Subsets of the Hexominoes

Like hexominoes, but are overwhelmed by the sheer number (35) (!) of them? Well, have I got just the puzzle for you. It's these:

The set consisting of the eight hexominoes that contain a 2x2 block is... less than fantastic. Sure, at first glance it looks interesting, a manageable set of cooperative pieces, but scratch the surface and you hit the snag that seems to ruin everything in the world of tetrominoes and hexominoes (how the octominoes avoid it is a mystery to me) - parity. The set has an imbalance of ±2, so kiss goodbye the hopes of being able to make any rectangle with an even edge length, and many other symmetrical constructions. To alleviate this problem somewhat, I've added to the set a single monomino which allows a 7x7 square to be filled. It's natural to want to solve it with the monomino in an aesthetically pleasing place - the centre or a corner - but due to the aforementioned parity constraints none of these is possible. In fact, the only places the monomino can go are the black squares on the diagram below, which means there are effectively four different positions for it excluding rotation and reflection.

All of these are solvable - holes at positions 1 through 4 have 11, 22, 11 and 5 solutions respectively. But the ones with the notch at the centre of an edge are undeniably the best.

If you take the pieces off-road, away from the confines of the tray, there are other shapes you can wring out of them too with or without the supplemental monomino. A 5x10 with two corners on the short side removed is possible with the eight hexominoes, and there are (if FlatPoly2 is to be believed) 47 ways of doing this.

Fig. 1: Here's one of them

There are also a few configurations of the 5x10 with two holes down its long line of symmetry which have solutions, and I'm beginning to think that maybe I should have made the wooden puzzle a 10x5 frame with two monominoes and it would have been more interesting.

Again, numbers are excluding rotations and reflections. And these numbers are of unique solutions, it doesn't guarantee interesting solutions. Several may be related to each other in that reflecting a symmetrical subpart of one solution made of two or three hexominoes could lead to another. Still counts. I always felt a little bit cheated by this as a kid, looking at the 2,339 distinct solutions of the pentominoes in a 6x10 rectangle. But I grudgingly came to accept that cases like these were in fact solutions in their own right.

Fig. 3: Like this. This would have made nine-year-old me's blood boil. (Not really.)

In fact, that example solution before is even more egregious an example than I'd first realised. The two pieces at the top (the fish and the one that looks like a backwards δ) can also be flipped over, giving four possible solutions (two for each of the configurations of the orange part). And then the entire non-orange section of the image can be reflected and rotated, giving a total of 2x2x4 = 16 solutions for the price of one.

The Remaining 27 Hexominoes

Perversely, taking a parity-unbalanced subset away from the hexominoes doesn't leave a nice balanced set remaining. Of the 27 hexominoes that don't include a 2x2 sub-square, seven of them have 4-2 balancing when checkerboard-coloured, which means we've still got all the same problems as with the full set. In addition, we've took out eight pieces which are some of the easiest to work with, meaning that the resulting set is going to be a real challenge to work with, at least if we're solving by hand.

Fig. 4: Here's the set crammed into the first shape I could think of, solved by computer because I got lazy. It's like 28°C here right now, it's lucky I can be arsed to even write a blog post.

The other subgroup of the hexominoes that suggests itself is the set of 15 symmetrical hexominoes (including both rotational and mirror symmetry). However, these pieces are just so uncooperative I've found it pretty much impossible to solve anything pretty or interesting with them, and I've tried on and off for the past couple of months.

Picking this back up after a long break

Man, I've missed this so much. Only one post so far this year and it was ages ago. Basically, stuff just got kind of busy, a surprise change of job in February/March took up a big chunk of my time, and with my attention divided between the blog and the shiny new website I ended up not contributing a great deal to either. But~! Enough excuses. A few nights ago - just for the hell of it, just for old times' sake - I dug out the heptominoes and just had a crack at solving something with them. Just a 29x29 rectangle with a central 9x9 hole (and some other unit holes). Knowing full well it wouldn't be something interesting enough to write to the blog or the site about.

And I remembered why I started doing this in the first place.

Spring 2019 - when this blog started) - was, for me, not exactly the most fun of times. And I think in those evenings of stumbling into the world of polyforms, retreading the steps people like David Bird and Michael Keller had made way before me, it became a sort of meditative thing. When I was knee deep in a hexomino or heptomino thing all I was able to focus on were the pieces at hand (and occasionally the Andrew W.K. I had blasting on the iPod) and it was a welcome break from everything else going on. Kept me sane (or at least, kept me from getting any less sane than I already was...)

And I think that's what I need now.

Fig. 1: I finally made good on that promise to buy a proper camera instead of just using the one on my phone. Problem now is, I don't know how to use it (and I scale down the photos for the site anyway to save space) so they still look just as bad.

It's not like I've got a lack of things to write about on the blog, anyway. In the past 6 months or so I've been getting things made (generally laser cut) like no-one's business, though these generally lie in the more 'out there' realms of polyforms - sets where the base shape is something weird like a domino sliced in half diagonally, or assembly puzzles of mathematically incomplete polycube sets. Think Soma cube, and its many variations and relatives. Enough to keep churning posts out on here anyhow.

Here's a picture of some actual polyominoes, as way of apology for turning this place into LiveJournal up there before:

Fig. 2: Four-colouring these adds an interesting challenge.

Oh, and I made this too.

Add caption

It's the nine enneominoes that have either the 2x4 octomino or the 3x3 with a missing corner as a subset. There's one solution (excluding rotations and reflections) for getting them into that 9x9 box, and I'll leave it as a puzzle for the reader (if indeed there are any).

I made this stupidly small - the pieces are at a scale of 5mm/edge meaning the entire thing including tray is a mere 65mm wide. I was trialling how small I could feasibly make pieces and have them still be nice to play with. Because I have plans for a certain large set of pieces. And evidently 5mm/edge is just a tad too small. Might have to do 8mm.