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.

Finding shapes to make (Hexahexes Edition)

This isn't touched on nearly enough on other sites: the process of finding which shapes are possible to construct with a given set of pieces. Sure, with squares or parallelograms it's easy enough - just factorise all the numbers from the total area of the pieces upwards and hope there's one that breaks nicely into two large enough side lengths to work with. And hope that the number of holes lets you do something nice with them too. Nothing worse than realising your potential even x even rectangle needs an odd number of holes, guaranteeing at least one of them will be hideously off-centre.

But when it comes to more adventurous shapes, you need to get a little bit creative. Generally, I try to find a formula that gives the total area when fed in the values for however many edge lengths. In the case of hexahexes, for example, a hexagon with two lines of symmetry should be workable, and the formula A = ab(a-1) + (a-1)² gives the total area for a hexagon with a unit hexagons in its shorter diagonal and b unit hexagons in the longer diagonal.

Determining these formulae is an art unto itself. The usual approach is to break your shape up into squares or rectangles (in the case of polyominoes), or parallelograms and triangles (in the case of polyhexes), then get expressions for the areas of those separate bits individually. All of this goes to hell when you try it on polyiamonds though.

Once I've got the formula it's then a case of making a big, ugly and confusing-to-the-untrained-eye Excel spreadsheet (actually, it's an OpenOffice spreadsheet because I'm cheap) where every possible combination of values for a and b are evaluated. Then it's just a case of going through and finding the values that are equal to or slightly greater than the total area you want. For hexahexes it's 492 (6x82), and all the likely candidates will fall on roughly a nice curve like in the picture below.

Click the image for bigger (if you're into dull spreadsheets)

Then after this step it's easy. Take a punt at sketching out the shape with a configuration of holes that looks half way presentable, then dig out the hexahexes and clear a flat surface and get some solving done. Admittedly, feeding the shape and the pieces into some solver software will yield much the same results in a fraction of the time, but where's the fun in that?

Here's the case where a = 13 and b = 14. And below we've got a = 11 and b = 19 with seven holes.

Merry Christmas! (feat. some heptominoes)

This is the problem since getting the new website set up: Every time I have something I want to post I'm unsure of whether to do it here or put it there. And in the indecision I end up doing neither. So the blog looks abandoned and the site doesn't get updated. You'd think after like what, eight months, I'd have figured out a strategy for doing both, like updating here first then working the new stuff into the web page at a later date, but nope. I'll have to add that to the ever-growing list of New Year's Resolutions.

I'm still solving things with the heptominoes and such like. (And still sobbing into large sets of polyiamonds 'cause the pieces just won't go...) But there's just not so much happening that I feel the need to tell the world about. Some day there will be - some day there'll be a little momentarily lapse in judgement and I'll find myself ordering a set of enneominoes so big they'd practically tile the entire floorspace of my flat - but until then it's just making similar shapes with the same sets of pieces and there's nothing to say about that which the past fifty blog posts haven't already said.

So for now, have this square ring with the internal holes not quite evenly spaced. They're spaced enough so that it looks pretty at first glance, but then you realise the spacings are 5-6-5 and once you notice that it'll just irk you for ever more.

And now it's telling me 'spacings' isn't a word. Lovely.

Hexahexes

I always sort of overlooked polyhexes in the past. Polyominoes were the main event, so to speak, and the first polyforms I really got properly into, and on the rare days I wanted a really infuriating challenge I would usually turn to the polyiamonds. But polyhexes for whatever reason just weren't really on my radar. Sure, I had a physical set of the 1- through 5-hexes from Kadon, and I solved a couple of things with them. And I even made a half-arsed blog post a while back. But that was about it really. Until now.

A few weeks ago on a whim I got a set of hexahexes cut out of the cheapest MDF money could buy. I didn't even shell out the extra two quid for the laser cutting people to cover the wood with protective masking tape, instead opting to let the bits get gently toasted around the edges by the laser. And then I took them on holiday, to a chilly weekend in a caravan in Northumberland where I knew I'd be a captive audience in the evenings. And while there I slowly began to realise that I'd missed out... Polyhexes were fun. In fact they weren't just fun, but were in fact... very fun.

This photo doesn't really give any sense of scale, but each hexagon is 8mm to an edge, and the full solution has a diameter of about 40cm or so on average. I think. Nice and chunky. I actually checked the scale this time before cutting unlike my positively tiny enneiamonds.

The slight browning of the edges turned out to be something of a blessing in disguise - it makes the borders between adjacent 'hexes stand out a bit in photos which is handy. Sometimes I'm too lazy to draw up a pen-and-paper record of a solution, so just being able to take an aerial photo that I can work from to create a digital image is a nice time-saver. And talking of digital images:

Here are solutions to two different hexagons, the more compact one is the shape of the solution Kadon uses for Hexnut II; the larger thinner hexagon I haven't seen anywhere before. I haven't ran the numbers for hexagons larger than this; it could be that there is an even bigger thinner (and therefore harder to solve) hexagon ring out there waiting to be found.

Speaking of, solve difficulty is the best thing about the hexahexes. It's somewhere between that of hexominoes and heptominoes, I'd say. A good, meaty challenge but one that I don't need to set aside a whole evening for. There are a couple of kinks to be ironed out with my solving technique, though, mostly the fact I'm not used to hexagons so it's often not immediately obvious whether a piece will fit in a certain place without actually trying it a few different ways.

Unholey Hexahexes

If you discard the holey hexahex (as we sort of unintentionally did for the rings above) you get 81 pieces and a total area of 486 hexagons, which divides up very nicely indeed. So far all I've done with this set is the really easy stuff - a couple of approximations of parallelograms, of which one is shown below for your perusal.

The 81 unholey hexahexes squeezed into an 18x21 parallelogram. Solve time approx. 45 minutes manually.

But there's a lot more out there than just parallelograms. It's fairly easy to work out formulae tying the edge lengths to the area for various hexagons, triangles and other such shapes that hexagons lend themselves well to. And from there just a little bit of searching for edge lengths that give the magic number, 486.

Which will all be a nice excuse to post a bunch more blog posts. I need to pick up the pace - this year my posting rate on here has gone right down. That's partly because I've been putting some things directly to polyominoes.co.uk (and discovering the joys of trying to display characters like '°' in html), but it's also partly because my interest in polyforms seems to come and go in phases. And summer this year I've just been preoccupied with other things (recording an album, teaching myself to read Japanese, and dusting off the Rubik's cubes and getting back into speedsolving). But now with winter drawing in, and with its long cold rainy evenings with nowhere else to go and not much else to do, there's a non-zero chance I'll dedicate a bit more time to the sacred art of polyform-ing. And to the subsequent rambling about it on here.

Tetracubes, revisited

A while ago I wrote a post exploring polycubes up to and including the tetracubes. And I didn't really go too far in depth with it. I found a few things with the tetracubes then just kind of stuffed them in a cupboard and forgot about them. But recently I dug them out and all it took to rekindle my interest with them was a few minutes scribbling on them in felt tip pen so that the individual pieces could be told apart in a construction. Behold:

It's messy and it looks like a six year old did it, but it's a slight improvement on how they looked before. From a solving point of view anyway.

After doing this I decided I needed to solve a few things to test out how they looked. And this quickly showed me that there was a lot more overlooked potential in this set of shapes than I ever realised.

Scaled-up tetracubes

The total volume of the tetracubes is 32 units, which is just enough to construct a big tetracube scaled up by a factor of two. For four of the five planar tetracubes and two of the non-planar ones this is trivial once we've got the two 2x2x4 blocks above solved, just put them together in whichever configuration. For the two remaining tetracubes - the T-tetromino and the cyan one bottom left in this image - it's a little more tricky.

For the diagram on the right, a dot indicates that that piece extends into the layer below there, and a square indicates that it extends into the layer above.

The solution for the T-tetromino is shown above, but the final tetracube is a challenge for the reader. It's possible, but I can't be bothered to draw out another diagram for it so you'll have to find it yourself.

Almost-cuboids

There are also the two cuboids-with-holes-in-them that can be done. There's the 2x3x6 with a 1x1x4 hole, and the 3x3x4 with a 1x2x2 hole. I imagine there are several ways to solve each, but I've included once possibility for each, in that not immediately readable notation everyone uses for polycube constructions.

Octomino towers

Imagine an octomino. Any octomino you like. Then imagine it made of cubes as opposed to squares, the planar octacube equivalent of the octomino. Now imagine stacking four of these perfectly on top of each other, creating a prism with volume 32 units squared and the octomino as its cross section. There is a chance that this resulting shape can be filled with the tetracubes. I mean, sometimes there's not, the I-octomino in this instance corresponds to a 1x4x8 rectangle which clearly can't fit the non-planar tetracubes. But The tetracubes show a surprising versatility when it comes to most of the other octominoes.

Some highlights shown in the diagrams below.

The stairstep octomino came as a surprise to me, I really didn't think it would be possible.

A useful solution is the one above to the doughnut-shaped one. This can be broken into three small 4-unit high towers as shown below, and these can be pushed together in various ways to make a whole host of octomino stacks.

Which begs the question: which of the octominoes can we solve this way, and which ones can't be? The above 'kit' of three pieces probably covers the majority, but it'd be interesting to see the set that just can't be done, either through the kind of impossibility the I-tetromino demonstrates immediately, or just cases where the pieces won't go despite there not being a clear reason for it.

This felt like the kind of problem where if I posted it to the Puzzle Fun Facebook group someone would get back with the results of an exhaustive search by the end of the day. So I tried that.

Pretty quickly Edo Timmermans had found all of the octominoes that could be created by combining the monomino, triomino and L-tetromino that make up the solution to the ring octomino way up there in the previous bit. These are marked in green in the above image. He also showed that any 'L' shaped octomino with a single bend in it was impossible, as the three nonplanar pieces would all have to occupy that bend.

I then found that my solution for the zigzag octomino could be similarly partitioned into a monomino, triomino and Z-tetromino which allowed solutions for a further five octominoes, those in dark blue.

George Sicherman then found the set of the 48 octominoes for which the prism has no solution, these were marked in red in the diagram. Which left nine octominoes, all of which had solutions, but which hadn't yet been found. I managed to pick off some of these by hand, as shown in the diagrams below, and the remaining five I verified in Aad van de Wetering's 'Poly3D' software. But I'll not put the solutions here, just in case anyone reading has a set of tetracubes themselves and fancies a nice challenge.

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I've been updating this blog and the website sort of in tandem; I prefer the site for its more flexible formatting and the fact I can interlink and break things into pages a bit better, but this blog probably gets more traffic (it's not much, but it's traffic nonetheless) so I'll keep adding stuff here too. I'll figure something out.

One-Sided Hexominoes

It's been a long time since I last posted anything on here. Too long. But sadly there just hasn't been that much in the way of polyomino-related goodness to post. Partly because I've just been too busy doing other things, and partly because there's only so much you can do with polyominoes - the easy stuff isn't interesting enough to write blog posts about, the hard stuff is too hard to do without the assistance of a beefy computer, and the stuff in the sweet spot is hard to come by.

I've never written up anything about the one-sided hexominoes before on here. Basically, earlier this year I got a second set of heptominoes laser cut, which opened me up to the possibility of solving shapes using the set of one-sided heptominoes by combining the sets and then trying not to flip the pieces over. Then I realised the same thing could be done using my two sets of hexominoes too. And the set of one sided hexominoes is a much more versatile set than the plain ol' regular hexominoes.

There are 60 one-sided hexominoes in comparison to the standard set's 35. But this new set doesn't have those pesky parity constraints which means that a lot more shapes are possible to tile - rectangles without unsightly internal holes, for example.

The total area is 60x6 = 360, which means that rectangles of size 4x90, 5x72, 6x60, 8x45, 9x40, 10x36, 12x30, 15x24 and 18x20 should all be possible. Sadly, due to the length of the perimeter compared to the amount of perimeter squares the pieces can provide, 4x90 isn't possible.

Solving manually is a little trickier than the normal hexominoes - each piece having only one accepted 'right side up' means that any given piece is slightly less practical than its two-sided equivalent, and you do get those cases where you're down to one piece left and the hole is the mirror-image of the piece you're holding.

Here's two 9x20's which can be combined to make either a 9x40 or an 18x20, in a two-rectangles-for-one type deal. As 360 divides up really nicely, this gives a lot of possibilities for tiling groups of congruent shapes, but that'll be a blog post for the future. Others have already solved congruent sets of ten or twelve shapes, so go look at those. Scroll about three-quarters of the way down the page for them. In fact just read the entire page, it's all good.

Something else nice you can do with the one-sided hexominoes is square rings, i.e. squares with a centred square hole. Here are three possibilities (these might be the only three actually), with the rings getting progressively thinner, and as a result a little harder to solve manually:

And here's one more shape, a diamond with a central hole and those tricky diagonal edges.

And I'm purposely leaving this post a bit less thorough than usual so I've got an excuse to post 'One sided Hexominoes - Part Deux' in a few weeks. Or a few months, if my recent posting schedule is anything to go by.