[I wrote this post to get my thoughts from yesterday’s talk down on paper. It was written very quickly, not edited, not formatted, and probably lots of other bad stuff, too! I just wanted to get put something on paper quickly before I forgot everything!]
I spoke to a math camp at Williams College yesterday (July 11, 2016). I’m incredibly grateful to professor Allison Pacelli for the invitation to speak at her camp. The talk was 45 min plus questions and I was super rusty at talking to a group of 20 kids! The topic were:
(1) Larry Guth’s “No Rectangles” problem:
I discussed the problem with my kids here:
This is a great problem for high school kids to think through. In fact, I learned about the problem from a lecture that Guth was giving for a high school camp at MIT last summer.
Unfortunately this problem needs a bit more than 1/3 of a 45 min talk. Still, there were some great comments by the students. The 3×3 case is a great way to get the discussion going. Pretty quickly the kids will see that if you cross out more than 6 squares you’ll get a rectangle. Also pretty quickly they’ll be able to come up with a reason why. It is fun to hear kids discover the pigeonhole principle!
If you have more time you can start to look for patterns. The most squares you can cross out in a 2×2 grid is 3, for a 3×3 grid it is 6, for a 4×4 it is 9×9, and for a 5×5 it is 12. I’d guess you could arrive at these numbers in 20 minutes with a group of 20 kids.
One incredible bit of math in the problem is that it has been solved when the grid size (I’ll call the grid size of an nxn grid n) is related to the power of a prime. If q is a power of a prime, then the most squares you can cross out without forming a rectangle in a square grid with size is . I don’t know how to prove this fact to a group of high school students, but you can just state it.
Taking you’ll find that the maximum number of cross outs for a 7×7 grid is 21 and the “increase by 3” pattern above has broken. A good challenge for the students is to find the maximum number of cross outs for a 6×6 grid (it isn’t 15).
(2) Bjorn Poonen’s n-dimensional spheres problem
This is a fun (and actually pretty mind-blowing) problem about n-dimensional spheres. I happened to be sitting in the MIT math lounge after listening to a lecture and heard Bjorn Poonen discussing the problem with a student.
I have three blog posts about the problem – all linked below. You can actually talk through this one in 15 minutes with high school kids. The only math they need to know is the Pythagorean theorem.
The students were able to get the 2 and 3 dimensional cases pretty easily, but hardly any of them (maybe even none) had seen higher dimensions. Luckily the geometry from 2 and 3 dimensions makes the pattern pretty clear and extending the geometric ideas to higher dimensions at least seems plausible.
I got two great questions from the students about this problem:
(1) How do we know that you calculate volume in higher dimensions the same way as you do in 3 dimensions?
I loved this question so much, but my teaching rust played a role and I didn’t give the most satisfying answer. I wish that I would have at least mentioned the Minkowski metric and Einstein’s theory of relativity. Here you are, indeed, calculating distance in 4 dimensions is a totally different way! Unluckily that idea didn’t occur to me as I was driving home.
Instead my first thought was sort of dive into some calculus-like ideas about how you can construct higher dimensional spheres by stacking up lower dimensional ones. My snap decision was that this would lead us too far of on a tangent.
My next thought was mentioning Grothendieck asking questions like “what is length?” and “what is area?” but again I rejected this approach as being a little too philosophical and it would look as though I was trying to side step the student’s question.
Then my thoughts turned to cubes (which were part of Poonen’s problem anyway). But thinking that was too simplistic and wouldn’t give the student as satisfying an answer as the question deserved, I went a tiny bit beyond cubes. After telling him that this was a *great* question, I told him that he might really enjoy looking up some information on the 4-dimensional platonic solids and how you find their volume. We’ve played around with Matt Parker’s video, for example, which is probably why this idea popped into my head:
Anyway, while I do think that looking up and playing around with the 4-d platonic solids would be a neat way to learn about 4-d geometry, I really wish that I had provided an example of non-Euclidean 4-d geometry. Darn rust!
(2) The answer to Poonen’s 3rd question – when does the volume of the inside sphere actually get bigger than the volume of the box? – is dimension 1206.
The kids were baffled that it was even possible to calculate the volume of a sphere in that dimension, and I’m really glad that they asked me how to do it.
In a 15 minute talk about the problem I’m not going to derive sphere volume formulas, but . . . the formula for the volume of even dimensional spheres is actually pretty easy:
The volume of a sphere in dimensions is .
They were amazed that the formula could be so simple 🙂 This question also led quite naturally to the fun result in my 3rd blog post about the problem – the surprising relationship between and hiding in the problem.
3. Ann-Marie Ison’s Modular Arithmetic Art
The last 15 min of the talk was about Ann-Marie Ison’s art and it blew the kids away!
I first saw her work on twitter here:
You can see all of her work here:
The great too for sharing her work is this interactive Desmos activity that Martin Holtham made and Ison shared with me here:
The various images of multiplying by N in mod M completely captivated the students. They wanted to see more and more and more and more. They would offer conjectures – some would seem to work while other’s wouldn’t – and we just played and played.
It was a great way to show kids how much fun a math exploration can be. Can’t wait to use it again the next time I’m speaking to students!