A first foray into Multiple Congruent Rectangles with Octominoes

This one had been on my to-solve list for ages now.
369 is 3x3x41, which means that the octominoes can be partitioned into equal-sized groups of 3, 9 and 41 pieces. And also 123 groups if you're so inclined, but in this case each group would contain only 3 octominoes so you can't really do a lot with them. In fact it sounds like there's an impossibility proof lurking there - prove there's no shape that allows 123 congruent copies to be constructed with the set of octominoes. It would have to have a hole, and be able to accommodate the 1x8 piece, so if there is a shape it's likely going to be a real odd-looking beast. Anyway...

Splitting the set into 3 yields the most manageable challenge (nine looks possible but a little intimidating*) so of course it was the one I attempted first:

This was a rare case of one of those blissfully hassle-free solves where there's pretty much no back-tracking and the pieces just fit first time, followed by momentarily sitting back stunned because I now had a free afternoon I hadn't previously banked on. The whole thing came together in three hours, tops.
I've still got gripes with octominoes though. Mainly, that there are too many of the damn things. One thing I kept encountering was I'd have a space that could be nicely filled by one specific octomino that I knew I hadn't used yet (keeping track of which pieces have already been used is something you just begin to get a feel for after a while without really trying to), but then not being able to locate that piece. Or not without several minutes of scrabbling around the little box I keep them in looking for it. They're transparent acrylic too, which makes spotting the one you want even harder since the edges of any piece and the pieces under it all kind of blur together. But between these frantic moments of digging around for a piece which may or may not be there, the solving process gets strangely therapeutic in a way.

Next time, nine 15x22 rectangles, two holes each?

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* 41 is pushing it, massively, but it is doable - I've seen a solution for 41 copies of a rotationally symmetrical near round-looking shape with a central hole, but a.) that was (as far as I'm aware) found with computer search, and b.) I seem to be unable to re-find the image right now.

Crinkly wrinkly shapes with Heptominoes

Seeing as I have an excess of spare time on my hands recently I've been attacking some constructions that I had previously written off as 'too hard' or at the very least on the boundary of what I could do without 'cheating' and using a computer solver. One of these is the following 29x29 rectangle with wrinkly crenelated edges around the outside, and a little 5x5 hole in the centre. And a couple of other little holes too because the harbour heptomino needs something to chow down on.

This wasn't the easiest thing in the world to put together. The first challenge was the outside corners. There are a very limited number of pieces that can occupy those corners, and I only discovered this a little way into the solution - too far in to just backtrack and start again, corners-first. Miraculously I hadn't used up too many of those precious corner pieces when I realised, so I was able to keep on trucking.
The whole perimeter of this was a completer pain in the arse to do. I started out with the left-hand edge, using pieces derived from the C- (or U- if you're weird) pentomino but they ran out fast and by the time I got to the final edge (the right-hand side) I was stuffing whatever pieces I could in, in the desperate hope they'd fit.

And they did. But they didn't leave a nice set of pieces remaining. And they didn't leave a nice shaped internal hole either. Still, I pressed on.

Fig. 2: Approximately what I had left to contend with after completing the perimeter.

From here I filled in the remainder of the space going around clockwise starting from about the 4 o' clock position. In fact, here's a needlessly detailed step-by-step of what I did. I usually refrain from this on the grounds that it's not very interesting, but really, is any of this very interesting?

Firstly, that question mark-shaped piece was just crying out to be put in that question mark-shaped hole, so I did that. Then there were a bunch of really hideous pieces (the other orange ones) that I just felt needed to be used up as early as possible, just so I didn't have to look at them again. Those two sort-of cross shaped bits especially. Winding up with one of those near the end of a solution when you're down to a handful of pieces is the stuff of nightmares.

Then, using up all those long S and L-shaped pieces because they're no fun either. Sadly, in order to do this we had to sacrifice that nice 2x4-with-a-notch-taken-out-of-it piece, but it was for the greater good. And now the edges of the hole in that top-right section are all relatively smooth and straight - ideal for for the very end game when you're solving with relatively squarish blocky pieces.

This is where the going started to get even tougher. Those two red pieces especially, I have a personal vendetta against now. I managed to get them all packed in there - but at what cost? So many potentially useful end-game bits used up in their prime.

Got that top-left corner squared off. And at this point we're left with not the worst hole in the world (that award would have to go to... I don't know, the ozone hole? Or the holes in the end of trombones that allows them to make their noise), but not the easiest to solve either. That two-cell-deep well at the far left was to haunt me for the next hour or so.
And the pieces I was left with to try and plug this hole? Check out this ragtag bunch of misfits:

Admittedly not the worst selection imaginable, but still... especially that one that kind of looks like a number 4 or a heavily-damaged tuning fork (kindly marked in orange for your identifying pleasure), that can suck a fat one. You'd think that piece and the 2-cell well would be a match made in heaven, but try as I might, it just was not to be...

At this stage in the game, there really is no more technique to speak of, no prioritisation of pieces or synergies between two pieces that are painful to deal with on their own but make a well-behaved 14-omino when coupled together. It's just a case of try something then if it fails tear it out and try something else. And that makes solution times vary wildly. Sometimes just through blind luck you'll stumble upon a valid solution after five minutes; other days it's hour after grueling hour trying various configurations in vain.  I guess the factor that determines how long this takes will be how many solutions to this particular endgame there are lurking in the search space as you wander randomly through it.

I got all but one piece in several times as I was doing this one. That's always the worst, that feeling of so near yet so far... And there's another weird thing that happens occasionally when finding a correct solution. It's when you get that 'eureka' moment, that "Yes! I've done it!" spark of excitement a fraction of a second before you've actually understood how the last two or three pieces are going to fit in there. I'd write it off as premature celebration if not for the fact it only ever seems to happen when I've got an actual solution on my hands - maybe I'm subconsciously spotting the solution and the rest of my mind takes that split-second to catch up? 'Tis a mystery.

Here's the eventual configuration I found. A quick consultation of some solving software tells me that there are 9 solutions to this, and that 5 of them do fit that '4'-shaped piece into the two-cell well, so goodness knows how I managed to miss 'em all.

In fact, here's all nine, since I'm being so generous with the pictures today. Mine is number 7.

And while we're on the subject of wicked tough constructions, here's another recent one that might be a contender for 'hardest construction so far', or at the very least most time-consuming:

As I've probably mentioned at some point before, octiamonds are just not a pleasant bunch of shapes to work with.
Sorry, no step-by-step account for this one. Partly because I solved this about two weeks ago so I can't remember the solution process clearly (other than that it was all over the place), and partly because I'd rather not re-live the experience. All I know is I solved the bottom edge first, and the rest followed, four hours later...

The cruel irony of all this is I'm awful at Tetris

I've been playing with a set of hexominoes a lot recently. It seems the bottom of the barrel wasn't as thoroughly scraped as I'd previously thought.
It turns out there are a family of shapes that approximate hexagons with angles of 90 and 135 degrees that are just as versatile as rectangles, if a little less pleasing to the eye (especially if you're not too fussy about them having a few holes in 'em.) These also have an added bonus: the diagonal edges make these a slightly more challenging solution than squares and rectangles.

Again, odd x odd bounding boxes seem to work best with this, because an even width or height would mean a line of symmetry bisecting the entire shape into two congruent 110-cell sub-shapes, which would mean the overall shape has balanced checkerboard parity. So with the aid of a hideous and confusing spreadsheet I knocked together, I found the following set of solvable problems:

3x73 - 9 holes
Yeah no. I think we can generally rule out the possibility of any purely 3xn constructions with hexominoes. I roughly outlined my reasoning here (but that's not a rigorous mathematical proof so it could well be wildly incorrect...)

5x54 - 3 holes
Pros: At this width solving these shapes feels not much different from solving a normal rectangle.
Cons: Solving a width 5 rectangle isn't exactly a walk in the park.

7x35 - 11 holes
At this point there's a sort of 'choose your own difficulty' option built in - with the difficulty levels being hard, harder and hardest, naturally. Two immediate options for arranging the 11 internal holes are having them as one big 11-omino, or as a line of 11 individual monominoes. The long 11-omino option creates two stretches of treacherous 3xn space which takes a bit of doing:

Whereas the individual holes introduce all manner of feisty challenges.

This one's left as an exercise for the reader. I spent a little while trying to solve it and got fairly close, but not close enough. I think I gave it like 45 minutes or so and wound up with like 6 pieces remaining a few times. But they were never nice pieces.

9x29 - 11 holes

This one was tough, although in hindsight this could be because of a complete lack of solving strategy. The obvious (looking back now) thing to do would have been to solve the ends first, then either the top half or the bottom half of the middle (if we think of the row of holes as a dividing line), trying to leave a fairly convex-looking endgame to be filled in last with the 2x2-block-y pieces. Instead I just went at it the way a nine-year old would his first all-you-can-eat buffet, and I paid for it dearly by spending maybe close to two hours (!) getting those last few bits in.
You know you've utterly stuffed it when you're down to the last five pieces and one of them is the stair-step hexomino (as it's known in Conway's Game of Life terminology).

11x25 - 5 holes
This is the start of the sweet spot, so to speak. Wide enough so that the central holes don't get in the way too much, but not so wide that there's miles of diagonal edge to contend with.

Of course, the fact that there's only 5 holes maybe makes this a little easier too. You could just as easily do this with the holes arranged vertically. Or with one bigger hole shaped like a pentomino of your choice.

13x23 - 5 holes

Can be done with the five holes the other way on. (So can the 11-height one but I didn't think of it at the time.)

I think there is a 15xn possible but it's going to have like 17 holes or something, and I think there's a point where unless you can arrange them in a particularly pleasing way it just gets silly. So I'll skip over this and go straight to...

17x21 - 3 holes and 19x21 - 9 holes
No solution images here, for the simple reason that I just plain can't be arsed. If you want them that badly, acquire a set of hexominoes and have a go at finding them yourself. More fun than looking at a picture of someone else's solution, guaranteed. Or feed them into a solver program, they'll eat these for breakfast.

And finally, 21x21 - 11 holes
This is just a rotated square at this point. And I've seen a solution for this on the internet somewhere, with the 11 holes as the straight 11-omino dead centre. As far as I can remember it was illustrating something to do with parity imbalance, and this was the example where the difference between black and white squares is exactly ±22, the maximum permissible. I assume that the restrictions that places on parity-imbalanced pieces means it's a head-bendingly difficult solution.

[EDIT: The solution I saw somewhere was here, on page 9 of Chessics, issue 28]

(The above solution was found using a computer search. Because it sounded like it might be difficult to do, and I'm always one to back down from a difficult challenge when one presents itself.)