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.