The Pipe cleaner Problem
Imagine the rings below in three dimensions. They’re linked: you cannot separate them without cutting one open.
These two rings are not linked; they come apart easily.
Your goal in this problem is to make a very special collection of three linked rings that will not come apart without being cut, but where if you remove any one of the linked rings, the other two are no longer linked at all!
the solution
Written, and constructed with pipe cleaners, by BEAM’s CEO Dan Zaharopol
What's great about this problem is the way it makes you think.
Some people might look at the problem and just ask themselves, "what's the trick?" Find the trick, and once you do, it doesn't feel like there's any math here. You know it or you don't.
That's not how math really happens, though. Every "trick" comes from careful thinking and being able to reason through what, specifically, you need your "trick" to accomplish! That's what opens the door for creativity to solve new kinds of problems.
So, instead of just showing you a trick, I want to walk you through how you might come up with the trick yourself. As I tell our students: The more you learn how to think, the more you will be able to do, in math as well as in life.
To start off, you might just experiment. Three rings linked together in the standard way doesn't work, because if you remove any one of them, the other two stay linked:
Similarly, you could connect them in a line, and if you remove the middle one, sure enough the other two go free. But remove either one on the ends, and the other two stay linked, so pairwise links in a line does not work!
If you're being methodical, you'll realize that these are the only ways to link three rings using only pairwise linking. In other words, if we're going to answer this problem, we must come up with a different way of linking rings, one that does three all together, and is not just pairwise linking.
What, then, can that be? Methodical thinking doesn't just reveal that we need a new idea: it can reveal what we must accomplish with that new idea, which will open up the whole problem. Let's see how that works.
In this case, here's what I'd like to try to do: to connect three rings where no two are directly linked, but where two are almost linked and the third prevents them from moving past each other. That's still somewhat vague, but it might be enough.
So now I fiddle. I try several ideas, and eventually, I find something new. Here are two rings that are not actually linked, but to move them apart they must still slide past each other:
Then all I need to do is block them from sliding past each other. Perhaps a third ring could do that! Something through the center of the yellow ring could do that, because it would keep the blue one from sliding out. Sure enough, we can accomplish it with a third ring (in brown) that keeps the other two in place:
The brown ring is not linked to yellow or blue; if either yellow or blue was gone, it could move out of the other. But all together, these cannot separate, even though if you remove any one, the others come apart easily. Done!
And so our methodical approach, plus a lot of playing around and fiddling, led us to the moment of genuine creativity that solved the problem. It's also just incredibly satisfying to find; BEAM students are always delighted when they come across this idea.
After that moment of celebration, however, BEAM students also realize that there is another problem: problem 74 on the 100 Problem Challenge asks us to do the same for four rings!
the four ring challenge
So how can we do four rings? Where all four are linked, but if you remove any one, the other three come apart?
If we've done the careful approach above, then we can actually extend it fairly naturally. After a bit of thinking, you realize that you can connect as many loops in a row as you want with this method of doubling them over and hooking them together. Then, connect the ends with a final loop making one big circle, and it holds all of them in place until one is removed. So we start with three folded-over circles that could slide past each other:
And we add a fourth loop (in brown) that keeps them from sliding, but is not pairwise linked to anything, making one big loop-of-loops:
And with that, we have completed all four! In fact, we could keep this up as long as we'd like, making longer and longer chains of rings that are connected but where, if any one is removed, all of them come loose!
This really is one of those places where, when you see it, it feels like a trick: "oh, bend the circles over so they're not actually linked. Why didn't I see that?" Of course, the answer is that it's not just a trick. It came out of careful thought, defining the problem precisely, and then well-applied creativity. It also showcases how important it can be to start with an easier question; the four-ring question would have been much harder without our work on the three-ring question!
For more on this: check out the Borromean rings (the three-ring version we started with, which also has some fun history), and the more general multi-ring version, the Brunnian link (which uses a very slightly different method from above, although we're pretty sure ours still works!). It turns out that it's a good thing we gave you pipe cleaners: as an interesting geometry note, mathematicians have proven that you can't do these with perfect circles. You must have the flexibility of pipe cleaners, or be able to stretch the circles into ellipses, or something that changes the geometry of a perfect circle to make everything fit.