"There's No Way I'd've Found That By Hand!" Vol. 1

At work I have a notebook. And in it there's surprisingly little pertaining to actual work, but lots of ideas, calculations and sketches for possible shapes that could be filled up with various sets of polyominoes or other polyforms. Most of these end up solved at home, then drawn up all pretty and posted on here, but there are some that I either try and can't do, or that just look so intimidating I don't even try. It's those jagged diagonal edges... I just have a right job doing them.

So, here's a little collection of solutions found with various programs, that I was too weak to suss out by myself.

In hindsight, this one doesn't look that bad, and I've solved similar in the past. I guess at the time I was just not feeling up for the challenge. I think (judging by the colour and scale of the image) this one was found using the solver on Peter Esser's site here.

I'll be honest, I didn't even attempt this one. I did the calculations to make sure that it was permitted by the parity constraints, then just despaired at the thought of having to actually solve it, central holes and all. But FlatPoly2 made short work of it, finding this in about five seconds flat.

Then there's this family of solutions I have no recollection of looking for but are in the folder called 'Polyominoes' so I may as well just post them for the hell of it.

 I think at this point, hexomino solutions just aren't the most impressive thing in the world any more, computer-found or not. But more interesting things like heptominoes take a while to do, and I've got driving lessons and band-related business to contend with on most evenings, so that stuff tends to take a back seat.
Actually, I've got a nice new shiny set of octiamonds that I've yet to do anything of note with... I'll have to have a little play around with those, see if I can create anything worth posting on here with them.

Hexominoes - Scraping the Bottom of the Barrel

In a previous post I made some pretty shapes with the hexominoes and had a good whinge about parity while I was at it. And at the time I thought I'd pretty much exhausted all the possibilities for nice things that could be done with hexominoes. But then a few weeks after that, I found myself on holiday in a little caravan, with the weather being utterly atrocious outside. Luckily, I had decided to bring my little homemade hexomino set along with me, so in the absence of anything else to do I uh... constructed some more constructions, I suppose.

First off, a good (but probably not 100% sure-fire) technique for finding things that are actually constructible (and aren't ruled out by parity constraints) is to start with a rectangle that has odd x odd dimensions (and an area > 210 units) then carve out a central hole until the total remaining area is 210 units. As long as you're fairly careful here, the resulting shape should be solvable.
(Some day I'll do a write up of parity the way I understand it on here... honestly more for my own benefit than anyone else's.)

Fig. 1: Three rectangles with nice symmetrical cross-shaped holes hacked out of them. That third one, I've seen a different solution to before (I think it might be in Solomon Golomb's book).

The third solution, the 15x17 one, took an outrageous amount of time and attempts to solve. It's a shame that processing-power-wise we're a long way off being able to enumerate solutions for a given shape with hexominoes (like we can for pentominoes) - it'd be interesting to see if the ones that take me ages have fewer solutions and are genuinely harder.

I also found a solution to a 13x21 rectangle with a 7x9 removed from the centre. With the central hole being 63 unit squares, that means there's enough space to get all the pentominoes stuffed in there with a little bit of wiggle room to spare.

Fig. 2: This took a while too, as far as I remember. Hexominoes and thin sections (the 3xn bits above and below the pentominoes) don't seem to mix too well.

Then I realised that I didn't need to restrict myself to taking the excess squares out of the middle of the rectangle - I could just as easily round off the edges, and make some stretched-out octagons.
Which I did.

Fig. 3: A way stretched-out octagon.

And here's a slightly more regular, more octagonal-looking one I found a little while later:

Fig. 4: I bet there's a way of doing this with the six internal holes all in a diagonal line, but I'll leave that as an exercise for the reader 'cause I can't be arsed right now.

And just for an extra Brucie bonus, while we're on the subject of octagons, here's one made out of heptominoes that I found a few weeks ago:

Fig. 5: For whatever reason I have no recollection whatsoever of the actual solving process that led to this.