Have you ever played “human knot”? It seems to be a common icebreaker/team-building sort of exercise (just do a Google search for “human knot” and you’ll find quite a few pages discussing it). Everyone stands in a circle, then reaches across with both hands to randomly grab the hands of two others. The goal is then to untangle the “knot” into a circle, without letting go of any hands.
I played this game several times growing up and I never thought twice about it. Many of the materials that came up first when I searched online seem to indicate that it will always be possible for the participants to untangle themselves into a circle. I found a few places that acknowledged it is not always possible, since instead you “might get two or three separate circles”. One site suggested that if the participants seemed really stuck, the facilitator might allow two of them to let go of their hands and then rejoin them on the other side of some obstruction; but it was phrased as more of a concession—to prevent the participants from getting too frustrated—than as a necessity.
But mathematically, this is all hogwash: it is possible to get arbitrarily complicated knots (or, more generally, links, which are knots formed from several separate loops) by this process, where no matter how hard the participants try to untangle themselves, they will never reach a circle. For example, here is a figure eight knot:
It may not be immediately obvious you can’t turn this into a circle just by fiddling around with it, but you don’t have to take my word for it: it can be mathematically proven that you can’t. (Perhaps that’s a subject for another post.) I leave it as an exercise to figure out how people standing around a circle could grab one another’s hands in order to form a figure eight knot. The game really ought be called “Human Unknot”, since if you actually form a knot the game is no fun at all!
Knots can also have arbitrarily high unknotting number, which is the minimum number of times the knot would need to pass through itself (corresponding to breaking and rejoining one link in the chain) in order to become unknotted; so allowing the participants one break/relink might not be enough.
People play this game all the time, and yet most of the descriptions of the game grossly misunderstand the possible outcomes. How can this be? For me, it raises the following question:
If n people stand around a circle and randomly take one another’s hands, what is the probability that they form a knot?
There are a lot more questions one could ask, and I’ve left the question sort of vague on purpose; to really answer it would require pinning down what we mean much more precisely. But for now I’ll just open it up to discussion!
The Math Less Traveled 
Even ignoring the knotting, you immediately have a lot of structure because a “human knot” is a 2-regular graph with participants as vertices and holding hands as edges. Then you could use the Erdős–Rényi random graph model to see what happens when the number of participants gets large. Unless I’ve made a mistake (or mistakes), there will be a giant connected component, in this case a cycle, and almost certainly some clusters of size O(log n). According to the model, there’d almost certainly be isolated vertices, but by construction there shouldn’t be any due to 2-regularity.
Very interesting post, I remember playing the know game quite often in elementary and middle school. After reading this I did some poking around on the internet and found this power point from a capstone course at the California Lutheran University.
http://www.callutheran.edu/schools/cas/programs/mathematics/people/documents/humanknotfinalfinal.ppt
Not sure it about “if you actually form a knot the game is no fun at all!”. That’s a bit like saying twister is no fun because you always end up falling over trying some impossible move.
Well, Fergal, I think one problem is that in Twister you expect it to end with a fall, but in the knotting people expect (as evidenced by the net instructions descriptions) to end in success. Furthermore, a sufficiently tight knot would not even allow the players to move very much.
As a side, I’d bet anything that people’s grabbing tendencies are far from random, preferring choices that allow some personal space, comfortable grip, and perhaps even prefer to reach a particular part of the circle relative to one’s self. And of course, if there is a cutie in the circle, everyone will try
to go for his/her hand. It is crucial to take this all into account when calculating probability to knot type formation 🙂
Great post, Brent; nice find, Brittany! Colin Adams of Williams College cited in the powerpoint!
Wow, this makes me feel so much better for the few times that I was in a group that was unable to untie! (I was pretty sure it was impossible at the time.)
Thanks, everyone, for the thoughts so far. And great find, Brittany! I actually took two classes with Colin Adams (one on knot theory and one on hyperbolic 3-manifolds) which is definitely where I’m coming from in thinking about this. =) My first exposure to the fact that the human knot might not be what I had always thought was actually when I attended a calculus course taught by Adams when visiting Williams — at the beginning of the class he had two groups of volunteers come up and arrange themselves identically in human knots according to a diagram he had drawn on the board — but then he strategically moved just one grip in one of the groups. Then one group proceeded to easily untangle themselves, and the other group struggled to no avail to get out of their figure-8 knot. =)
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I ran across this page doing a search to see how to untangle the human knot. I once saw where you start with players holding hands in a circle and then making them weave in and under arms (not letting go of the person’s hands) and making a knot, then the team has to untangle. Because they started with a circle, then it is possible to untangle the knot. It still was difficult with 12 people. But I can’t seem to find anything to show me again how to make them get tangled.
I think I may use this with my class. Thanks for the post!