Some Math I’m Playing With

I’m trying to spend more time playing with math. Because it’s fun, because I learn about learning, and because I want to practice the discipline I teach. Here are a few problems I’m playing with right now. No spoilers please, but feel free to play along!

The two diagrams above are different pictures of “Borromean Rings.” They are each a link of three components, which basically means three loops of string arranged together. The Borromean Rings have the property that the components cannot be separated as pictured, but removing just one of the components means that the other two can be separated. A link with this property is called “Brunnian.” Can you find a Brunnian link of four components? Of more components? (problem from Colin Adams’ The Knot Book)

This is the most recent problem from Play With Your Math and I have already had a ton of fun with it. It’s neat because the number of possible mountain ranges grows quickly, so it is very hard to count each possible mountain range individually for large numbers. I think it is possible to organize my work to find patterns that help me to make better predictions. I thought I had a breakthrough yesterday and then realized it didn’t work. So fun!

The Fibonacci numbers are really cool. They go like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,… where each number is the sum of the two numbers before it. They seem to have a neat property. 21 is the 8th Fibonacci number. 8 is divisible by 2 and 4, and 21 is divisible by the 2nd and 4th Fibonacci numbers, 1 and 3. Is this property true in general? Why? What other divisibility properties do Fibonacci numbers have? (problem inspired by this book, which was based on the PCMI 2012 math course)

I played a neat game recently. In a group of at least three people, each person randomly chooses two other people. Your goal is to stay equidistant from (though not necessarily at the midpoint of) your two people. Some positions act as an “equilibrium.” For instance, with three people, an equilateral triangle is an equilibrium position. What do some equilibrium positions look like for larger numbers of players? What is the probability that equilibrium is possible with randomly chosen people? What about a different version where each person chooses a “hero” and a “villain” and tries to keep their hero between them and their villain?


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