More Octominoes

Here's a few more (okay, two) octomino constructions from over the past few weeks. Mainly boring shapes like rectangles, just because I'm limited by the size of the desk in my room. A big diamond with diagonal edges would be awesome but it's just not going to fit on the available working space, in fact this one with height 57 really pushes the limit of what I can do, it was practically overhanging the edge of the desk:

Fig. 1: A 57x63 frame that fits the 23x33 heptomino rectangle perfectly inside it.

An odd thing with n-ominoes is this - as n increases, not only does the number of n-ominoes with a particular feature increase, but the proportion of all n-ominoes with that feature increases too. Take polyominoes with holes for instance. With heptominoes there's one, giving a proportion of 1/108 = 0.93%. With octominoes it jumps to 6/369 = 1.63%. And this trend seems to keep going, as shown below.

Counts for the really big polyominoes taken from here. Sorry it's an image and not a copy and paste-able format, blogger hates tables for some reason.

My gut feeling is that this keeps going, approaching 100% but never quite getting there, instead tapering off in a big s-shaped curve.

Here's a graph that has everything wrong with it. Default OpenOffice design, tidied up in MSPaint, uses percentages which I assume isn't good form for actual scientific purposes... I may as well have gone the whole hog and done the captions using WordArt.

The same trend seems to hold true for other features too, such as polyominoes containing a 2x2 block inside them somewhere as a subset of their squares, or pieces with a two unit deep well (like the pi heptomino.) I guess it could be in part down to the fact that as n increases there's more scope for a given piece to exhibit several of these features at the same time, e.g. have an internal hole as well as a 2x2 sub-section. Maybe? This feels like the kind of territory that would need approaching from a rigorous mathematical angle, but I have no idea how to do that so the best you're going to get is me guessing wildly based on a hunch I got from spending too long playing with glorified jigsaw puzzles.

Here's a 29x102 rectangle with the octominoes. David Bird beat me to the punch by several decades but I'm still going to share it here anyway, dammit.

Ooh, and we get a final bonus guest solution this time too... I had shared an older polyomino solution (the one with concentric layers of 5- through 8-ominoes) to the Puzzle Fun facebook group, mentioning that there was a possibility for a 121x129 layer of enneominoes to surround the entire construction. Sadly, there was no arrangement of holes that would preserve the symmetry of the hole distributions in the inner layers, but I settled on something close enough and Patrick Hamlyn found the following solution for it:

(Click for full-size.)

Interesting to note are the pieces his search program saved for last - the ones just below the bottom-left corner of the octominoes section. There's a lot of the kind of pieces I'd expect, blocky, rectangular pieces, but also a lot of long thin bits with a completely smooth edge on one side. I'd never considered holding onto these pieces that late in a solution, but clearly it gets results so it's maybe something worth me experimenting with while solving by hand.

And yeah, if anyone fancies wrapping a set of dekominoes around that, that would be grand.