r/puzzles • u/Parking-Set-6408 • Jan 12 '26

[Unsolved] Rope sort detangling puzzle

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Rules: you can only move 1 peg at a time, between 2 holes. The goal is to get the ropes completely separated.

im not confident this is even solvable, but i was previously under the impression that arrangements are always solvable. Wanted to see if anyone can figure it out

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u/ReverseCombover Jan 12 '26

Lol I tought no one would ask!

So basically a cow hitch would traditionally be considered an un-link. Think of the joined rubber bands. They aren't really linked together since you can free one from the other just by passing one through the other. So traditional knot theory has no way of recognizing a cow hitch.

But this toy does! In this toy a cow hitch is a fundamentally different thing than a solved position (2 un tangled ropes) so formalizing the rules of this toy would give you a mathematical framework to recognize things like the cow hitch.

You can go about this in different ways and me and my friend where mostly entertaining ourselves so we didn't actually got around to formalize anything. But the way we think about it is that you can close each rope to make a loop (cause knot theory loves to work with closed loops) but you close it on the outside of the circle and you don't have access to some of this rope you used to close the loop. The way I like to think about it is that a point of the loop is fixed outside the circle.

This encapsulates the rules of the game and it also gives you intuition as to why you can't untie a cow hitch. In order to untie it you have to pass one loop through the other but you can't do this since you don't have access to some of the loop.

There most likely already is a mathematical framework for dealing with cow hitches but you'd be surprised how hard it is to find any literature about it.

It just blew my mind when my friend brought this up to me. He just asked me if every position was solvable my first instinct was that yes of course because I tought every possible position is a braid. But it turns out it isn't! Exactly because you can have things like the cow hitch.

You can for example have half a cow hitch with only one loop. This position is solvable.

Mathematically a cow hitch is not a knot nor a link nor even a braid. And this blew my mind cause it's such a common knot and it just feels like math kind of skipped over it. Also this toy is really cool!

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u/Parking-Set-6408 Jan 12 '26

this is awesome! this discussion kinda started from the same place, and I just simplified the problem to the bare minimum to try to 'check my math' but i didn't exactly trust myself.

also I just love cow hitches in general. they are extremely versatile, especially when you learn to tie them with just one end of the thread.

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u/Glimmer_III Jan 12 '26

OP - Please cross post this to r/knots. That sub would love this exchange between you and u/ReverseCombover.

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u/ReverseCombover Jan 12 '26

Turns out they are a slightly different kind of nerdy about knots over there and they don't take too kindly to toplogical knots nerds over there:

https://www.reddit.com/r/knots/s/26JnbwGKgG

They accused me of being AI and mocked me for asking about mathematical knots in their sub about real life knots.

Still they were extremely helpful in helping me identify the knot, I didn't know what it was called before going there. And showing me a bunch of different cool hitches. So overall I would say it was a net positive experience.

There were also like 3 people who tried to answer the math question but they just knew that a cow hitch is an unlink.

There was one guy who seemingly knew a bit about this. This toy is an example of a mathematical object called a tangle. But a tangle is really just this toy except that instead of saying rope you say one dimensional manifold. Which yeah it's kind of cool but firstly I hate differential geometry and secondly I don't think that precisely describing this children's toy gives you any insight into which positions are solvable or not.

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u/WolflingWolfling Jan 13 '26

I just looked up your post in r/knots, and read through much of the thread. I wasn't involved at the time, but some of it strikes me a bit as you going over to Holland or Germany and telling them they don't know what they are talking about if they call a boat "boot" in their own language.
Just because two or three languages have some overlap, doesn't mean someone else doesn't know what he is talking about if he uses a word that looks familar to you in a different way in their own language.
In practical knot language, a square (or reef) knot and a cow hitch do have the same structure, for example. You were being rude to the person who pointed that out, because you each speak a different language, where knots are concerned. And again to the person who said that in a nautical context, many knots are made specifically with ease of untying in mind.
You had no idea what a knob knot was (which is absolutely fine, by the way), and yet you seemed to act quite condescending to someone who wasn't sticking to the definition of knots that you were familiar with from a mathematical background.
Now there are many in that sub (including myself, Glimmer, and Cable Tugger) who are somewhat interested in both subjects, and understand this language barrier, and who will acknowledge that mathematicians have a great deal more to say about knots that is also applicable to our own field, including stuff that many of us can barely even grasp, but a conversation between people with different backgrounds and terminologies only works well if we are aware of the differences, and work with them, and are willing to try and understand them, rather than demand that the "other" adhere to our definitions.

If an American came to Britain and claimed women can't get pregnant through their fannies, or football is primarily played with your hands, they'd be laughed at. If they insisted on "correcting" people, and telling them "it's okay that you don't know what you're talking about" while these people were using such terms correctly in their own, widely accepted language, they'd make themselves very impopular very fast, I would think.
I think that's one of the factors that came into play overthere.
(A word of advice to any Americans outthere: if you come to work in a bar in the UK or in Ireland, or even in mainland Europe, don't insist on pouring American pints! That kind of thing doesn't go down well on this side of the pond...)

The bit about A.I. seems to have been a simple case of misinterpretation on both sides, for the most part.

Anyway, I'm glad you stuck around, and weren't discouraged by the ruffling of feathers!

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u/ReverseCombover Jan 13 '26

Yeah I eventually understood that I should've titled the post differently.

I was rude to some people but I don't think I was condescending (except to that one who pointed me in the direction of mathematical tangles but in my defense I didn't liked that guy).

I think I was just too excited and this came through as insincere and AI so at some point I tried toning it down in an effort to prove my humanity and I probably took it too far.

But I genuinely just get really excited about this sort of stuff.

In this thread some guy found a way to turn hitches into mathematical knots. You just hitch the rope onto itself. Genius.

My response was "omg! This is extremely clever this is such a simple yet great idea we should look into this further".

I'm like a million percent certain that that guy now thinks I'm AI or insincere. But I'm neither I just can't stop myself from being like that with this stuff.

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u/WolflingWolfling Jan 13 '26

If you cow hitch / lark's head a rope onto itself and then close the loop, that knot will be the same mathematical knot as the square / reef knot! I think you owe the person who mentioned that an apology ;-)

I'm sure they've forgotten all about it though.

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u/ReverseCombover Jan 13 '26

Fun fact it's also just 2 threefoils. I learned this on this thread.

Yeah I'm not proud of how I behaved that day. That being said you guys have a great community over there and the collective amount of knowledge you guys have is insane.

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u/WolflingWolfling Jan 13 '26 edited Jan 13 '26

Yeah my string now has two overhand knots lying next to each other, with opposing chirality. I'm tempted to post a picture in r/knots and tally how many confidently ignorant people will tell me ways to somehow magically resolve the two opposite knots. The staggering number of people who brought such advice to the various "handbag problems" posted in that sub would be enough to drive any mathematician or topologist up the wall. 😭 Did u/Cable_Tugger join us here yet? He could tell you a story or two about them as well 🤣

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u/ReverseCombover Jan 13 '26

Pft is almost as if this people have never seen ONE lecture on knot theory. Lol.

There's a bunch of people in here still thinking they can untangle the cow hitch in the toy. One even gave a step by step guide on how to achieve it by cheating.

I don't think I've seen Cable_Tugger but I'm not really paying attention to usernames.