More Heptomino Constructions

I've been playing with the heptominoes a hell of a lot recently. Actually, thinking about it, it's not really surprising that these pieces have the same kind of addictive properties as Tetris does, given that they are pretty much just Tetris pieces on steroids.
First big success was about a week ago. I'd spent most of my day at work distracted, thinking about what other shapes I could try and knock out with the heptominoes (rectangles were getting a tad old at this point). I can't quite remember how I came to the conclusion that a 29x29 rectangle with a missing central 9x9 gap and 4 extra holes (to allow the harbour heptomino to fit) seemed doable, but I did. Of course, actually doing it was another matter entirely.

Basically, the long and short of it is by the time I've been at work all day then got back in in the evening, my mind's not exactly firing on all cylinders. So (this is becoming a common theme) after having placed about 60 pieces and thinking I was well on my way, I noticed I'd made one of the 'walls' of the square ring too narrow, off-centering the central hole. And it just ain't a central hole if it's off-centered, is it, so I had to tear huge chunks of construction up and start again, not quite from scratch but not far off.

Fig. 2 - The 29x29 square with symmetrical square holes.

Finally got it though! Took about two hours of my life I'll never get back, but there it is.

Spurred on by the above success, I decided later in the week to revisit the other construction I'd made-but-not-quite a little while back: the 20x38 from here.

Fig. 1 - New, improved 20x38 rectangle, with the four holes placed in the centre this time...

This was a lesson in taking care which order you use the pieces in. The usual technique is to leave the chunky blocky bits until the end (see the bottom-left corner of this one, where all the easier pieces are hanging out). But equally important is getting rid of the more awkward pieces earlier on, so you're not stuck trying to place them in the frustrating repeated-backtracking stages that is the endgame of a construction like this.
Look at the 'Z'-shaped piece and the cross bit touching it, about halfway up the left hand edge of this one. These pieces are nightmares, and being stuck having to place them twenty times each as I went full trial-and-error completing the last 5% of the puzzle was something I wouldn't wish on anybody.

Then today I managed to do this:

Fig. 3 - Three 11x23 rectangles with central holes.

This was more of a personal milestone for me than anything. I remember my first introduction to heptominoes was seeing the construction on the Kadon Enterprises website, three 11x23s with centrally-placed holes. That and Michael Keller's 51x58 octominoes further down the page just absolutely floored me when I was like 9 or however old I was, struggling to make rectangles with my little set of pentominoes hacked out of the back of a Weetabix box. These solutions seemed like superhuman achievements to me back then, so it's just a nice feeling having found my own alternate solution to this one - I mean, I'm still nowhere near the level of that octomino rectangle, or some of the even crazier solutions, but I'm getting there. Slowly.

Of course now I'm itching to have a go at something with octominoes... The only thing really stopping me is the fact that I don't have a physical set of 'em handy. The thought of tackling something like an octomino construction as a purely pencil-and-paper thing just terrifies me.

Various Hexomino Constructions

The hexominoes are an infuriating bunch. In many respects they seem like they ought to be perfect - none of them have holes, there's just a nice amount of them (enough to be a challenging solve, but not enough to be overwhelming), and they cover 35x6=210 unit squares which could in theory make all manner of fun rectangles. But because of that whole checkerboard-colouring parity imbalance thing, the shapes you can actually make with them are shockingly limited.

First off the bat, all rectangles (5x42, 6x35, 7x30, you name it) are out. The closest to a solid rectangle you can really get is an 11x13 with a gnarly appendage sticking out of it.

Fig. 1 - Like this...

Fig. 2 - ...or this. Eww.

Which isn't particularly pretty.

Next up, parallelograms. For some weird parity reasons that I haven't fully wrapped my head around, you can do parallelograms with odd base lengths, but not even ones. So the following parallelogram with width 35 is fine, but a similar one with base width 30 or 42 or 14? Forget it.

Fig. 3 -The hexominoes in a 35x6 parallelogram.

As far as I can tell, an isosceles trapezoid shape similar to this would be impossible for all dimensions, odd or even. Bummer, eh?

But~! If you don't mind great big gaping holes in your constructions, then there are a number of fun things you can make with hexominoes. Very fun indeed. Below are two possibilities, a 15x15 with a 3x5 hole, and a 13x17 with a long, skinny 1x11 hole.

Fig. 4 - There's probably other similar rectangles - I reckon 5x43 with a horizontal 1x5 hole is doable - but I'm still scared of attempting long skinny rectangles after a grim incident involving the 3x20 pentomino rectangle.

That second one, the first of the 13x17s... that took some real blood, sweat and tears to solve. It doesn't look like it should have be any harder than the 15x15, but crikey... I had taken my little home-made plastic set of hexominoes along with me on holiday a little while back, and spent several evenings trying to do it but getting nowhere. Admittedly, part of the problem was my habit of building three-quarters of the construction, then realising I've gone and put the central hole in the wrong place...

Then there's a final category, things that aren't really proper shapes, but are still symmetrical (or just look nice):

Fig. 5 - A sort of approximation of an octagon.

Fig. 6 - Whatever this is.

In truth though, a far more interesting set of pieces (though sadly less mathematically 'pure' a set) is created by combining the hexominoes with the pentominoes. This yields a set of 47 pieces, which cover a nice 270 square units and don't have any weird parity issues. There are all sorts of rectangles and what-not which can be constructed - and even better, solutions tend to be slightly easier to find because you've got the nicer pentominoes to use alongside the nice hexominoes once you're nearing completion. But that's a post for another time.