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Volledige versie: Partial solution
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I've come up with a partially-automated solution to Jigidi puzzles, which I've documented on my blog. Here's a demo video (as soon as YouTube make it live...):





The essence of it is: you use the Javascript debugger in your web browser to intercept the bit of code where normally Jigidi loads your saved progress, if you've worked on this jigsaw before. If you're new to this jigsaw, then it runs a bit of code that "shuffles" the pieces; if you're loading a saved one, they're left in their previous positions. So all you do is make it load the pieces as if you were doing the jigsaw from the start, but then trick Jigidi into not shuffling them! This results in the pieces being "piled" in the middle of the screen, stacked from top-left to bottom-right.

It's not a completely automated solver, but it really speeds things up because all you have to do is hold the bottom-right piece and keep moving it "right" and then "down", scooping-up all the other pieces (in the correct positions) as you go. I was able to solve 12 jigsaws (part of a series of puzzle caches) with hundreds of pieces in about 2-3 minutes each.

If somebody can automate the final "clicking and dragging" step, this could be the basis of a fully-automated userscript. But for now, being able to solve any Jigidi puzzle in under 3 minutes is a big help!

(First post on the forum; hope I'm doing this right!)
Possibly in response to me publishing this on my blog, Jigidi have changed their code. But it's still got the same fundamental problem, so I've changed the instructions on my blog post and it still works just as well.
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