Our first activity was about the problem of writing numbers as sums of three cubes. This Numberphile video with Tim Browning explained the problem. We tried solving for a few numbers like 39.

We saw the proof that numbers of the form $9k+4$ or $9k+5$ can’t be written as a sum of 3 cubes.

Then we learned that the problem for 33 has been solved… yesterday! It was pretty neat to learn about a genuinely new mathematical discovery. We talked about how the problem itself isn’t important for applications, or even for other areas of math, but the way to solve it is kind of interesting.

Instead of a brute force search, you can look for rational points that are near the curve $x^3+y^3=1$.

Our second activity was based on Fun With Folding and Pouring by Steve Dunbar and Anne Schmidt. We worked out the folding part, where you start with a strip of paper with a crease at the midpoint. Fold one end to the crease, make a new crease, and unfold the strip; then repeat with the other end, and go back and forth between the two ends, each time folding it to the most recent crease, to make a new crease. The creases seem to converge to two limits. We also tried starting differently (with a different location instead of the midpoint.) Some groups explored it by working out the locations of creases in fractions, with results like $5/16$, $21/64$, $85/256$… These were close to $1/3$, in each case with the denominator off by $1$ or $-1$. It was neat to see “converging to $1/3$” in such clear terms. Other groups used algebra to write formulas with $x$. Those formulas led us to something reminiscent of continued fractions and also to base 2 decimal (“binimal”?) expressions. Our old friend, geometric series showed up. Specifically, we had $1/4 + 1/4^2 + 1/4^3 + \dotsb = 1/3$. We came up with some really neat and colorful dissections of triangles and squares that showed why that worked.

Finally, for upcoming Pi Day, we watched some 3blue1brown videos about an unexpected counting problem, where collisions of sliding blocks are counted by the digites of pi. The first video posed an interesting problem. The second video gave a really nice answer, touching on phase space, changing coordinates, circle geometry, the tangent line approximation $\tan x \approx x$, and more. The third video also was interesting, especially the very striking visuals of the final answer, but it was a lot more complicated and involved, in the work to get to that final answer. We thought high school students would probably enjoy the first two videos. And a lot of students would enjoy the third video too, but also a lot of students might be a little overwhelmed by it. It would have to be deployed judiciously.

Today Zach wore a “Mathematicians For Equality” t-shirt, designed by Kate Poirier on Zazzle.