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.