Originally published at: https://boingboing.net/2024/07/10/can-you-solve-the-six-box-puzzle.html
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Neat. Took me a moment, but neat. That one might have to go on a whiteboard here…
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There are easier solutions but they break an unwritten rule: The line connecting two boxes cannot pass through another numbered box.
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This reminds me of this problem:
Hang a picture using two nails, s.t. if either nail is removed, the picture falls.
ETA: this is a math problem, not a lateral thinking problem. I was too lazy to write out all the caveats, e.g. “the nails go in the wall”. Non lateral answer:
ETA: here’s a video solving the 4-nail version. Skip to 1:10.
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And an easy one depending how you want to parse “within”.
If that includes “on”, the 3 boxes are already connected.
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This one was pretty quick for me, guess it matches my brain’s puzzle-solving approach.
Spoiler
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That’s exactly the solution that came to my mind. The use of singular “rectangle” in the rules (not rectangles) made it 100% legal to my mind to cross through the smaller rectangles.
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I immediately solved it by going through the boxes, and then re-read the rules twice to make sure I wasn’t missing something.
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You can generate such a solution for any configuration of any number of paired boxes. What you do is start by placing the boxes paired up vertically like this:
123...n
|||...|
123...n
Then, imagining the lines to be infinitely bendable and stretchable, you drag the boxes into the desired configuration dragging the connecting lines along with them.
Here’s how it works in the specific example. The start:
Then you drag things around to this:
Now we need to do a half-twist around the blue circle shown here:
Part of the way along:
The rest of the way:
You can also do the twist with the other handedness, or do it multiple times. I think that the boxes touching the boundaries will mean that the boundary 1, 2, and both boundary 3’s can’t be braided (b/c you’d get arcs that want to go between them and the boundary), but that’s about the only issue.
This argument comes from topology. Specifically the topology of the space of marked points on the disc and is related to the braid group.
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I went a different direction with that one. Put both nails in the wall but leave each protruding far enough that they form a shelf. Place the painting on the “shelf” formed by the nails. Remove the left nail and the painting falls, lower-left corner first. Remove the right nail and it falls lower-right corner first.
Though I suppose you might not consider that “hanging” the picture.
Welcome to the Stargate program 
(Or was that the Wormhole X-Treme! project, since the Stargate is still highly classified?)
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I put one nail in the wall and another in the picture frame, connected by a string.
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It took me less than a minute.
But it’s my job. I design printed circuit boards for a living.
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Ah, you jogged my memory! I was addicted to Flow Free on Android for a long time, that’s why this puzzle was so quick for me. So for anyone who likes this kind of puzzle (and wants to waste some brain plasticity on layout problems):
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Yeah, my first solution was going through the boxes. But I did an alt solution that was basically the “accepted” solution.
It was fun!
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No.
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Yeah, I looked at it and thought “it depends on the trace width.”
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Neat. I’d seen it before, but (thankfully?) had forgotten the solution, because it was fun to work out.
I have a notion to turn this into a puzzle for my D&D game.
The idea is that the party will come upon a little walled sand garden with stones set in it that represent the boxes. The stones have matching icons instead of the numbers.
An inscription reads:
The Bee returns unto the Hive,
And Rabbit to his Burrow;
The Rain must fall upon the Flower
Awaiting in its furrow.
These three paths your hand must find
That do not cross each other;
They must not touch, or cross at all
That great grey ring, the garden wall.
The idea is that the party has to draw the paths in the sand (a stick will be conveniently nearby) in order to unlock a door, or reveal some secret or such.
Image attached. Feedback appreciated.
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Too easy. You have to 
Floating encircled bun alert.
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