with a frame to hold the pieces and keep them neatly aligned, which adds considerably to the time and expense per copy. But even though I won’t be able to give this away at G4G13, I hope to be able to be able to sell a few copies at my vendor table there!