Enneominoes, revisited

First off some unusual news in the world of big tilings with big sets of polyominoes. It turns out that Sumio Baba of (presumably) Japan had been putting together shapes using sets of 10- 11 and 12-ominoes in the mid-2000's unknown to the rest of the world. Part of the reason for this is that he'd shared them with the world via a little Kindle eBook on Amazon, which is the last place you'd look really. Sadly, (and I think this is a fault of Amazon rather than the author; cheers Bezos ya berk) when you buy these there's no way of zooming in or doing anything with the pictures - the 11-omino and 12-omino squares are just big fuzzy unreadable grey lumps.

The smaller ones (if you can call heptomino and octomino sets smaller that is) are legit though, from what I can see snooping through the free preview samples so I'll take his word for it with the bigger ones. (Oh, and there's loads of big polyiamond and polyhex sets there too, the larger ones probably just as unreadable due to the file format too...)

So needless to say after this big tease and subsequent disappointment I was left fiending for some big tilings with unwieldy polyform sets. And with no other choice, my only option was to dig out the biggest set I had, the enneominoes, and have another crack at making something with them myself.

Phase 1: The Design

The main constraint I had on my choice of shape to fill was the size of the table in my room. Its dimensions are 68x120cm, which corresponds to a maximum rectangle size of 102x180 units, with my enneomino set being made at 6.666mm/edge. This sort of forced something rectangular, just because the bounding box of a diamond or parallelogram or something is way bigger than that of a rectangle of the same area. So I knocked together a python script that listed the various rectangles with an area slightly above 11565 square units (along with the corresponding hole count), and I picked one that

a.) I hadn't already done last time, and

b.) was odd x odd, which makes it slightly easier to achieve high symmetry placing the holes.

I settled on 93x125 with 60 holes - two domino holes and 56 monomino ones. Using MS Paint zoomed right in so you can see the grid and the individual pixels is a godsend for fleshing out things like this, finding a nice configuration, checking it's symmetrical and counting how far from the borders the holes all are for ease of construction. the end result was this:

and with that I could sweep everything off my table onto the floor and begin building.

Phase 2: The Construction

First off I tried something a little different. Something I don't really bother with with heptominoes or octominoes, but felt necessary this time. I painstakingly sorted all the enneominoes into a half dozen boxes by 'category'. First, all the holey pieces came out - fairly easy, they're white plastic so stand out a mile in the box. Then anything with a 2x2 sub-rectangle in the piece got put in a separate box (2x3 sub-rectangles got their own tiny box for the very end) and I also split out any 'snakes', i.e. pieces with no branches that are just a chain of squares.

Fig. 1: This kind of thing.

These pieces have no special significance when it comes to packing them, it just makes looking for an individual piece a little easier (and it was easy to pick them out by eye). You know which of the smaller boxes a piece is in so that's looking through ~400 pieces rather than a thousand. I kind of wish I'd filtered down further, maybe taking out all the pieces that fit in a 2xn or 3xn box. But I was running out of tupperware by this point so that categories I had would have to do.

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The grille I'd picked for the holey pieces was tougher than I'd anticipated - the pieces just needed to be too close together and that sort of forced specific pieces to go between them. And looking for those specific pieces was a tedious affair. I then realised that with like 56 holes and only 37 or whatever holey pieces, I didn't need to pack them so closely, but I'd already done like 20 by that point (it was late and I was tired, okay) so they were staying like that. Then a case of counting and checking like 5 times to make sure the number of units between the end of the domino-shaped hole and the edge of the rectangle was the same as that in the blueprint. I wasn't about to have a repeat of last time.

A Random Aside

All this time I was wondering another thing - how exactly Sumio Baba was constructiong his creations. As in, by hand or by computer search. There certainly seems to be a look of hand-solved-ness to them, the David Bird-esque clumping together of the big chunky pieces in one corner. And some mention in the accompanying text about the difficulty of solving (I think - but my command of the Japanese language is pretty dire considering how long I've been studying it).

Which then begs the question, what kind of technique was used for keeping track of which pieces were already in the construction vs. were still free to use? That is essentially what all my physical sets of pieces are, a way of ensuring I don't re-use a piece or miss one out. For smaller sets like hexominoes where you quickly commit to memory all the pieces they're not needed - I've sketched out solutions freehand or solved via drawing using the table border tools in Excel on particularly slow days at work. But for a set like the 12-ominoes...

Actually I can sort of think of something that could be manageable - I'd need to throw together something that could convert between a graphical representation of a polyomino and a simple textual form, something like APGcode for storing objects in Conway's Game of Life. Then it'd be possible to have a searchable list of the pieces and a flag for each as to whether it'd been used or not. (Provided you had a nice means of generating all the pieces, that's another headache in itself!) And as for creating the actual solution, honestly I'd probably do it in Microsoft Paint, the old classic. Worked last time...

The Construction (again)

The remainder of the solve was knocked out in practically one go - a single day (07/02/25) from about half 9 in the morning to just before 4, with breaks here and there to have lunch and learn the keys to '5 Minutes' by The Stranglers for a rehearsal that evening. But whenever there was a gap I was drawn back to the table with the 'ominoes on it, intending to do 'just one more' or to fill one little tantalising gap. And each time it would turn into another 30 minute sesh where I'd put like a hundred pieces in and have to tear myself away to do normal person things.

These are markedly trickier than octominoes. I probably mentioned it last time but that one extra square in each piece, you really feel it. Everything's a bit wigglier and weirder, and I notice I have to take much more care finding a piece that not only fits the hole I want to plug but that doesn't also wriggle around and interfere with another area somewhere else. The first 75% of a heptomino construction I can generally sleep through pretty much, unless it's something with weird diagonal edges or a really thin rectangle, or multiple congruent shapes. But these it's tougher. Not something I can do so well while tired either.

Another issue I ran into was due to sorting the pieces into boxes before hand. I prioritised the 'normal regular pieces' box, only to near the end find I had an abundance of snakes. And snakes don't tend to tile well with other snakes, unless you've got a lucky situation where two of them bend in perfect unison and can just be stacked together. The top-left of the solution has a bit of a gathering of these pieces, as I furiously used them up before getting to the 2x2-sub-square pieces. Which are (marginally...) nicer to tile.

Again I got really lucky with the very endgame - I could I suppose say I made my own luck by being careful with the order I used the pieces, but there was still a big degree of trial and error involved too. I only had to spend about half an hour or 45 minutes max on the repeated backtracking phase getting the last 15 or 20 pieces to play nicely.

I wondered a little bit if it's not any particular property of the 2x2 or 2x3 sub-boxes that make those end game pieces fit together nicely as much as it's due to the lower perimeter that results from a more compact shape. Less units of edge that need to be matched perfectly?

Fig. 2: The completed 93x125 rectangle. I'm scared there might be mistakes in this drawing, because I caught one of them while finishing it and now I'm thinking could there be any more in there?

Total solve time, somewhere between 10 and 12 hours, probably closer to the 12 side but once you hit that flow state it's really hard to keep track of how long you've been sat there for. And bonus points for not building to the wrong dimensions, or with off-centered holes and having to tear out and redo any huge chunks. That's a first for enneominoes, and a rarity in general for me.

3x3x3 Dissection Puzzles - The Ultimate Guide

It's been too long since I posted on here. I was reading the blog over at Puzzle Mad, and noticed that the posts there are weekly, every week like clockwork. And they're decent length posts as well. And while that blog's wider scope means there's probably a lot more to talk about, it still kind of shamed me into putting something on here to make this blog a little less abandoned-looking.

(Also a reminder, I have a site at https://polyominoes.co.uk/ which while for now it looks similarly abandoned as here, will eventually hopefully be the main place for updates just because I like how much tidier I can make things with HTML. Tidier and very very dated...)

The Puzzles

I haven't really done much with any standard polyomino/iamond sets the way I used to - university took up a lot of time and makes me feel guilty using what little free time I have for fun things...) but my foray into polycubes in 2024 led me down a little rabbit hole of cubic dissection puzzles. That is, puzzles where the pieces are a selection of polycubes and the goal is to assemble them into a cube, generally a 3x3x3 or 4x4x4 one. The sets of pieces are generally not mathematically complete sets in any way, with any one unifying property, more they're just picked to give the solution process of the puzzle certain features.

Soma Cube

The classic. 

The original 3x3x3 cubic dissection puzzle (I think), designed way back in 1933 by Piet Hein. I've had several of these at various points, including one I made myself in a high school woodworking lesson that needed constant regluing because of how clumsily I initially made it. This one's a proper shop bought tidy wood one though.

Solving the pieces into a 3x3x3 cube is fairly straightforward (a little trial and error but not too much), but the real fun for the Soma cube is the fact the pieces are small and simple enough to facilitate construction of other shapes too. They're kind of like tetracubes in that regard. There's probably another blog post there actually. Maybe. Soma pieces are simple enough and few enough in number that full enumeration of the solutions for a given shape is possible, rather like the tetrominoes. It's an avenue I'm tempted to look into.

Overall rating: 4/5

Stewart Coffin's Serially Interlocking Cube

I've already talked about this one at length here, but I thought I'd mention it again, a.) for completeness, and b.) because it's such a good puzzle.

It's only four pieces, but they're shaped in such a way that they can only be assembled into the cube in a certain order, and the cube can only be disassembled in one way too. Each piece sort of holds the others in place meaning the solved cube can be picked up and generally chucked around without coming apart. Hence, 'interlocking' I suppose. It makes a good challenge figuring this out the first time, but subsequent solves you end up sort of remembering how it goes and the challenge wears off pretty quickly. The biggest hint is knowing that the finished cube needs 8 corner cubes, and that each of the pieces can supply a maximum of 1, 1, 3 and 3 corners respectively. And that narrows things down a little bit.

Overall rating: 4.5/5 (These ratings are calculated by the rigorous scientific process of pulling a number out of thin air and seeing if it kinda sort of feels right.)

Mikusiński's Cube

As well as the above design, there are several other puzzles of this sort in Stewart Coffin's book, 'The Puzzling World of Polyhedral Dissections'. Rather than make the lot out of wood (it's slow gluing them together, and with the cheap wood cubes I buy they're never a 100% fit even after copious sanding) I looked for other ways of constructing them. And hit upon these. It was a bag of like 1000 little 1cm² cubes probably designed as classroom teaching materials, but I found that by clicking them together then using nail clippers to bite off the extra remaining nub I was able to make solid enough prototype pieces. Yeah, if you exert a lot of force on them they'll come apart, and the little scar left where the nub was removed doesn't look too tidy, but for general experimentation into the world of polycube puzzles they do the job. Quick n' dirty. And if I get bored of a puzzle I can just take the pieces apart and build a different one.

Mikusiński's Cube is made up of five pieces, three pentacubes and two tetracubes. It's also maddeningly difficult - easy enough to assemble something that's one mispositioned cube away from completion but hitting upon one of the two actual solutions took me a couple of weeks, off and on. Mainly off, to be honest. But still, a good couple of hours were spent swearing at this one in pure frustration. To the point I was seriously doubting there was a solution, and that I may have just assembled one of the pieces incorrectly or the illustration in Coffin's book could have been wrong. It wasn't though, it was just a tough puzzle. The individual pieces are shown below, so as not to give away the solution to anyone who fancies making their own copy. Because with puzzles like these, once you know the solution it sticks in your memory fairly well and lowers the replay appeal somewhat.

The black piece is a mirror image of the white piece, by the way. The photo isn't great quality so it might not be clear.

Overall rating: 5/5

'Five-Piece Solid Block'

Another design by Stewart Coffin. This started life as a prototype made out of the above plastic cubes, but it's such a fun puzzle I went ahead and made a bigger more permanent wooden version. Five pieces, all of them asymmetric and wiggly. Again, finding the solution is tricky the first couple of times, but then you start to remember what goes where, and associate different pieces with the ones they interlock with...

Overall rating: 4.5/5

'Diabolical Cube'

Honestly, this one's anything but diabolical. Its main selling point is that the pieces are of sizes 2,3,4,5,6 and 7 cubes, but in terms of solving it it's not particularly satisfying a challenge. I made a version of this from the lazy plastic prototype pieces but didn't feel it was worth the wood making a proper copy.

Overall rating: 2/5

'Wooden 3D Puzzle Cube'

This is just 9 identical copies of the V-shaped tricube. Solving it is insultingly easy, but the pieces are all painted pretty colours, it comes in a nice metal box and it cost me like £1.99 from Aldi, so I can't complain too much.

Overall rating: 3/5

This random blue plastic one I own

I got this in the gift shop of some museum (I think it was the optical illusion place in Keswick, Cumbria) or other when I was like 7 or 8, along with a similar red one with different pieces which I've since lost. There's not really any rhyme or reason to the pieces, but their chirality does kind of throw you off when solving so it's never as straightforward as I imagine it'll be. Still not difficult difficult though, like there's a guaranteed solve after a few minutes playing around.

(If I recall correctly, the red one was a lot harder, but that could just have been because the last time I owned the complete thing with no missing bits I was 11 and not quite as good at puzzles.)

Overall rating: 2.5/5.

EDIT: I did some looking around and I think these are two from the 'Impuzzables' series, designed by Gerard D'Arcey and mentioned in Martin Gardner's 'Knotted Doughnuts and Other Mathematical Entertainments'.

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Conclusion

If you think about how many ways its possible to break a 3x3x3 cube into two or more pieces, its clear that these half-dozen specimens are only scratching the surface of the world of polycube dissection puzzles. A few seconds of internet searching will find you tons more, I'm sure, and I have been on occasion attempting to come up with my own designs. but they all just end up being a bunch of random wiggly tetra- and pentacubes that happen to make a cube in an unknown number of ways. There's never been that satisfying feeling that there's something interesting or deliberate about the set thus far. Nothing that has a peculiarity about the solution like the serially interlocking pieces, or a pleasing completeness to the set of pieces themselves. But that's the joy of those plastic cubes, isn't it? The ability to rip the pieces up and try again.

Tune in next time, and I'll dig out all the 4x4x4 dissection puzzles I own and bitch about those.

Enneiamonds Again

A breakthrough, of sorts. This was the first time I managed to solve something with the full set of enneiamonds without having to resort to any kind of help from computer solvers or anything to knock in the last few pieces. The last few solves I did with them there was always a little bit of 'cheating' going on, not necessarily using the computer so fully solve the end of the puzzle but to check that a solution existed with the pieces I had remaining. But none of that this time. The training wheels are off.

Solve time took a while, but it was lots of short bursts spread out over a week or so. The first 80% of it (solving left to right) came in one sitting, probably an hour or so, then the end was just twenty minutes or so here and there, between university and other things, often getting all but one piece in there. And then today, by fluke more than design I'm sure, I hit upon a solution.

(That one little green unit triangle is there to mark 50 units along the top edge - I'd started off with more there, marking out every ten rows so I didn't build to the wrong dimensions like I have a habit of doing - but as the region between 50 and 60 units was the hardest to complete that marker stuck around and I guess I forgot to tidy it up.)

Here's a nicer image of the full thing, because the photo isn't exactly clear when there's two or more pieces of the same colour bordering each other. Clear acrylic would have solved this, but hindsight and all that.

Fig. 1: The completed trapezium.

Pentacubes (Planar ones mostly)

 A while ago when I made my set of tetracubes I mentioned the possibility of making the next set up, the pentacubes, if I ever felt in the mood for a lot of gluing and sanding. Well at some point between then and now (probably last year when I wasn't really doing a lot of blog post or website writing) I was obviously up for it because now I have a set of pentacubes to call my own. Very rough ones, admittedly - I glued but didn't bother to sand - but they work and can be fitted together without issue so they'll do.

The plan was (and still is, I suppose) to colour the edges of each piece just like I did for the unit cubes so the boundaries are a bit more distinct. But to do that I'd need to sand them, and to sand them I'd need to bring them to somewhere without a bit of outside space. So it's not happened yet.

The total volume of the pentacubes is 5x29 = 145 which clearly can't be factorised into three numbers greater than 1 to be the length, width and girth of a solid cuboid. The closest thing I could immediately find was adding two monocubes (or, you know, cubes) to the set to bump up the volume to 147 which can be factored as 3x7x7. And it turns out that packing them into this shape (as in the above picture) is a fun and satisfying challenge. And it can be made more challenging by specifying the position of the unit cubes before hand to yield a symmetrical configuration.

Schematic for a symmetrical solution to the 3x7x7. A square indicates the piece extends up into the layer above, and a dot indicates it extends down into the layer below.

Isometric diagram of the above solution.

Planar Pentacubes

These are the twelve pentacubes whose cubes all lie in the same plane - essentially, just chunky pentominoes. As well as solving into cuboids of sizes 1x6x10, 1x5x12 etc., these can also be solved into 2x3x10, 2x5x6 and 3x4x5 cuboids. And these, from my experience, are way harder than the polyomino rectangle counterparts. There are apparently 3,940 solutions to the 3x4x5 but you wouldn't think it trying to find one manually. I had tried on and off for several months and never really even got that close until a few nights ago when I found the below solution.

The 2x3x10 cuboid is even more of a nightmare, and it's mainly because of the I-pentomino (or rather, I-pentacube). Wherever you stick it it creates a narrow little space that seems to severely limit the pieces it can accommodate. I managed to stumbled across one of the 12 solutions, and it took several hours I'll never get back.

The 2x5x6 cuboid is interesting. It has 264 solutions, of which one is special in that none of the pieces extend into both 'layers' of the shape. Meaning that it's essentially the pentominoes solved into two congruent 5x6 rectangles then stacked up on top of each other. Like this:

I've noticed the shading on these images seems to subconsciously reflect where I am relative to the light source in the room. Here in the day time I have a window to my right, so I drew the surfaces facing it the lightest, but when I drew these ones a while back I must have been working at night with artificial light.

Another configuration possible with the planar pentacubes is the following sort of 'ring' shape. This can be made trivially in a few ways by adapting the solutions to the 3x20 pentomino rectangle, effectively folding it around on itself and bringing the ends together.

The ring shape. That 1x7 void in the top goes right through and out the bottom.

A better diagram of the assembly.

There's clearly a lot of possibilities with this set and with the full set of pentacubes that I've only just started to dip a toe into here. So I imagine there'll probably be further posts on here as I keep playing about and maybe gaining a little bit of competence with them. Right now it feels very trial and error.

In the mean time, check out David Goodger's page which has a way more in-depth exploration of what these sets can do.

Solving Technique: 'Piece Substitution'

By sheer chance when solving a hexomino thing a few days ago I was presented with a really nice clear example of a technique that I'm sure I always just referred to in blog posts and things as 'piece substitution' but never actually ever bothered to clarify. So here it is. Consider yonder picture:

The two pieces in red to the side are the two I'm left with, and the hole remaining just won't accommodate them in any way short of physically snapping the pieces apart. The best we can do is getting the more irregular piece in there in the obvious place, leaving a longer thinner 'L' shaped gap than we're capable of filling. Like this.

The trick here is to look at the two pieces in light blue, one of which is the long skinny 'L' piece we need. Notice that we can do this:

which uses up our unusable P-shaped piece, and at the same time frees up out long piece, allowing us to fill the other hole and complete the puzzle.

Of course, there's no guarantee that it'll fall into place as nicely as that. Sometimes it's two pairs of pieces that can make the same shape that need to be swapped, or sometimes it's even uglier, like a chain of substitutions that free up one particular piece then use that piece to free up another. But it's a viable technique surprisingly often given how much of an utter fluke it looks.

Miscellaneous Solutions That Didn't Deserve Their Own Posts, Volume 2

First here's an interesting coincidence. The total areas covered by the hexominoes and heptominoes are 210 and 756 respectively. In both cases these numbers can be written in the form n² − n where n is a triangular number (15 and 28). This can be illustrated as in the image below:

Sadly the areas of the other polyomino sets don't continue the pattern - the trominoes get closest: they have area 6 = 3² − 3 with 3 a triangular number, but a 3x3 square isn't big enough to hold three internal holes.

Secondly here's a construction with the heptominoes that I did a few days ago as a means of easing myself back into the whole solving lark. Like all skills it works like a muscle, don't use it for a few months and it's really difficult when you get back into it.

Solved top rectangle to bottom, right to left for the first one then left to right for the second and third. And I relearnt the importance of piece ordering, i.e. using the really difficult pieces up as soon as you possibly can. Because I'd accidentally held onto the 'F' and square bracket shaped pieces in the bottom right for way too long, and they made finishing that last corner really really difficult.

Finally, here's another heptomino thing. The idea/design was a challenge posed by Livio Zucca on the Puzzle Fun Facebook group, and the solve was not easy. So not easy, in fact, that I had to resort to FlayPoly2 to place the last 10 or so pieces. The four little green rectangles took up a lot of nice corner pieces for a start, and then those four 3xn limbs on the blue and yellow sections that reach towards the centre were massively restrictive too in terms of which pieces would actually fit there.

So if you already read the Puzzle Fun group religiously then 2/3 of this blog post will have been old news by now. And the other 1/3 isn't particularly interesting news either.

Oh, and here's an octomino rectangle that I solved about a year ago then never bothered to post anywhere as far as I know. Forty-seven by sixty-three.

EDIT: And this one too. This one has a 28x28 internal void that fits the heptominoes. It could even fit that 28x28 with the holes arranged all triangular from the very top of this post if you switched it out. But here it's got a heptomino pattern where the diagonal lines of holes sort of kind of line up with four of the holes in the octominoes.

Again, both heptomino and octomino sections all done entirely by hand (well, using a set of plastic octominoes, which is a little bit easier than pure pen and paper but still takes some doing).

For all the effort it takes to digitise these solutions, I have no idea why I just sort of let them fester on a folder called 'Polyominoes' on my desktop and don't do anything with them. Until the time comes when I notice it's been ages since I've put anything up on the blog or website, and I hastily chuck together a post like this one with them all in.

More Hexahaxes

Been a bit busy the past couple of months, and also I lost my main polyform-solving table to a sort of part time working-from-home setup. So things worth blogging about have been a little thin on the ground. But here goes:

polyformer.py - A Substitute for Creativity

A little while ago I made this:

Basically, you tell it what size your set of polyforms covers, how many holes you want and how big an individual piece is, and it calculates possible shapes that can be tiled - squares, rectangles, triangles, diamonds, groups of congruent rectangles, etc. It doesn't tile them, that's left up to the user, it just gives some suggestions as to what can be done. Essentially, I was sick of doing all the area calculations by hand so I automated it. And every so often I'll think of a new class of shapes that might be interesting and it's not a lot of work to add that into the program as and when. It doesn't handle parity (yet), and sometimes it'll just suggest something completely impossible because I overlooked something, but on the whole it does its job.

Solving Hexahexes

Hexahexes are a relatively unexplored territory to me - partly just because I haven't had the pieces very long, and partly because I just generally overlook polyhexes for whatever reason. And partly because there's a piece with a hole and that's just another little irritating detail you have to plan for when designing constructions (or constructing designs, as the case may be). But the program said that there was lots of fun things to do with the set (once I'd deciphered the confusing shorthand that is the code's output - it made sense on the day I wrote it but I quickly forgot which sides of the parallelogram etc. the lengths all referred to.)

Fig. 1: Here's one of the aforementioned constructions.

Solving with hexahexes is a piece of piss. sort of. There's not really anything weird like parity to deal with, and the proportion of friendly easy pieces that work at the end of the solution is quite high. After you've burnt through the stack of hideous wiggly wormy pieces that look like diagrams escaped from some cursed organic chemistry textbook then it's a solve that I'd rank somewhere between the hexominoes and heptominoes in terms of challenge.

Fig. 2: Here's some example nice co-operative pieces. Just for reference, or if you own a set of these yourself and want some handy tips, but also want to be spared the pain of many trial-and-error hard solves while you work out an optimal piece order.

Here's a couple of other miscellaneous solves.

A Final Random Thing

I get the feeling that a set of heptahexes (on a smaller scale than these ones) wouldn't be outside the realms of possibility. There's 333 of them, so less than the number of octominoes, and I've seen from the enneominoes that scaling down pieces even by quite a bit doesn't have a massive effect on their usability. Sure it just feels more satisfying solving with big meaty pieces that have some weight to them, but in terms of practicality (and cost!) some half or two-thirds scale 'hexes would be more sensible. So I'll see. I'm holding off on the laser cutting right now - still letting my wallet recover after the enneominoes - but some day...