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.