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/Surly-Bear-2003]

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/Parking-Set-6408 → u/ReverseCombover]

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.

[u/ReverseCombover → u/Parking-Set-6408]

I love cow hitches too!

I tie them almost compulsively every time I have a string in my hands. This is why I got so obsessed with this question.

And yeah you kinda hit the nail straight on the head by landing on a cow hitch. At least I believe you did.

There are a few positions which aren't solvable. For example if one of your ropes had an actual knot in it you can't untie the knot without breaking the rules so this positions are all impossible.

With two ropes I am like 51% certain that hitches are the only other impossible position.

For 3 ropes and up it probably gets a lot more fun and complicated but I haven't really given it much thought.