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!)