How can a jigsaw have two distinct solutions? [video] | Hacker News Reader

How can a jigsaw have two distinct solutions? [video]

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Is it regionally common somewhere to call a "jigsaw puzzle" just a "jigsaw"? Or is this just done by fanatics? From the title I couldn't figure out what this was going to be about.

I notice that Wikipedia[1] says "with context, sometimes just jigsaw" but as jigsaw owning American who doesn't do jigsaw puzzles I don't think I've ever heard it used this way.

[1] https://en.wikipedia.org/wiki/Jigsaw_puzzle

It's very common in the UK to just call them jigsaws, as most people are more likely to interact with the puzzle than the tool.

Interestingly, in Germany, they tend to just be called puzzles, so they went in the other direction.

The nordic languages are similar. Danish and Norwegian have the full word, "puzzle-game":

Danish Puslespil, Bokmål Puslespill, Nynorsk Puslespel. Swedish, (like German) goes for just Pussel; Finnish says Palapeli ("piece-game", probably related to the older Swedish läggspel, "lay-game" or "put-game").

I feel like using AI to make optical illusions have been some of the more interesting things to come out of the generative image AI thing.

I suppose when it comes down to it, image AI could be viewed as a fancy way to interopolate between human meaningful images where the inbetween values are still human meaningful. I wish that the AI discussions had more of a focus on that, as it seems unique and novel, and much more interesting than the tired moral panic around "what is art?"

There was a post a while back about making QR codes that look like other things that was also super cool.

Not remotely the same idea, but the Magic Puzzle Company [1] makes fantastic puzzles with "two solutions." They're traditional puzzles with hand-drawn art, but the first solution is in four distinct pieces that can be reassembled to make a slightly larger rectangle with a hole in the middle. (Similar to Sam Loyd's Missing Square puzzle. [2])

Then, a second set of pieces (in a separate envelope) allow you to fill in the hole with an image that (typically) wildly alters the interpretation of the original image.

[1] https://magicpuzzlecompany.com/ [2] https://proofwiki.org/wiki/Sam_Loyd%27s_Missing_Square

It’s kind of a shame they had to use AI to get the images to tessellate properly. Seems like there should be some math there.

Escher would do something more with this if he were alive.

I'm not sure how jigsaws are usually modelled in maths. I feel like there should be a graph model? Perhaps each piece could be labelled and have 'dangling' half-edges that are coloured such that they can only be connected to the appropriate partner.

Then a two solution jigsaw would be a set of these pieces that reconstruct into two different labelled graphs.

Hmm ok I realise this is all very dumb, as it's the labelling (the image!) that is the tricky part here ...

I'm not sure how to do it in math, but from the video it looked like they modeled it simply

Each square/piece has 4 sides. Each side has a different type of cut. Cut 1 matches with cut 2. (one is the in the other is the out). Cut 3 matches with cut 4.

Each square is a list of 4 numbers where each number is the cut on it's N, S, E and W side.

As Steve Mould describes it, solution is very mechanical. You start from chaos and gradually improve it so that it matches two different image classes. Pattern-matching is not considered AI nowadays.

Uh, that's literally how Stable Diffusion works, and I think it's considered GenAI?

But with way more constraints than those provided by prompts themselves. Not sure how to term that but it's considerably different from typical GenAI that solely depends on prompts.

It’s not like they just wrote a prompt and got a result. It sounded like a very hands-on process.

While they used the same process generative AI uses, it doesn’t sound like they really used AI.

They literally explained how they used GenAI and to generate the image.

Yes they explain it very clearly and it’s not just a single prompt in an off the shelf genAI.

Decades ago, I had fun with jigsaw puzzles where all pieces were exactly the same shape, so you could combine them any way you wanted! The goal was to actually produce an image that made sense.

The shape of each piece was a salamander, well-known from M.C.Escher's works.

I've just searched the name of the company, and the website is still up: https://shmuzzles.com/

It seems like there's a simpler, lazier solution to having a puzzle with just two solutions.

Just make two corners diagonal from each other interchangeable, then simply print "this side up" on one of the corner pieces and have the image be a rotational illusion that look like different items when rotated 180°

Trivial solutions aren't interesting solutions.

Or just flip over a normal jigsaw and give yourself a challenge.

The most recent one I did had a grid of letters on the back, to make sure things align. I wonder if that would make it easier or harder to solve than if it were a single color

There's a jigsaw brand (impossipuzzle, I think) that does something like that to make them as difficult as possible. They're printed on both sides, and the rows/columns are cut from different sides.

I have aunts that took almost a year solving one that was an 5*7 foot picture of baked beans and chips. Most of the patterns are repetitive and non-distinct on purpose.

What do you do with such a monstrosity when it's finished?

They glued it to a sheet of plywood and hung it on the wall.

Long before this Matt video, I've done just that. A generator of devilish no-image jigsaw puzzles :-) which I indeed wanted to do a little Show HN someday. I made a small python script call a SAT solver to create a MxN puzzle with only 1 solution but only a few jig shapes, so maaany 'almost solutions'. Then I laser cut a plastic board and made it real, and gave one copy to a friend. At least a nice use of SAT solvers that is more efficient than Matts brute force approach.

Yes please, show your script!

And in the video Matt talks about the effort he went to in order to avoid trivial solutions like that. He wanted to find the least trivial one he could.

Or don't print an image at all, leave it a solid color, and then you have four solutions - two landscape, two portrait - if you leave yourself restricted to 90° angles.

(Technically eight, if you also flip the puzzle upside down.)

Check out Stave Puzzles if you want to get into insanely difficult jigsaw puzzles with multiple solutions. Some have multiple solutions (3+) and some go 3 dimensional. I have one that is either a 3 or 4 leaf clover.

That appears to be $1k for a single puzzle. Yikes.

This could probably be extended to more than two images. Don't know how far you can push it before the images look too contrived to be interesting.

In thinking how I would start, I imagined starting with only one kind of shape, except filling/flattening out the edges/corners. Then come up with a dual picture that depends on 2 arrangements of the pieces. Finally adjust the edges of the piece shapes without breaking the 2 selected arrangements but allowing all the other possible arrangements to collapse down.

It's like making a repeating/tiling shape where you start with a blank tiled plane then start drawing anywhere, all the repeated places also get drawn in, and you keep drawing/undoing until you like what you see. i.e. 'maintain the constraint from the beginning'. The tricky bit is in the drawing/painting. I imagine the same could be done with this jigsaw, blank the pieces, show both arranged blanks, and let someone paint on the pieces updating both as you go. When you like what you see, you're done.

This is covered in the video.

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