Congruent Rectangles with Heptominoes

(This post might be Part 1 of many, depending on how much free time I have to look for other sets of rectangles.)

A few weeks ago I found this:

Fig. 1 - Four 10x19 rectangles using one complete set of heptominoes.

There's a number of ways of making sets of same-size identical holey rectangles with the heptominoes. First off, there's three 11x23s which I've done before in an earlier post (and since 11 and 23 are both odd you can get the hole slap bang in the centre of each rectangle.)

Then, there's four 10x19s as above. Then it looks like the next set of rectangles that might be possible is nine 5x17s (with potential for centered holes too!) but whether that's even solvable is another question entirely. I mean, it goes without saying it'll be a frustrating ordeal knocking it out by hand. Actually, Patrick Hamlyn managed twelve 8x8 squares (three quarters of the way down this page) so anything's possible.

***

Also, a thing I've found that works really well for heptominoes if you're just after solutions and not the glory of finding them entirely by hand. Solving the first 90% of a shape by hand, saving the nice clumpy blocky bits of course (that goes without saying), then crack open some software like FlatPoly2 to brute-force the last 10 or so pieces. This works with heptominoes especially, since the really wide search tree that is placing the first ninety-odd pieces doesn't seem to play particularly nice with my computer. (More expensive computers may get better results!)

Then there's that other thing. The one that feels partially like cheating, just a teensy little bit. Picture the scene, you're just spend two hours slogging away at a particularly gnarly construction and just can't get the last few pieces in. There's a point at which you start to wonder if what you've got left here is even possible. So occasionally I've succumbed to the temptation to input in the shape of the remaining hole and the last five or six pieces, not to find a solution, but just to check if one exists. Then if it transpires you're looking for a solution in a space where none exist, I tear out one or two pieces, input the new, bigger hole and keep checking until we've got a gap where there is definitely at least one possible solution. Finding that will still be an ordeal though.

So it's computer-assisted solving. I don't know whether it counts as truly properly solving by hand, but when I've just sunk a whole evening into a construction and it's pushing eleven and I've got work tomorrow, I'm usually not too bothered.