# Sharing an idea from Alissa Crans’s JMM talk with kids

Yesterday videos of several of the invited JMM talks were published:

I was skimming through the talks to see if there might be anything fun to share with kids and came to this slide in Alissa Crans’s talk “Quintessential quandle queries”

I had a hunch that kids might find it strange that an expression like $A B A^{-1}$ would be something of interest to mathematicians, so I decided to see what the boys thought:

When we played with numbers, the expression $A B A^{-1}$ was just $B$. So now I tried a few non-number ideas with A and B representing certain moves on a grid. The first set of moves seemed to behave just like the numbers did, but the second set of moves produced a little surprise:

Now we looked at a completely new situation -> in the video below A and B will represent moves of a Rubik’s cube.

Here I got a really fun surprise when the boys saw that doing repeated applications of the “move” $A B A^{-1}$ was actually really easy to describe mathematically 🙂

Definitely a fun little project. There’s no real need to show the boys the complete talk – they don’t need to learn the complete content of the talk. It was fun to show them an idea that is interesting to mathematicians, though, and especially fun to give them a peek at some simple operations that don’t commute.

# Sharing a new result about the Cantor set with kids

Earlier in the week I saw a tweet announcing a new (and really cool!) result about the Cantor set:

The new result is that any number in the intrrval [0,1] can be written as the product $x^2 * y$ where $x$ and $y$ are members of the Cantor set.

After reading the paper, I thought that it would be really fun to try to share some of the ideas with kids. The two ideas I wanted to highlight in the project today were (i) the geometric ideas in the construction of the Cantor set, and (ii) the interpretation of the Cantor set in base 3.

I started with a question about base 3 -> how do you write 1/2 in base 3?

Now we looked at constructing the Cantor set by removing intervals. The boys had lots of interesting ideas about what was going on

Next we looked at the incredible property that you can make any number in the interval [0,2] by adding two numbers in the Cantor set. This ideas here were a little harder for my younger son to understand than I was expecting, so I ended up breaking the discussion into two parts.

I think the ideas here are fun for kids to think through – how do I pick a number from one set and a second number (possibly from a different set) to add up to a specific number.

Here’s part 1:

and part 2:

Finally, we took a peek at the result from the paper -> how does multiplication work? This was also a fun discussion. The ideas necessary to see why you can find three numbers from the Cantor set that multiply to any number in [0,1] are obviously way out of reach for kids. However, seeing why the multiplication problem is difficult is within reach.

It is always a real treat to find math that is interesting to mathematicians to share with kids. I think talking through some of the ideas related to this new result about the Cantor set makes for an amazing math project for kids!

# Christopher Long’s fun generalization of an Expii problem

Twitter is really great place to see fun math. Before showing the fun generalization, though, just to avoid spoilers I want to show the original problem. Here’s the direct link:

https://www.expii.com/solve/69/5

and here’s the problem itself:

So, I’ll pause here to not ruin the problem for anyone who wants to work on it.

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Ok . . . here’s the really cool set of tweets I saw from Christopher Long this morning:

which continued as follows:

To see how delightful Long’s general solution is, maybe walking through my chicken scratch solution which happened to be still sitting on my desk will be helpful:

Here’s a sketch of my approach.

In order to maximize your chance of winning a bet like the one in the problem (one that you expect to lose) you should bet as much as you can at each step (subject to a maximum of the amount you need to win) at each stage.

So,

(i) at step 1 the probability of getting the tree to grow to 40 feet is 1/5 and probability of losing is 4/5.

(ii) Assuming you win, you now have a 1/5 probability of getting the tree to grow to 80 feet and a 4/5 probability of losing.

(iii) Assuming you win, you now have a 1/5 probability of getting the tree to grow to 100 feet and a 4/5 probability of having it shrink to 60 feet.

(iv) If you win on stage (iii) you win (1 out of 125 times). If you lose, you now have a 1/5 probability of having the tree now grow to 100 feet and a 4/5 probability of having the tree shrink to 20 feet.

So, after the 4th branch in my picture you’ve either won (probability 1/125 + 4/625), returned to 20 feet (probability 16/625) or lost (the only other case).

Thus, your probability of winning the game from the start, $x$, satisfies the equation:

$x = 1/125 + 4/625 + (16/625)*x$

We can solve this pretty easily to see that $x = 3/203$.

The really fun – and honestly, amazing – thing about Long’s solution is that he notices that the pattern in the branches of the binary tree corresponds exactly to the pattern in the digits of the binary expansion of 1/5. For clarity, the 1/5 here comes from the growth multiple – 20 feet growing to 100 feet – and not from the probability which, by coincidence, also has a 1/5 in it.

Anyway, Long’s solution also allows you to immediately see how to solve any problem like the Expii one, and, for extra fun, problems where the growth multiple is irrational:

The answer is in Long’s timeline, but it is a good challenge to see if you can work out the answer just from the tweets I’ve included here. Since he skips a bit of algebra in his tweets, working through his tweets is also an important way to make sure that you really understand his work.

I think the sequence of tweets from Long are a great thing to show kids who are learning math – especially kids learning probability and stats. Those tweets really show how a mathematician thinks about a problem.

# A challenge / plea to math folks

Two interesting math contest related articles this week. First in the Atlantic an article about different participation rates in math contests for girls and for boys:

Math for Girls, Math for Boys

By the way, the 2009 study by Ellison and Swanson mentioned in the article has one of the most interesting (and frankly baffling) statements about math education that I’ve ever seen. From the top of page 3:

“Whereas the boys come from a variety of backgrounds, the top-scoring girls
are almost exclusively drawn from a remarkably small set of super-elite schools: as many girls come from the top 20 AMC schools as from all other high schools in the U.S. combined.”

As far as I know, no one has done a follow up study to see what’s going on at those 20 schools. Seems like a very interesting topic to study.

The second article was about the US team finishing 2nd at the European Girls’ Math Olympiad:

US Team Takes Second at European Girls’ Mathematical Olympiad

After reading this article I clicked through to see the problems, and problem #1 caught my attention:

I found the problem to be fun to solve and also pretty interesting mathematically. It struck me as a great problem to use to show how mathematicians think.

Tim Gowers did an fantastic “live blog” showing his own thought process in solving a problem from the International Mathematics Olympiad a few years ago:

Tim Gowers walks through an IMO problem

I love this self-depricating line: “You idiot Gowers, read the question: the a_n have to be positive integers.”

Richard Rusczyk also has a fantastic example showing mathematical thinking involved in solving a contest problem – in this case problem #24 from the 2013 AMC 12.

So, my challenge to math folks is this – live blog your solution to problem #1 from the 2016 European Girls’ Mathematical Olympiad. Share your thinking, share your false starts, share the “aha” moments. I think this problem provides a great opportunity for people to see how people in math think, and, importantly, that the path to the solution of a problem isn’t always a straight line.

# A “new to me” proof that there are infinitely many primes

Last evening I saw Patrick Honner link to a Richard Green post discussing a prime number theorem he saw on Cut the Knot’s website:

So, thanks to Patrick Honner for pointing out Richard Green’s post, thanks to Richard Green for a great post and also for pointing out the Cut the Knot piece, and thanks to Alexander Bogomolny for maintaining the amazing Cut the Knot site.

By chance I’m between chapters in our Introduction to Geometry book with my older son, and since he’s going on a short trip today I didn’t want to start talking about circles just yet. This “new to me” proof that there are infinitely many primes was perfect for a discussion today.

Both Green’s piece and the Cut the Knot piece show how delightful this proof is, but there’s a lot going on in the background – especially when you are trying to talk through the ideas with a kid. We spent about 45 minutes going through things slowly and then started in again with the camera on.

First up, an introduction to the problem and my son discussing the “standard” proof that the number of primes is infinite:

Next up, we begin to dig into one of the critical ideas in the proof – the factorization of the polynomial $x^4 + x^2 + 1$, and the idea that the two polynomial factors will never have common factors when $x$ is an integer. The factorization idea relies on the Euclidean algorithm.

The next step is looking at how we construct the sequence of numbers that have progressively more prime factors. I was trying to talk through this part of the proof without referring to mathematical induction, but if a student is familiar with that concept, there’s probably an easier way to explain this portion of the proof.

Finally, a last little discussion of how the powers of 2 help us construct the sequence of numbers we are looking for. Sorry we went off camera a little bit on this part – I didn’t realize the camera was zoomed in so much. One small point about exponents confused my son a little, but once we got past that little hiccup I think he was able to get his arms around how we walk down the path of creating the sequence of integers.

This project made for a really fun morning. I like how some basic ideas from math come into play – prime factorization, factoring polynomials, and the Euclidean algorithm, for example. I also really like the idea of seeing a known result from a completely new angle. Seems like a really great example for kids interested in learning about proof and mathematical thinking in general. Thanks again to Patrick Honner, Richard Green, and Alexander Bogomolny for the inspiration.