Heptominoes Miscellany

A few more heptominoes type things I've done recently, since lots of small posts is easier than one big post. Again, nothing particularly groundbreaking here.

Here's a 11x71 with 25 holes arranged in a central grille:

I solved the middle bit first seeing as it put the most restriction on pieces that could be used. Then built the left side (using up a handful of nice useful 2x2-blocky pieces in the process), then the right hand side. I don't know whether it's possible to do a 23x35 with a grid of 49 internal holes. The maths says yes but the pieces themselves might not allow it.

And here's an 11x69 rectangle with 3 holes.

I think I'm getting the hang of rectangles now; the biggest challenge here seemed to be finding a surface long enough to construct it on. Next time I think I'm going to have to try my hand at putting together some more involved figures, and confront my fear of building things with diagonal edges. (I mean, it can't be that hard can it? Look at how many heptominoes have got those wiggly zig-zag edges anyway.)

Congruent Rectangles with Heptominoes - Part 2

Part 1 is here. It's dead disappointing, mind, there's only one set of rectangles and it's the 4x 10x19s. But in that post I mentioned a couple of other possibilities for sets of rectangles that use the entire set of heptominoes between them. Here are some (coincidentally none of these are the ones mentioned in the above post...)

First there's this set of three 15x17s using 36 pieces each which I found ages ago, but didn't feel like it deserved an entire blog post to itself at the time

Fig. 1: Three 15x17 rectangles. You can almost tell by the pieces used in each which order I built them in.

In the first post or Part 1 or whatever we're calling it, I had somehow overlooked the fact that 108 divides by 6, and that a set of 6 rectangles should therefore be possible. The minimum number of holes that works is 12 (I think), 2 per rectangle, and that gives each an area of 128 cells which can be done as 8x16 (I can't rule out 4x32, but at the same time I'm utterly terrified of the prospect of trying to fit heptominoes into a 4-cell-wide anything.)

Fig. 2: Six 8x16 rectangles.

There wasn't really any pleasing ways of distributing the holes, this is about as good as it gets. Also, just look at those four hideous stretched-F-pentomino-looking pieces clustered in the middle of the top-right rectangle. I had accidentally forgotten to use those up sooner, and wound up stuck with them at the very end. That last rectangle (top-right, the one with all the supposedly easy to work with pieces) was partially a computer search job too. I think it was getting late and I needed the table space for something else so I had to speed up the solving process somehow.

Still not touching the nine 5x17s with centered holes though. It's gonna take one hell of a rainy day to drive me to attempt that.

Octominoes!

Well, this was an utter ordeal. I used the same method as my last little failed attempt at an octomino construction, the whole two MS Paint windows deal, but with a slightly differently shaped rectangle (20x148 this time, with eight holes, six of which are due to the holey octominoes.) And this time I ran into a different set of problems. I got about 90% of the way through the construction and did the usual sanity check, counting the amount of free area left and praying it divides by eight and I haven't drawn a heptomino (or a nonomino* for that matter) in there anywhere by mistake.

I counted 168 unit squares remaining. So far so good. But then I counted the remaining octominoes. And got 21. And 23x8 is not 168. Something was afoot, but it being fairly late on Friday night (because I lead such an exciting life...) I was far too tired to work out exactly what was up. so I hit the hay and resolved to see what was up tomorrow.

Oddly enough, as I was falling asleep I had some kind of major tetris effect going on, seeing endless visions of octominoes fitting together in various ways all wiggly like; a veritable kama sutra for tetris blocks. And, as I'd been reading a fair bit on organic chemistry recently, in my bizarre sleep-deprived state I was also fruitlessly trying to assign the 'systematic name' to each octomino based on the way it branched and twisted, not quite awake enough to notice that the octomino was not in fact a molecule.

Well, the next morning and with a fresh pair of eyes, I took another look at the almost-complete rectangle and couldn't immediately spot any foul play just by eyeballing it. My guess was that I'd somehow used a piece twice, forgot to cross it off the 'used bits' list the first time round. And so began the laborious task of verifying this - getting a fresh image of all the octominoes up, then crossing each one off as I highlighted it in my construction. And if I found one that had been previously used... well, I'd cross that bridge when I came to it.

And so it transpired I had inadvertently duplicated two pieces. Thankfully, they were both right near the bottom of the construction; I only had to backtrack about 15 pieces to be back in a state where the rest was solvable. I guess I had started to get a bit careless just as I was becoming too tired to think properly and in hindsight it was lucky I called it a night when I did on the Friday. So I continued (being just a little more careful this time) and managed to get the rest of the pieces in without incident.

Of course I used FlatPoly2 as a further sanity check when I had about 12 pieces left, just to make sure I hadn't solved myself into an impossible endgame. And in running that check I might have accidentally glimpsed the position of two or three pieces that allowed a solution. But I did the rest of it. All by hand! And it only took me, what, six hours or so? (Actually, putting it that way, it feels like a colossal waste of time I could have spent doing something useful, but...)

Anyway, here it is, in all its glory:

Fig. 1: All 369 octominoes in a 20x148 rectangle with symmetrically placed holes. Not pictured: enough blood, sweat and tears to fill an Olympic swimming pool.

All of this points to one thing - I need to get myself a physical set of octominoes by any means necessary. It's a little bit harder to use a shape twice when you've only got one of 'em to hand.

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* Or is it 'enneomino'? I think I've seen both but I can't decide which I prefer.

Combining Pentominoes and Hexominoes (into one nice big happy family)

In an earlier post, (this one in fact) I mentioned the possibility of combining the sets of hexominoes and pentominoes together, creating a nice (if mathematically incomplete) set of 47 pieces covering a total area of 270 square units and not having any of the irritating parity issues that make hexominoes on their own such a ball-ache to build shapes with.

And as an extra added bonus, you can hoard some of the nicer pentominoes until the end (ones like the P, L and V) which makes solving things with these generally fairly easy too. Nice practice before tackling pure hexomino things, if nothing else.

Rectangles

So... the rectangles. 270 divides by nearly anything so we've got a lot of choice here. 1x270 and 2x135 can be ruled out straight of the bat, because the pieces are too big. 3x90 I don't know, I mean, every piece fits within a 3xn box but I could see that getting ugly fast - look at how few 3xn solutions there are for just the pentominoes alone! Maybe it has a solution, like the thin solution with the pentahexes, but I don't have the balls to go looking for it.

5x54 is where things start to get definitely possible, and the rest of the possible rectangles (6x45, 9x30, 10x27 and 15 by whatever 270 divided by 15 is) are all easy enough if you don't mind a little trial and error.

Seriously, get yourself a set of these. Buy them, make them, hack them out of the back of a cereal box, whatever it takes. You won't regret it.*

Fig. 1: Some rectangles. Looking at the notebook I transcribed these from, some of these were done back in 2015 or so, before I'd really refined what little technique I have.

Now, moving onwards to some more tricky stuff...

Rectangles 2: The Revenge

Since 270 divides up so well, not only can you make rectangles but you can do sets of congruent smaller rectangles too! Check it out:

Fig. 2: Three 5x18s, two 9x15s, three 9x10s and five 6x9s.

The challenge here is making sure you have the right balance of pent- and hexominoes in each rectangle. For example, for the bottom row (the five pink 6x9s) each rectangle is 54 units, and the only way to get 54 by adding 5's and 6's is either nine hexominoes (9x6) or six pentominoes and four hexominoes (6x5 + 4x6 = 54).

For the same sort of reason, we can rule out six 5x9s. Each rectangle would have 45 units, which can only be filled by 3 pentominoes and 5 hexominoes (since 3x5 + 5x6 = 45) but that would require 15 pentominoes total so it can't be done :(

Nine 5x6s is out too. A 30-cell rectangle must have either 0 or 6 pentominoes so that the remainder can be filled with hexominoes - that is, it must be either all pent or all hex. This would mean that all the hexominoes would end up together in seven 5x6 blocks... but then you could just push those together and make a 6x21 rectangle of pure hexominoes, which ain't possible because of parity constraints.

Parallelograms

Moving on to some less rectangular shapes...
Again, we've got a lot of variety possible here -  the base lengths 6, 9, 10, 15, 18, 27, 30, 45 and 54 are solvable in theory, but in reality it's a little bit of a different story.

Of these, I've only done a selection of these by hand, mainly because they're not massively challenging or interesting (then again, if I didn't bang on about uninteresting things here this blog would have like no posts.) The longer the diagonal sides, the trickier it is, just because you run out of wiggly pieces building one side then have to use the nice easy pieces on the second side.

Fig. 3: Two example parallelograms. You can solve the rest yourself (with your newly-acquired set of pieces!) because it takes so long to transfer them from notebook sketches into pretty diagrams and it's really not worth it for not-that-interesting solutions like these.

Of course, you don't need to stop there. With the aid of computer search ('cause I'm lazy like that) I found the following two even taller skinnier parallelograms

Fig. 4: Just look at these freaks of nature. And I'm willing to bet the even thinner 6x45 is possible too, but so far FlatPoly2 has failed me.

Rectangles with little bits missing from inside 'em

...because not only does 270 factorise well, but some of the numbers just above it factorise fairly nicely too!

Fig. 5: Holey rectangles. (I fought the urge to follow that with 'Batman')

Getting all Fancy

Depending on how much you like solving wiggly diagonal edges there are all sorts of other possibilities too, so long as you don't mind the odd hole slap-bang in the middle of everything. Check out the following bad lads, sorted in order of how many tears were shed before finding the solution:

Fig. 6: How many images have I produced for this blog? And I still haven't settled on a colour scheme... sheesh!

That central heptomino is completely arbitrary really, it's just the most fitting one that has a few axes of symmetry. The same patterns could be done with a hole the shape of the H-looking heptomino (for example) but it just doesn't look as neat.

Another option (which to be honest I haven't really looked at myself on account of being proper lazy) is putting additional restrictions on the way the pentominoes are distributed, for example, making sure that each pentomino isn't touching any others, or that they are all bunched together in particular ways. The solution below has the pentominoes grouped into two symmetrical end bits. Actually, I didn't solve this one thinking about it as a pent+hex construction, I did the middle hexomino bit and only realised the two pentomino end caps were possible a while later.

Fig. 7: A nice five-cell high pattern with the pentominoes bunched together at each end all nice and symmetrical.

Another Possibility

A slightly more complete-feeling set would be the entire range of 1 through 6-ominoes, but sadly, their total area is 299 units, which you can't really do a lot with. There's a 13x23 rectangle, and by extension two 13x23 parallelograms, and adding a unit cell hole allows things like the 24x24 right-angled triangle.
But then you've got monominoes and monomino-sized holes together in the same construction, which just isn't very aesthetically pleasing for some reason. And since you've got all the tiny little triomino, domino and monomino pieces to work with, these aren't that much of a challenge either. So it's not such a good set after all.

Heptominoes next time. I promise.

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* Polyominoes the Blog accepts no responsibility for any regret caused by the acquisition of any polyform sets.

"There's No Way I'd've Found That By Hand!" Vol. 1

At work I have a notebook. And in it there's surprisingly little pertaining to actual work, but lots of ideas, calculations and sketches for possible shapes that could be filled up with various sets of polyominoes or other polyforms. Most of these end up solved at home, then drawn up all pretty and posted on here, but there are some that I either try and can't do, or that just look so intimidating I don't even try. It's those jagged diagonal edges... I just have a right job doing them.

So, here's a little collection of solutions found with various programs, that I was too weak to suss out by myself.

In hindsight, this one doesn't look that bad, and I've solved similar in the past. I guess at the time I was just not feeling up for the challenge. I think (judging by the colour and scale of the image) this one was found using the solver on Peter Esser's site here.

I'll be honest, I didn't even attempt this one. I did the calculations to make sure that it was permitted by the parity constraints, then just despaired at the thought of having to actually solve it, central holes and all. But FlatPoly2 made short work of it, finding this in about five seconds flat.

Then there's this family of solutions I have no recollection of looking for but are in the folder called 'Polyominoes' so I may as well just post them for the hell of it.

 I think at this point, hexomino solutions just aren't the most impressive thing in the world any more, computer-found or not. But more interesting things like heptominoes take a while to do, and I've got driving lessons and band-related business to contend with on most evenings, so that stuff tends to take a back seat.
Actually, I've got a nice new shiny set of octiamonds that I've yet to do anything of note with... I'll have to have a little play around with those, see if I can create anything worth posting on here with them.

Hexominoes - Scraping the Bottom of the Barrel

In a previous post I made some pretty shapes with the hexominoes and had a good whinge about parity while I was at it. And at the time I thought I'd pretty much exhausted all the possibilities for nice things that could be done with hexominoes. But then a few weeks after that, I found myself on holiday in a little caravan, with the weather being utterly atrocious outside. Luckily, I had decided to bring my little homemade hexomino set along with me, so in the absence of anything else to do I uh... constructed some more constructions, I suppose.

First off, a good (but probably not 100% sure-fire) technique for finding things that are actually constructible (and aren't ruled out by parity constraints) is to start with a rectangle that has odd x odd dimensions (and an area > 210 units) then carve out a central hole until the total remaining area is 210 units. As long as you're fairly careful here, the resulting shape should be solvable.
(Some day I'll do a write up of parity the way I understand it on here... honestly more for my own benefit than anyone else's.)

Fig. 1: Three rectangles with nice symmetrical cross-shaped holes hacked out of them. That third one, I've seen a different solution to before (I think it might be in Solomon Golomb's book).

The third solution, the 15x17 one, took an outrageous amount of time and attempts to solve. It's a shame that processing-power-wise we're a long way off being able to enumerate solutions for a given shape with hexominoes (like we can for pentominoes) - it'd be interesting to see if the ones that take me ages have fewer solutions and are genuinely harder.

I also found a solution to a 13x21 rectangle with a 7x9 removed from the centre. With the central hole being 63 unit squares, that means there's enough space to get all the pentominoes stuffed in there with a little bit of wiggle room to spare.

Fig. 2: This took a while too, as far as I remember. Hexominoes and thin sections (the 3xn bits above and below the pentominoes) don't seem to mix too well.

Then I realised that I didn't need to restrict myself to taking the excess squares out of the middle of the rectangle - I could just as easily round off the edges, and make some stretched-out octagons.
Which I did.

Fig. 3: A way stretched-out octagon.

And here's a slightly more regular, more octagonal-looking one I found a little while later:

Fig. 4: I bet there's a way of doing this with the six internal holes all in a diagonal line, but I'll leave that as an exercise for the reader 'cause I can't be arsed right now.

And just for an extra Brucie bonus, while we're on the subject of octagons, here's one made out of heptominoes that I found a few weeks ago:

Fig. 5: For whatever reason I have no recollection whatsoever of the actual solving process that led to this.

Congruent Rectangles with Heptominoes

(This post might be Part 1 of many, depending on how much free time I have to look for other sets of rectangles.)

A few weeks ago I found this:

Fig. 1 - Four 10x19 rectangles using one complete set of heptominoes.

There's a number of ways of making sets of same-size identical holey rectangles with the heptominoes. First off, there's three 11x23s which I've done before in an earlier post (and since 11 and 23 are both odd you can get the hole slap bang in the centre of each rectangle.)

Then, there's four 10x19s as above. Then it looks like the next set of rectangles that might be possible is nine 5x17s (with potential for centered holes too!) but whether that's even solvable is another question entirely. I mean, it goes without saying it'll be a frustrating ordeal knocking it out by hand. Actually, Patrick Hamlyn managed twelve 8x8 squares (three quarters of the way down this page) so anything's possible.

***

Also, a thing I've found that works really well for heptominoes if you're just after solutions and not the glory of finding them entirely by hand. Solving the first 90% of a shape by hand, saving the nice clumpy blocky bits of course (that goes without saying), then crack open some software like FlatPoly2 to brute-force the last 10 or so pieces. This works with heptominoes especially, since the really wide search tree that is placing the first ninety-odd pieces doesn't seem to play particularly nice with my computer. (More expensive computers may get better results!)

Then there's that other thing. The one that feels partially like cheating, just a teensy little bit. Picture the scene, you're just spend two hours slogging away at a particularly gnarly construction and just can't get the last few pieces in. There's a point at which you start to wonder if what you've got left here is even possible. So occasionally I've succumbed to the temptation to input in the shape of the remaining hole and the last five or six pieces, not to find a solution, but just to check if one exists. Then if it transpires you're looking for a solution in a space where none exist, I tear out one or two pieces, input the new, bigger hole and keep checking until we've got a gap where there is definitely at least one possible solution. Finding that will still be an ordeal though.

So it's computer-assisted solving. I don't know whether it counts as truly properly solving by hand, but when I've just sunk a whole evening into a construction and it's pushing eleven and I've got work tomorrow, I'm usually not too bothered.