# Nine 9x37 Rectangles with Octominoes

The second in the 'Nine congruent rectangles with the octominoes' series. Last time the rectangles were closer to square and had less holes. but this time I decided to tackle 9x37 rectangles with 5-hole configurations in the centre, which would make it even harder than the last batch. And that last batch was hardly a walk in the cake.

### The First Eight

The more skinny a rectangle is, the harder it gets to tile with polyominoes. Because thin rectangles favour pieces with a long flat side, and these pieces run out fast. Then, every time you get to the end of a rectangle, the two corners there usually demand a piece with a clear right angle. And there aren't too many of those either. Well, there are a few, but they're generally the simpler more cooperative pieces, the kind you want to save for the dreaded final rectangle. It was painful: when I'd put all but two pieces into a rectangle I'd then mentally dissect the remaining 16-omino shaped hole, trying to partition it in such a way that both halves were awkward wiggly pieces. But it pretty much never happens that way - it's either one nice, practically rectangular, piece and a wiggly piece, or two wigglies, one of which has been already used elsewhere in the construction. So inevitably, each completion of a rectangle left the pool of nice endgame pieces a little emptier than before. And that means when you get to the final rectangle, you get problems.

Fig. 1: Getting the easy bit out of the way with.

### The Last Rectangle

I'm sure I've said before on here that for solutions like these 20% of the pieces take 80% of the time. In this case it was more that 10% of the pieces take 90% of the time, or possibly more.

The pieces I was left with after tackling the first eight were... less than ideal, let's just say. Some were the kind of pieces it's easy enough to use up in a large, wide rectangle, but that don't play so nicely in a construction like this where it's all edges. Some were just hideous. Check out numbers 2, 6, 7 and 41 in the image below.

Fig. 2: The remaining pieces, and the shape to fit them into.

I gave the last rectangle a valiant shot bare-handed but really struggled. I only ever managed to get about two thirds of the rectangle done, just past the central dots, and even then I'd find myself at either a dead end or with some really unappealing pieces left. Accepting that I wasn't going to get anywhere continuing in this vein, I threw the pieces and board into mops.exe (from Peter Esser's website) and let it run for a while, to see if it fared any better than I did.

It didn't. The best it managed was positioning 39 of the 41 pieces - which it had managed several times - but no better. Which concerned me. The lack of all-but-one-piece solutions was a concern, especially because given the amount of 39s I was seeing it seemed statistically likely that we'd see at least one 40. It was at this point I thought maybe there could be some additional restriction, something like parity, that I'd overlooked.

I posted the above image of the 41 pieces and the board to the Puzzle Fun Facebook group, asking if there was any restrictions that made it impossible but that I had missed. And not long after, it turned out that not only was there no such issue preventing it from being possible, but Patrick Hamlyn (I don't think he has a website or I'd link it here) had managed to find a solution using his search program! It's the rectangle in orange at the bottom of the following image:

Fig. 3: The full construction, with the rectangle solved by Patrick Hamlyn shown in orange. Apparently his software hit upon 866,000 almost-solutions with 40 of the 41 pieces placed before finding this, so I would have had to have been very patient with mops.exe to even have a hope of turning this up.

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Lewis Patterson. Last updated 25/07/21.