Miscellaneous Polyform-related Puzzles, Part 1

It's been ages since I last wrote anything on here (so much for that promise I'd update this with a little more frequency) so have some first-draft-quality rambling about polyform-ish puzzles that are sort of just close enough to fit the theme of the blog.

3x3x3 Serially Interlocking Cube

I built this one after seeing a diagram of the pieces in Stewart T. Coffin's The Puzzling World of Polyhedral Dissections and not being able to visualise how on earth they fit together. It's a four-piece puzzle with the unusual property that it must be assembled and disassembled in a certain order (hence 'serially interlocking'); the pieces all kind of hold each other in place. It's not too hard a solve; the very first assembly takes a little bit of thinking but any assemblies and disassemblies after that aren't hard at all because you end up just sort of memorising what goes where, which spoils the fun a bit.

The two visible red bits are two parts of the same piece. Likewise with the blue bits. It's a weird puzzle.

Building this was something of a trial run for other polycube puzzles. I wanted a full set of pentacubes for the longest time (and still do), and I decided the cheapest way to make polycubes was probably to buy the cubes and attach 'em together myself. So I went out (okay, stayed in and fired up the internet) and ordered in enough wooden cubes to make Mio Naganohara seethe with envy, then set about making the tetracubes you can see half way down this page as well as these pieces. The problem is, the combination of my own shoddy handiwork and the cubes not being exactly cubic meant that the resulting pieces didn't quite fit together as snugly as I'd hoped. Not so bad with the relatively simple tetracube shapes, but with these pieces there's C-pentomino-esque indentations that need to be able to fit round another cube and those require a degree of exactness. So I spent a good couple of hours sanding down the glued-up pieces just to make them actually fit together and then come apart afterwards without getting stuck.

And then I scribbled on them with felt tips because I couldn't be arsed to look for paint. It's eye-catching from a distance, but when you actually hold the pieces and examine them up close it's like something a child would make in their first woodworking lesson. But it's the best I can do without resorting to things like putting effort in...

Fig. 2: The pieces in all their terrifying glory. Between this photo and the one above there's probably enough info for readers to make their own set if desired.

Polyarcs

Another one I made myself. It's made of the same materials as my heptiamonds and their tray, which makes me think I just chucked the SVGs for these in the margins then sent the lot to the laser cutting place to get more puzzle for my money.

Polyarcs are the polyforms where the base units are the two shapes you get drawing a quarter circle inside a unit square. There's a bunch of information and constructions involving the 1- through 3-arcs over on Henri Picciotto's site.

Fig. 3: These bad boys were spared the humiliation of a felt tip paint job.

This is the set of two 1-arcs and seven 2-arcs, and they fit in that 2x4 tray in dozens of ways; it's not particularly hard to get them back in there after tipping them out. Putting them in so that the wood grain lines up works as a marginally harder challenge. I kind of hoped I'd be able to make some other shapes with these too, but they're kind of limiting in that there's such a small total area to work with. They do the rounded shape from that linked site in a few ways too.

Sadly, I don't think they can make the 4-way rotationally symmetrical shape on the right - I can get all but one piece in so I'm wondering if there's some parity-like constraint preventing this from working, or if it's just a case of 'the pieces won't go'.

Unnamed 8x8 polyomino puzzle

This one's a doozy. At least it is if you take your dictionary, scribble out the definition for the word 'doozy' and write 'pain in the ass' there instead.

It solves into an 8x8 tray, not unlike the pentominoes + O-tetromino set I probably thought this was when I first bought it (for the grand sum of £2). But the pieces are bizarre. Four of the most uncooperative pentominoes (U, X, T and W), the T- and U-hexominoes (!) and a handful of larger 'ominoes seemingly picked at random. And the worst of it is, it's a really infuriatingly tricky puzzle. There's not a time I've picked this up and not put it down feeling like uppercutting a nun. Even when I manage to solve the thing, the rag-tag assortment of pieces is so illogical and un-mathematical it winds me up anyway.

Going through my cupboard of polyomino stuff and failed laser-cutting experiments, there's probably enough stuff there to write a second one of these posts some time, so consider this a Part 1.

Enneominoes!

It had to happen sooner or later.

Ever since not long after I got my octominoes made, I'd wondered occasionally whether the next set up would even be feasible. For the longest time I dismissed the idea outright; octominoes was as high as I'd be able to go, for three reasons:-

Firstly, the total area of 1285 pieces as 10mm per unit edge was just far too much to handle. The octominoes already pushed the limit of what the table in my room could hold, and even if I resorted to building enneomino constructions on the floor I would struggle with particularly long and skinny rectangles. (That and the fact it's all carpet so not really a suitable surface.)

Secondly, the amount of pieces would just be too much to work with - I knew well the pain of searching through the box of 300+ octominoes for that one specific piece (usually only to find that I'd already used it in the construction...) and even making the enneominoes in 10 colours, that'd still be 128 or so pieces of each colour, so even if I knew the colour of the piece I was after it'd still be a tedious hunt every time. And more often than not I wouldn't even know the colour of the piece I was searching for.

And finally, making this set wasn't going to be cheap. And I'm averse to spending money at the best of times, so was very reluctant to look too hard into this if I was laser cutting something that was going to spend most of its life in a box in my room for the above two reasons.

But even then these reasons felt more like excuses than actual set-in-stone barriers; to paraphrase Lilly Satou, "I knew the possibility of it happening was there."

Making the Set

Gradually I realised a few things that sort of tipped the scales. Firstly, I let go of the idea that the pieces had to be 10mm/edge. This was the case for the smaller sets because it allowed me to line pieces up to the grid printed on the cutting mats, a handy help given my track record for misaligning holes and building things to the slight wrong dimensions. I realised that by making each unit edge 6 and two-thirds millimetres I'd still be able to line some of the pieces up with the grid, and the tradeoff for this slightly awkward compromise was that I'd be able to actually fit more pieces on the table. Potentially even all of them if I'd done my maths right.

I then found a construction by Patrick Hamlyn which had the entire set of unholey enneominoes in 48 congruent rectangles, which I could use as a basis for the CAD files when I got the pieces cut. The number of rectangles meant I had a bit of flexibility with how many batches I got them cut in, and as a result the number of different colours. Eventually I settled on eight (plus a ninth for the holey pieces), which was more influenced by the number of colours of perspex the laser cutting place offered than anything else. It gives 156 pieces of each colour, which is not ideal but it's marginally better than 1248 of each colour I suppose.

I drew up the .SVG files of the pieces while on holiday in Arran, on the evenings where the midges were out in full force and I couldn't do a great deal else. This wasn't the most fun task, just manually copying out the pieces from the reference image. Almost as tedious as digitising octomino solutions from a photo after finding them. And then when I'd finally uploaded all the files to the laser cutting people and parted with an eye-watering amount of money, I set about devising something to actually solve with the pieces once they got here. Something easy to start with, or at least as easy as enneominoes can be. A 79x147 rectangle with the holes all in the centre in a 4-fold symmetrical configuration, for instance.

The Solve

By this point I naively assumed that the enneominoes wouldn't be that much harder to solve than the octominoes - they were wigglier, sure, but there was more of them and therefore more possible pieces to fill a particular niche whenever one should arise in the boundary of the solved area. Right? And lots of pieces with inner 2x2 squares too. 436 of them in fact - a whole 34% of the pieces, up from 23% of the heptominoes and 30.6% of the octominoes.

I misunderestimated these bastards.

Fig. 1: The start of the peeling process. To call it gruelling would be an insult to gruel.

I started the solving and the peeling of the protective sticky plastic pretty much simultaneously.  It was less that the solve was too exciting to wait for, and more that the peeling process was too boring to do on its own. And at this point I made my first mistake.

In planning the boundaries and hole placements of the rectangle, I worked in units and units squared, and then had to convert back to mm in order to place markers on the table. But somewhere here clearly I cocked up, because I built the ring of holey pieces, filled it with 100+ pieces and built a good chunk of solution connecting the centre to the bottom edge before realising it was two thirds of a millimetre higher than it needed to be. So I had to rip out a strip of enneominoes, move a huge patch down carefully without disrupting everything, and then re-solve the little strip I'd taken out with a completely different set of pieces and some new ridiculous constraints on the pieces. Fun stuff. An afternoon well spent.

By this point it had dawned on me these were an entirely different beast to the octominoes - the difficulty plateau I'd envisioned after comparing the octominoes to the heptominoes just wasn't there, and this wasn't going to be as fast and easy a 'warm-up solution' as I'd initially hoped. Additionally, it was taking absolutely ages; by this point I must have sunk 6 or 7 hours into the solution and only placed about 350 pieces or so. And a lot of this time was spent digging around looking for specific pieces, just like I'd feared. Additionally, this being the first time with the pieces meant I wasn't familiar enough with them to know which colour I was looking for half the time either. So I began trying to organise things a little bit. A separate dish where I'd put all the 'really nice' bits - those containing a 2x3 rectangle as sub-polyomino, a dish for the pieces with 2x2 subsections, and a dish for pieces that would work to fill 1xn cavities at the edge of the board - which I internally refer to as edge pieces despite the fact you can use them anywhere really. And this made things a tiny bit easier.

Fig. 2: How the solution looked at about this point. The various trays were my method of sorting the pieces - pieces with 2x3 sections in the smallest tub, ones with 2x2 sections on that big tray on the floor, etc. This was also the last time I could use my desk for normal desk things like writing and eating for a while.

The going got tougher still when it came to filling those two narrow parts between the centre and the long edges. They're not even particularly narrow by most standards - like 20 squares or something, would have been a walk in the cake with heptominoes - but it was enough to make me sneakily reach into the pile of pieces marked 'save for later' multiple times. By the time I had extended the top edge along to the top-left corner I had pretty much no 'normal' pieces left, just 2x2-box ones. Plain sailing from here onwards, I thought.

A Cock-up Most Spectacular

The remainder of the solve was, just as I'd hoped, fairly simple - no worse than putting the last third of the octominoes in. Until I got down to about 19 or so pieces and slowly realised with a creeping sense of dread that the space I had left was concerningly smaller than it should be. I had checked and double-checked all the measurements as I went to prevent exactly this type of thing; I didn't want a repeat of the off-centre centre a few days back. But obviously I hadn't checked hard enough, and sure enough after another tedious bout of counting rows I discovered I had in fact built a 79x146 rectangle - one row shorter than it should be.

I'm still to this day unsure of how I managed to screw this up, I had left little post-it notes tucked under the top edge of the solution as I went denoting the number of columns to each edge in each direction. I must have just been exceptionally tired that day or something.

And so I ripped out a huge wedge of solved pieces right back up to the top left corner, probably 150 or so in total, and began again to the correct dimensions this time. Which was a lot harder than the first time around, given that I was working in a long, narrow space.

The Endgame

The final bit of the solution wasn't too bad all things considered. Once I've whittled the set down to a dozen or so squarish blocky shapes, and the space they need to go into doesn't have any tricky bits, it's just a matter of trial and error. And in this case it maybe took only an hour or so to find something that worked.

Fig. 3: Complete!

The total time to solve was probably 12-14 hours, spread out over several days. Many little solve sessions crowbarred in between work, sleep, gym and whatever else I had to do. And some of that was peeling the plastic backing off pieces too. In fact there's a few of the pink pieces where I don't know quite what was going on, but the stuff just wouldn't come off (or would come off in tiny bits) so I just left it.

Here's a nice clear image of the solution where you can actually see the individual pieces. Which took another couple of hours to draw up, that wasn't a fun task... Normally I find the digitising of solutions kind of therapeutic, but this one just took the piss.

Fig. 4: The completed rectangle in all its glory.

So I suppose the question now is, what next? Dekominoes are definitely out of the question, but I reckon it's only a matter of time before I try something else with the enneominoes, something a little more interesting. Sets of five congruent shapes are definitely possible. For next time I don't feel like using my desk for a week or so.

And there's another fun side effect of solving these I noticed. When you go back to hexominoes (or even heptominoes) afterwards, they seem like baby toys in comparison. There's barely any of them, and they're all such simple shapes... This holds for a little while, until you're about half way into a solution, and then they kindly remind you that they can still be fiendishly tricky.

Pentaboloes

Think of this as an unofficial continuation of this blog post from way earlier.

Polyaboloes (or polytans sometimes, depending on who you ask) are the set of shapes made from joining together edge to edge the triangles that result from chopping a unit square in half. The sets where n triangles are joined together for n = 1, 2, 3 and 4 have 1, 3, 4 and 14 pieces respectively, and these are discussed in detail at the above blog post link. And also at this website link. Which is just the same stuff as the blog post but with a few extra bits snuck in.

Pentaboloes are the next set in the sequence - the shapes made by attaching five of those triangles:

Notice how I was too lazy to trim the edges off the bottom of the tray. And too lazy to paint or varnish or do anything with the pieces.

There are 30 of these in total. Mine are made of MDF, lightly toasted by the laser that cut them. 30 multiplied by 2.5 gives a total area of 75 unit squares, which feels like it should be a good thing; it's not prime or anything. But it means that for rectangles (the most obvious shape to try and pack a set of pieces into) we've only got one option - 5x15 - and it just doesn't seem to be possible. I've tried by hand, and I've let Peter Esser's mops solver churn away at it in the background while writing this, and it seems to get a maximum of 29 pieces placed but no higher. I'm wondering if parity has a filthy triangular cousin, or maybe there's a lack of available orthogonal edge sections within the set of pieces to fully build a rectangle with 40 units of edge.

(This is the point where a proper polyform website would launch into an investigation of just why the 5x15 doesn't seem to be possible, but this is Polyominoes The Blog so don't kid yourselves, all you're getting from me is a shrug and a suggestion to look into it yourself.)

You get slightly better luck with a 7x11 rectangle with the corners snipped off. I have a solution saved from ages ago and I can't remember if it was found by hand or by computer. And there's no obvious way to tell either. With sets of pieces like the n-ominoes for n > 5, I generally save certain pieces for last, so there's a visible gradient across a human-found solution going from wiggly nasty pieces all the way to big blocky square pieces. But with these there are only like three or four really nice pieces - the three consisting of a domino with a half-square attached to it somewhere. And they all end up properly mixed in with the other pieces. Finishing a solution with these is much more perseverance than technique.

One other thing you can do with these that's fractionally easier than solving just using the pure pentaboloes set is to introduce the triaboloes (or tri-tans if you call them that) into the mix, and solve things with an area of 81 units². Here's one of many solutions to the 9x9; this was solved by hand, and it took an absolute age... I just left it unfinished on my desk then ever so often took a punt, desperately rearranging a few pieces. And then one day, after several cumulative hours of trying, the bits I happened to have left fell perfectly into place. This solution has the four triaboloes separated from each other, which looks nice but was entirely unintentional.

There'll be a solution where all four triaboloes are bunched together in one clump, but finding that can be a job for someone else.

I have been toying with the idea of getting the hexaboloes laser cut at some point, but I get the feeling they're going to be even more of an endurance test than these, which doesn't fill me with a lot of motivation.

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.