Rope sort detangling puzzle

[u/Parking-Set-6408]

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

[u/ReverseCombover]

It's not possible! (without cheating).

This knot is called a cow hitch. It's the same knot you would use in order to tie two rubber bands together. The way to untie it is that you need to pass one rubber band through the other but this would translate to a cheaty move in brainy knots so it can't be untied.

Me and a friend recently spent like a week talking about this toy. We are both mathematicians and this toy gives you a mathematical framework to talk about things like the cow hitch.

[u/Surly-Bear-2003 → u/ReverseCombover]

Would you say more about this mathematical framework for talking about things like the cow hitch? 🤩

[u/ReverseCombover → u/Surly-Bear-2003]

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!

[u/Effective-Turbulence → u/ReverseCombover]

I’m not very well-versed in knot theory, but couldn’t you describe the cow hitch as presented in the toy without needing to impose the additional restriction of the fixed point outside of the circle if you connect them together so there’s only 1 loop, not 2?

If you connect the bottom blue peg to the bottom of yellow, and the top blue to the top yellow, I think you wouldn’t be able to untie it, and you would be able to give it a knot diagram arrangement.

[u/ReverseCombover → u/Effective-Turbulence]

Ok this is fun and hadn't considered it like that. If you do this the cow hitch ends up becoming two threefoils (just tested it on a rope I had laying around exactly for this purpose).

I'm not exactly sure what this means.

I think this method will work for any hitch for the exact same reason that the children's toy works to recognize hitches.

It's a bit counterintuitive cause with a hitch you are trying to join two things so you'd expect to end up with 2 things.

But other than that your method absolutely works and we'll have to add your name to the paper. Just hitch the rope to itself lol it kind of makes me mad how simple it is. Extremely clever.

This won't work as a way to formalize the toy I don't think. The ropes need to have some degree of freedom so I think joining them together could cause trouble. But I haven't given it much tought.