Nine 15x22 Rectangles with Octominoes

As per the title really.

Fig. 1: I was going to say that this is approaching the limit of what can feasibly be solved by hand, by a human. But then again I probably said that about much simpler hexomino things when I was first starting out so who knows, eh?

The top-centre and middle-left rectangles had to be rebuilt fairly late on in the solve because I'd done them with the holes offset by one square the first time. Total solve time was about 6-7 hours, of which close to two was spent on the last half of the final (bottom-right) rectangle. There were just a few awkward pieces - the three which surround the right-hand hole especially - that I had unintentionally held onto far too late into the solution, and they caused all manner of ball-ache.

Nine 11x30's with similar hole configurations to this should be possible (also 6x55 rectangles, if you want to suffer...), but not right now. After solving something like this there's always that period of a few weeks where I feel like I'd rather be made to eat the octomino set than tackle another huge construction with them. Right now I'm still in that phase. Recovering.

Miscellaneous Solutions That Didn't Deserve Their Own Posts

Sometimes polyomino-related things have a decent story behind them (or, failing that, a really boring story that can be stretched out to blog-post proportions.) But sometimes they don't. Today it's a selection of the latter; digitised solution pictures that were just sat around cluttering up the folder named 'BLOG STUFF' on my desktop, to tide me over while I write up some actually half-decent posts.

Rhombus with Hexominoes

Difficulty level: Mild (approx. two chilies out of five)

I've seen this, or variations of it, done before so it's not really particularly groundbreaking as solutions go (although really, are any of them that groundbreaking?) Including it here because it was a hard-won battle - I kept building the edges wrong, accidentally adding in steps of size 1 or 3 then not realising until right near the end when I was left with an internal hole whose size wasn't a multiple of six.

Heptomino Rectangle with 21 Holes

Difficulty level: Breakin' a Sweat

Solved the middle first, since the closely-packed holes are quite restricting on what pieces can even go there. But then the rest just solved like a normal heptomino construction and I've banged on about those at length in other posts so it wasn't really worth doing another one.

And the harbour heptomino doesn't need to be in the very centre of solutions like these, but it just feels wrong any other way.

5x45 rounded Rectangle with 11 Holes

Difficulty level: Real Tears

This was an utter nightmare, combining two of my worst fears into one shape: 5xn with hexominoes is always an ordeal, and adding that row of holes just pushes it over the edge into the kind of territory where it's actually frustrating and unpleasant to solve. You can see by the way the eight pieces with 2x2 blocks in them are scattered all over the shop that my usual solving technique only got me so far before I was left to fend for myself, desperately applying trial-and-error for several hours of my life I'll never get back.

Recommendation: FlatPoly2 can probably crack this one in under 10 seconds, just do that instead.

Stay tuned, next time I might actually have something a bit more substantial.

Polytans, polyaboloes, whatever you want to call them

Polytans (also polyaboloes, depending on which website you're looking at) are the shapes made from joining isoceles right-angled triangles (45°-45°-90° triangles*) together edge-to-edge. Theres one 1-tan, which is just the triangle on its own, then three 2-tans, four 3-tans (tritans? triaboloes?) and 14 tetratans/tetraboloes. The numbers grow colossally fast compared to the numbers of polyominoes, polyiamonds or just about anything else; probably partly because there's often more than way to append a triangle to an edge.

Here's some pretty pictures of the 1- through 4-tans:

Fig. 1: Here they are, courtesy of a tedious as balls half-hour in Microsoft Paint. Upon completing this I realised I could have just generated them using Peter Esser's solver and took a screen shot. You live and learn.

Above this, there are 30 penta-tans, 107 hexa-tans, 318 hepta-tans and none of those look like real words, I can kinda see why the 'aboloes suffix gets used. Yeah, pentaboloes and hexaboloes rolls off the tongue a lot better.

I have made little acrylic sets of the 1- to 4-aboloes to play with. The larger sets I haven't gotten around to doing yet, partially because coronavirus and lockdown and all that, and also because the place I usually get them made have upped their prices and I've only got so much annual budget for polyomino-related spending. But tetraboloes are more than enough of a fiendish challenge in the mean time.

Surprise, surprise, I still can't take a decent photo for toffee.

The combined area covered by the triaboloes and tetraboloes is 34 which is a bit of an ugly number but it's still workable. For a start, we can do rectangles of area 36 with the corners snipped off, as in the image below. this woks for 6x6, 4x9 and 3x12 rectangles.

Fig. 2: The triaboloes are highlighted in a slightly lighter shade of yellow.

 Difficulty-wise, the thin rectangle doesn't seem noticeably easier or harder than the square, but then again they're all infuriatingly tricky for something so deceptively simple-looking. Best technique seems to be to try and use up pieces with lots of diagonal edges first. But that only gets you so far. Prepare for lots of trial and error.

And when you turn over the tray I made for them there's the following configuration, which is just unfairly difficult. The centre requires the square shaped bit, leaving the remaining 13 pieces to fill the square doughnut around it.

Fig. 3: The design on the other side of the tray. Finding a solution to this is left as an exercise for the reader.

Apparently there are 45 solutions to this. I've sunk literally hours into it by hand and found only one so far.
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* Fun fact: It's insanely hard to describe specific triangles without a diagram.