Concentric Rectangles

I'd like to blame the fact I'm stuck indoors avoiding the Coronavirus for this one, but realistically I'd have probably found the time to solve this anyway, given that I lead such an exciting life:

It's an awful photo, but here's the gist of it. Imagine a 7x9 pentomino rectangle (with a little 3-cell hole) inside a 13x21 hexomino rectangle, inside a heptomino rectangle I can't remember the dimensions of and can't be arsed to count, all surrounded by the octominoes (plus 8 holes) in a whopping 47x85 rectangle. And that's what I spent an entire afternoon doing. Well, two sittings with a break for dinner in the middle.

It would maybe have been a bit nicer if I'd started with a central monomino and worked my way up through all polyomino sizes, a la Karl Wilk's Polyominium, but there didn't seem to be a way of doing it that yielded such nice symmetrical layers like this. It's difficult to wrap a pentomino rectangle around a hole big enough for just the tetrominoes, let alone anything else.
In fact, when I had the original idea that became this, it was born out of the fact I'd built a hexomino pattern that just happened to be able to fit the 7x9 rectangle inside.

A sort of precursor solution found way back.

Peeps with a keen eye will have spotted that the pent-, hex- and heptomino sections of that early solution are totally different to the ones this time round - more a testament to how bored I was than anything else.

Fun fact about the 7x9 pentomino solution: when the triomino hole is vertical there are 360 possible solutions, whereas with the horizontal hole there are a mere 150 (excluding rotations, reflections and all that jazz.)

Here's the full solution drawn up so you can actually see where one piece ends and another begins:

And there's a little voice inside me saying "What about a layer of nonominoes?" but realistically I'm not going to be able to do that without a physical set of them and that ain't gonna be cheap. Besides, there's not a flat surface in my house big enough to hold all those pieces.

Octiamonds

Polyiamonds are tricky. Even the smaller sets seem harder than their similarly-sized polyomino counterparts. Hexiamonds, for example. There's 12 of them, same as the pentominoes, and in a geometric sense they don't seem any more jagged or otherwise unruly, but for whatever reason solving anything with them seems a lot harder.
I mean, it could just be that I'm more used to polyominoes and that a sort of intuition for other polyforms would build naturally with time. And with polyominoes I know all the handy tips and tricks, which pieces (or types of pieces) are most useful in which situations, whereas with polyiamonds I don't have that (yet).
With larger sets of polyominoes, (i.e. hexominoes and above) a technique emerges of saving the more cooperative shapes for the end game, and as the sets get larger this pool of 'nice' pieces increases rapidly in size. But with (say) octiamonds it's not so obvious which shapes are the most useful. They're all pretty hideous, actually, at least to the untrained eye. The little hexagon made of six triangles is the closest analogue I can think of to the 2x2 square block that makes for nicely-behaved polyominoes. But there are only 4 out of 66 octiamonds which contain it and even these 4 pieces don't play especially nicely together with each other.

So the solution below was the result of about an hour and a half of stumbling about cluelessly followed by a flash of pure luck.

Fig. 1: The 66 octiamonds in a 12x22 parallelogram.

 Another fun fact: Drawing these out neatly is really hard too. Pixel art and triangles don't mix too well.

Finally! Truncated 55x55 Square with Octominoes

Third time lucky, eh?
Getting that central 13-hole configuration to work was surprisingly tricky - they're too close together to just treat as individual holes, sling a holey octomino around some of them and be done with it.*

Then throughout the rest of the solve I had in the back of my mind a little nagging concern that maybe some issue like parity would render this solution impossible anyway. The octominoes as a full set have no glaring issues the way the tetrominoes and hexominoes do, but the 363 non-holey ones are imbalanced when checkerboard-coloured. And since I'd used the 6 holey pieces first I was in effect left with this imbalanced set. I assumed (well, hoped) that since the construction's dimensions were odd x odd this might negate the issue; I use this as a rule of thumb for hexomino constructions because it usually means that the overall structure is sufficiently unbalanced and therefore solvable.
Whether this makes any sense mathematically I have no idea.

Total solve time was approximately 5 hours spread over a few days. Total time drawing up the digitised image of the solution probably took another hour on top of that, come to think of it.

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* This construction by David Bird does something similar, there's probably only a handful of ways of accommodating those holes in that shape.

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After playing with octominoes for a bit, returning to the trusty ol' heptomino set is a strange feeling indeed. All the pieces are so tiny and simple in comparison, and suddenly 108 feels like such a reasonable, manageable number.
I made this, which has the maximum possible symmetry for a heptomino pattern.

Fig. 1: 28x28 with 28 holes.

This was a surprisingly painless solve too (although anything would feel that way after octominoes) - the last pieces all fitted into place on practically the first attempt and the whole solve couldn't have taken much more than half an hour. (And now there's an awful idea: speed-solving polyominoes!)

And here's some assorted hexomino things from 2019 that never made it onto the site because they weren't particularly challenging or interesting.

But these are the lengths you gotta go to when you've been too busy recently to do any interesting polyomino stuff but don't want to completely abandon the blog.

34x87 Octomino Rectangle

How to create a 34x87 rectangle with Octominoes (a handy guide)

Step 1: Solve 90% of a 37x84 rectangle, then wonder why there's so much space left over and not enough pieces to fill it (37x84 = 3108, which is slightly overshooting the octominoes' 2952+6 total area.)

Step 2: Despair for a little bit. Even with a physical set of pieces I find ways to screw things up.

Step 3: Salvage a nice big chunk from the starting corner of the failed solution (the corner where all the scariest, hardest to work with pieces live), and use this as the basis for a new rectangle with the correct dimensions this time...

Fig. 1: Success!

40x74 Rectangle with Octominoes

After the success of the new set of hexominoes I got made a while back, I decided to take the obvious next step and get a full set of octominoes cut too. They arrived about a week or so ago; three batches based on David Bird's three 29x34 rectangles construction*. Fluorescent green, fluorescent yellow, and a colour which is supposed to be fluorescent blue but doesn't really glow at all unless you turn a UV torch on it. Then it goes nuts. Again, all came with that sticky protective film on both sides of the acrylic so I spent the first few days just peeling it all off, a few pieces at a time. Fun times.

Fig. 1: The blue pieces look even worse here since the mat underneath them is dark green.

Once free of their protective sticky plastic stuff, I thought I'd break 'em in with a construction, choosing 40x74 simply because it was about the only thing that would fit nicely on the table. I like having the cutting mats there to make sure everything's all lined up and nice, and anything longer or wider than 40x74 wouldn't have fit on them properly.

Solving with physical pieces is a whole 'nother ball game to just solving the way I used to (or used to attempt to, I only ever got one complete solution that way.) For most of the solving process the hardest part by far is just finding the individual pieces you're after. Especially near the start when there are like 300 or more almost identical pieces to sift through.

Once I got to the last 10% of the construction the benefits of physical pieces really started to shine. The repeated backtracking that is just infeasible with drawing the pieces is now a lot more manageable. Which is just as well, because I must have spent close to two hours trying to get the last twenty or so pieces in. There were several times where I had 368 pieces down but the remaining hole was one cell out from the shape of the piece in my hand. That's the worst bit, those near-misses, but at least I suppose they mean I'm on the right track. If there was something terrifying like parity issues going on it would have at least alerted me.

Fig. 2: Getting there...

The entire solution took maybe about 4 or 5 hours total, a little last night and the rest this morning, which is a lot less than it usually took the old way. The little hole of unfinished pieces seemed to drift around as well during the endgame as I solved, eventually ending up right in the bottom-left corner of the photo above when all the pieces finally fell into place.

Fig. 3: Another side-effect of the colour scheme is that it photographs really badly. Especially when it's me holding the camera.

Here's the solution drawn out properly so you can actually make sense of it and tell where one piece ends and another begins:

Fig. 4: The finished rectangle.

Yeah, the spacing between the holes is imperfect. The gaps are 11-12-11 but it's about as good as you can get with an even-by-even rectangle.

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* Side note: Is there any information anywhere about how he constructed this (and his other solutions)? I always assumed he did them with a set of pieces but it's just as likely he did them using just pen and paper. Especially the nonomino patterns, making and using the full set of 1296 pieces would be a tad unwieldy.

Attempting 4xn Constructions with Hexominoes

4xn is about as tight as you can squeeze hexominoes.

3xn looks like it's pretty much impossible. Consider the blue pieces in the following image:

All four of these need to be in the final construction (and will only fit horizontally) but between them they create six 3-cell deep wells at the edge of the construction. And there's only five hexominoes (the red ones) that could fill those gaps. The I-hexomino could theoretically fill two 3-cell deep wells, but I think in every possible case the space between the two blue hexominoes either end of it would be less than 6 and therefore unfillable. Unless we're going for constructions with holes permitted, in which case that would be fine after all. But exceedingly difficult.

Then, check out these green cases that create two adjacent 2-cell deep wells. In filling one of them, either you have to use one of the red pieces from above, or you use a piece that has a 2-cell extension, which would then cover the square marked by the red 'X' and create a new 3 (or more) cell deep well.

I know this isn't a rigorous, mathematically watertight proof but it's enough of a deterrent to stop me spending ages looking for 3xn solutions.


So, back to 4-cell high...
Similar to how it is with pentominoes (the narrower the rectangle, the fewer solutions there are), finding 4xn rectangles with hexominoes has proven to be surprisingly challenging. The few search programs I know how to use don't seem to like really narrow rectangles very much either, which left me doubtful I'd be able to do much better by hand. (Although in hindsight it's more likely I just don't know how to use the programs as well as I think I do, or how to set them up so that they search efficiently.)

A few months back I'd found the solution below using the combined set of hexominoes and pentominoes. It's not really what I'm aiming for though; the addition of the smaller pentominoes makes this about a hundred times easier, and I was able to place the holes symmetrically as a result.

On 31/12/19 I had another crack at this, aiming for a 4x53 grid using just the hexominoes, but with no constraints on where the two holes would be. Just proving that a 4xn solution exists would be enough for the first step; making it all pretty could come later.

After far too long (about an hour, maybe? It's hard to say because I tend to lose track of time when doing things like this) I found a solution. This one:

It's butt ugly though; not only are the two holes not placed in any kind of order but one is on the edge of the rectangle too, which just doesn't look right to me. It's not really a hole now, is it? It's just some weird notch out of the side of the puzzle. Oh well, it's a start.

Side note, that 'T' piece near the right-hand end was the absolute worst piece to place. Had I used it up right near the start it might not have been such a pain in the arse, but somehow it escaped my attention until there wasn't a lot of long skinny pieces left that worked well with it. In order to place it vertically rather than horizontally, it needed to have one of the holes either side of it too, since it partitions the rectangle into two parts, both of odd size.

But, it's a proof of concept at least. A symmetrical 4x53 rectangle seems way more possible now than it did before. (Edit 06/09/2020: I found one!)