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.
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