Little mini post: 4x53 with Hexominoes, Revisited

 Yonks ago I wrote a post about 4xn constructions with the hexominoes, which ended with suggesting a 4x53 with symmetrical placement of holes should be possible. Well, I found one:

Amazingly, the entire solution took all of three minutes by hand. I must have just been insanely lucky and hit on a solution on practically the first try, because I was prepared for this to take quite a while.

I guess they're not technically 'holes' when they're on the very edge of the construction but it looks nice so I don't mind really. And for an additional bonus, it's three-colourable and has no 4-way crossroads where four pieces meet at a point. That wasn't even my intention while solving it, it's just how it turned out. Rrrresult!

Octomino Ring

A 57x57 square with a 17x17 square taken out of the middle leaves a nice square doughnut with a convenient area of 2960. This is the perfect size for all the octominoes as well as eight 1x1 holes sprinkled around for good measure.

A 15x15 hexomino pattern would have fit nicely into that central hole; I wish I'd thought of that before I'd finished drawing up the image because it'd probably look nice, but I can't be bothered to change it now.

I also found a four-colouring for it too, which took a little while and a little bit of trial and error. No two pieces sharing an edge are the same colour, and when four pieces meet at a point none are the same colour there either. (At least, I hope so. There's a chance I've slipped up somewhere and missed it.)

Full solve time must have been close to six hours, with the repeated backtracking phase of putting the last 15 or 20 pieces in taking unreasonably long for whatever reason. About two hours on the 22nd, and another half hour on the 23rd until I got that little 'eureka' moment and the last few bits went in perfectly.

 Edit:

I found a middle bit for it anyway - the 289 square central hole fits the pentominoes and hexominoes with 19 squares to spare, which doesn't allow for any particularly nice configurations. The picture below is as good as I could get; sadly it doesn't preserve the overall symmetry of the octominoes' outer shape.

 

Wide Tetrominoes, and a Different Kind of Parity

Note: This post was written ages ago (like, February or something) but for whatever reason I didn't publish it at the time, I think I'd decided it wasn't up to the standards of other blog posts. But today I've looked at the shambles that passes for other posts and this isn't really noticeably worse so here goes...

Introduction

Recently, it dawned on me that being able to get things laser cut has opened up a lot of possibilities for polyomino/polyform-related mischief. For example, for smaller sets of pieces if I was to cut the pieces in transparent acrylic then cut a tray for them from two opaque white bits, it would make a nice cute little self-contained puzzle. I decided to do a trial run, drawing up the .SVG files one evening for the hexiamonds set below:

Some day I'll invest in a proper camera. Today is not that day.

Which worked fantastically (read: I managed to glue the two parts of the tray together without attaching any of my fingers to the table.) It's a cute, if fiendishly tricky, little puzzle.

Wide Tetrominoes

So the obvious question was, what next, seeing as little puzzles like this are completely feasible? While brainstorming ideas I stumbled across a set of pieces I hadn't really seen mentioned or investigated anywhere else. These ones:

Like tetrominoes. But made of rectangles. Preferably ones where the aspect ratio means you have to join long side to long side, short side to short side, and can't have any pieces in at a 90° angle.
There are nine of these possible, and with an area of 4 square (well, rectangular) units each, giving a total of 36 units to play with. Promising! (If playing with units is your thing.)

I did the check for checkerboard parity, just to rule that out. Tetrominoes have this issue, as do one-sided tetrominoes (a.k.a. the Tetris pieces) so I wanted to be absolutely sure. And they looked fine. the presence of two variations of the T-tetromino meant overall parity was balanced. 6x6 and 4x9 rectangles, here we come!

But after several minutes of playing on with a set of these (well, simulating a set with pen and paper) I couldn't get all 9 in. A few near misses but no cigar. Maybe this is just one of those puzzles where there are no solutions and no good reason for it.

A Good Reason for It

At around the same time as all this I was balls deep in an octomino construction and was concerned that parity restrictions could rule out a solution for this shape and render the hours I'd sunk into it a bit of a waste. Sure, it would still have been valuable practice for other polyomino constructions in the future, but it'd still be a shame. Anyway, while worrying about this I found a page* detailing other restrictions on the set of 363 unholey octominoes, and various subsets of those. One of these was to consider not a checkerboard colouring of the pieces and solution shape, but to colour every second column, like this:

The table below shows the total imbalance each piece can contribute towards the total number of white and black squares in the construction. Five have no overall effect, three imbalance the amounts by two, and one (the vertical I-tetromino) either contributes four white squares or four black squares.

Very professional-looking MSPaint table here. LaTeX eat your heart out.

And for these to all fit the 6x6 box, the overall difference between the amounts of black squares and white squares must be zero. That is, ± 2 ± 2 ± 2 ± 4 = 0. Each ± can either be a plus or a minus depending on whether the unbalanced piece contains more white squares than black or vice versa.

In fact, no matter which combination of + and - you pick to stick in there, the equation never gives zero, meaning that there's no way the set of pieces can cover the same number of black and white squares.

Similarly, the difference in counts of each colour square for the 4x9 and 3x12 cases are 4 and 12 respectively**, meaning that the set of pieces will never fill them either. In fact, the 3x12 cases are even easier, it's just now dawned on me when writing this that the length-4 I tetrominoes wouldn't fit inside.

What next?

Well, maybe there are nice things to be done with the tetromino set but I'll leave that as an exercise for the reader (because I can't be arsed to do it myself, as per usual). The next thing worth looking into seems to be pentomino-analogs made of rectangles. In fact I'd be highly surprised if this hasn't already been independently thought of and subsequently done to death by others.
But I'll have a crack at it anyway. In another blog post, another time.

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* It's on The Poly Pages somewhere under 'More Octomino constructions', but I can't make a link to it work properly for some reason. Linking to Poly Pages gets weird sometimes.

** For the rotations of the rectangles shown in the diagram. the 9x4 and 12x3 cases, having an even number of columns each, both have parity 0, same as the 6x6.

A Few Parallelograms with Heptominoes

Edges with gradient 1:2 are surprisingly forgiving. When I started building this I had no idea just how long and skinny the resulting construction would be; I kind of pictured it in my head looking like the 45-degree parallelogram in this truly ancient blog post until I started building and it sprawled right out across the table.

Fig. 1: A 23x33 parallelogram

 

It turns out that the gentler slope here permits a much wider variety of pieces so it was easier than I'd anticipated. Solve order was like this: Started at the top-right and worked down the right-hand side, then continued around the perimeter of the shape until I reached the top of the left-hand side. This left a long thin internal section to be filled up last, finishing up at the top edge.

(A random sad fact: If some utter maverick decided to solve a heptomino construction the hard way, by using up all the blocky nice pieces at the start and finishing with the most hideous bits imaginable, everyone would just assume they'd solved it the regular way starting at the opposite end. All that effort for nothing.)

Edges with slope 3 should be easier still, since they're a step closer again to straight edges, and therefore even less restricting as to which pieces can be used to build them. But at this point the parallelograms themselves become loooong, and there's not a surface in my house that can comfortably accommodate them, short of the floor (although there's an idea...)

There's also another consideration here - the flatter the gradient, the fewer pieces can fit into the very thin ends of the construction. This limitation clearly rules out parallelograms with a 1:7 or greater gradient, but could make others impossible too. There aren't too many heptominoes kicking around that can fill a six-cell well, that's for sure.

And of course the sloping edges can go the other way, two squares up for one square along, as in the example below:

Fig. 2: A 19x40 parallelogram that doesn't lean quite as much as the other one.

I know, I know, the central holes aren't perfectly aligned with the long edges, but they're about as good as you can get I think. Holes positioned to match the gradient of the edges tend to like odd-number length vertical spacing between them, which doesn't play nicely with the (even) total height of the construction.

I could find more. I really should do. But I didn't want to go more than a month without a blog post so here. Have this lazy half-baked one.

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.