Mathematics, applications of mathematics to life in general, and my life as a mathematician.
Archive for October 2023
My friend, Alexander Karabegov, sent me one of his puzzles. I love the mixture of algebra and calculus.
Puzzle. Describe a real quadratic function f such that the graph of its derivative f′ is tangent to the graph of f.
My former student, Xiaoyu He, invented this elegant puzzle and shared it with me.
Share:Puzzle. We’ve got a murder mystery on our hands. There are four suspects, and it’s pretty clear that one of them is the actual murderer. But here’s the twist: there are also four witnesses who know who the killer is. Now, three of these witnesses are the honest type, always telling the truth, but the fourth one always lies.
You get to ask each of these witnesses a single yes-or-no question, and your question must be, “Is the murderer among this group of suspects?” You can choose any group of suspects you want. The challenge is to figure out who the murderer is.
Can you take it up a notch and determine the murderer if you have to list all your questions before getting any of the answers?
The Museum of Mathematics organizes a biannual conference in recreational mathematics called MOVES: Mathematics Of Various Entertaining Subjects. Being a math-recreatinalist myself, I attend all of them. The last one was held in August 2023 and was devoted to the mathematics of Fiber Arts. A few years ago, I wouldn’t have believed I had something to do with the arts. However, in recent years, I started crocheting mathematical objects for my classes, so I not only decided to attend, but also submitted a talk proposal.
The talk proposal was accepted. And, when I finished my slides, I realized that I created too many objects for a 25-minute talk. So, I divided my talk into seven sections and asked the audience to choose. I was lucky that they chose what I was most excited about, but I only had time to cover 4 out of 7 section.
Today I am posting a couple of pictures from my slides that showcase Brunnian links.
A Brunnian link is a set of loops that can’t be separated, but if any one loop is removed, they all fall apart. The most famous example of a Brunnian link is Borromean rings which I already wrote about. In the first picture, the Borromean rings are on the right. They are famous because they are the simplest Brunnian link.
There are two natural ways to generalize Borromean rings. One way is to use the same three loops, but make them twist around each other more. The second way is to increase the number of loops.
The left side of the first picture shows three tangled loops, which are more intertwined than Borromean rings. The crossing number of the link on the left is 12, while the crossing number of the Borromean rings is 6.
The second picture shows two Brunnian links with more than three loops. The left link has four loops, while the right link has five.
