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!