# One-sided Hexominoes

One-sided hexominoes are to regular hexominoes what the seven Tetris pieces are the tetrominoes. That is, the set of pieces you get when you treat mirror reflections of a piece as two separate pieces. There are 60 one-sided hexominoes in comparison to the standard set's 35 - eleven hexominoes are mirror-symmetric and so don't contribute a second 'evil twin' piece.

Basically, earlier this year I got a second set of heptominoes laser cut, which opened me up to the possibility of solving shapes using the set of one-sided heptominoes by combining the sets and then trying really hard not to flip the pieces over while solving. Then I realised the same thing could be done using my two sets of hexominoes too. And the set of one sided hexominoes is a **much** more versatile set than the plain ol' regular hexominoes. It's amazing. There's no longer any of those pesky parity constraints, which means that a lot more shapes are possible to tile - rectangles without unsightly internal holes, for example.

(The blue pieces are the symmetrical ones, the yellow set are the mirror pieces of the green set.)

The total area is 60x6 = 360, which means that rectangles of size 4x90, 5x72, 6x60, 8x45, 9x40, 10x36, 12x30, 15x24 and 18x20 should all be possible. Sadly, due to the length of the perimeter compared to the amount of perimeter squares the pieces can provide, 4x90 isn't possible. I found this out after sinking a few hours into trying. Hey, I'd managed a 4x*n* thing with the free hexominoes so it felt like it might be possible.

Solving manually is a little trickier than the normal hexominoes - each piece having only one accepted 'right side up' means that any given piece is slightly less practical than its two-sided equivalent, and you do get those cases where you're down to one piece left and the hole is the mirror-image of the piece you're holding.

Here's two 9x20's which can be combined to make either a 9x40 or an 18x20, in a two-rectangles-for-one type deal. As 360 divides up really nicely, this gives a lot of possibilities for tiling groups of congruent shapes, but that'll be a post for the future. Others have already solved congruent sets of ten or twelve shapes, so go look at those. Scroll about three-quarters of the way down the page for them. In fact just read the entire page, it's all good.

Something else nice you can do with the one-sided hexominoes is square rings, i.e. squares with a centred square hole. Here are three possibilities (these might be the only three actually), with the rings getting progressively thinner, and as a result a little harder to solve manually:

And here's one more shape just for funsies, a diamond with a central hole and those tricky diagonal edges. Notice the south, east and west points utilising those pairs of chiral pieces.

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Lewis Patterson. Last updated 06/10/21.