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...

Planar Heptacubes the Lazy Way

Usually polycube constructions are beyond my capability, since I don't own any physical sets beyond the tetracubes and have a computer that would probably struggle to find things with sets much larger than pentacubes. (I think hexacubes have a checkerboard parity issue just like the hexominoes as well, which makes the prospect all the more terrifying.) But recently I realised something about the set of thee 108 planar heptacubes that could make a few solutions possible despite only owning a set of bog-standard heptominoes.

Of the planar heptacubes the only one that really needs any special attention is the harbour heptomino equivalent. Unlike the case with regular heptominoes it doesn't force a hole in the construction, but it does sort of require that it must be placed perpendicular to whichever heptacube is filling its hole. At some point a few days ago I realised that if the harbour heptacube (it feels weird calling it that somehow) was 'neutralised' by a small number of pieces then I could solve the rest of the heptacube construction purely in layers using my existing set of heptominoes. Saving me from having to cut up and glue 108 pieces. Which I'll do eventually, just not now. The tetracubes were already pushing the limits of what I can make.

It turns out that two other heptacubes are enough to make a little bullet-heptomino-prism that can be slipped into the corner of a 3x14x18 cuboid, leaving three layers with a multiple of 7 cubes that can be solved manually. Something similar is probably possible for a 6-layer cuboid, 6x9x14 maybe.

Or a 4x9x21... And solving into four rectangles is a lot less painful than solving six... Anyway, here it is:

As usual, dots indicate that a piece extends down into the layer below, and a little square means it extends up into the later above. In this case the three heptacubes that lie perpendicular to the plane so to speak are the analogues of the pi and harbour heptomino and the one that looks like a little H inside a 3x3 box.

Here's a view of it as a nice isometric image, a needless victory lap purely because I learnt how to use Inkscape recently and want to show off:

If I can ever manage to solve anything with the larger polyiamond sets, at least it'll guarantee that the images on that blog post will look better than the old ones do.

EDIT sort of: A few hours after writing this, the urge to solve the 4x9x21 became too difficult to resist, and after about an hour or so I had the following solution using the exact same trick as the 3x14x18 for dealing with the harbour piece:

If I ever do make a set of these pieces, a box that holds them in a formation like this would be really nice probably. EDIT: Just after I first posted this post I got a comment pointing out something I should really have spotted sooner - the 'H' heptomino in these constructions goes through the central hole of the harbour heptomino, so unless the pieces were made of something really flexible (like rubber or very soft foam or something) there's no way you'd actually be able to assemble any of these physically.

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.