Simon teaching his sister Neva from the Mathematical Fundamentals course on Brilliant:
Simon loves challenging other people with math problems. Most often it’s his younger sister Neva who gets served a new portion of colourful riddles, but guests visiting our home also get their share, as do Simon’s Russian grandparents via FaceTime. Simon picks many of his teaching materials in the Mathematical Fundamentals course on Brilliant.org, and now Neva actually associates “fundamentals” with “fun”!
“What is the chance that two people in a group of, say, 30 people would have their birthday on the same day?” I asked Simon as we were sitting on a bench by the river Schelde late last night, waiting for his Dad and sister to arrive by boat. The reason for this question was that one of the professors at Simon’s MathsJam club turned out to have celebrated his birthday exactly on the same day as I the week before. Besides I was afraid of Simon getting bored just sitting there, “enjoying the warm evening”. At first, I thought he didn’t hear my question and repeated myself a couple of times. Then I noticed he was so silent simply because he was completely immersed in the birthday problem.
Eventually, at that time already on Antwerp’s central square, Simon screamed with joy as he told me the formula he came up with involved triangle numbers! “It’s one minus 364/365 to the power of the 29th triangle number!” he shouted. “It’s a binomial coefficient, the choose function!”
Here we are, on the day of my 40th birthday, while recording a lesson of drawing with turtle in Python. It was meant to be my birthday present, a beginner-friendly hour of code, in which Neva would also be able to take an active part. We ended up recording two beautiful sessions only to find out later that our screen capture video was irreparably corrupted (never record in mp4 in OBS). Simon was inconsolable. We also thought this webcam recording was gone but rediscovered it a day later. So nice to have it as a memory.
And I’m relieved to say that we have managed to redo the whole project from scratch today (sadly without Neva’s participation this time as she had better things to do, so I look rather redundant sitting there next to Simon giving the lesson). Once Simon is done with the editing (which is another two days of work I’m afraid), he will upload the hour of code on YouTube. He also plans to create a website for this project to enable his “students” to draw in a built-in application.
From our session today:
And here is an example of Simon drawing with turtle for his own pleasure, a Serpinski triangle in Python (a few days ago):
Walking home from the swimming pool (where he and Neva had been jumping into the water exactly 24 times, calling out all the permutations of 1,2,3 and 4), Simon suddenly stopped to tell me that some day, mathematics may become engulfed by computer science. Apparently, this was what he was thinking about the whole time he kept silent on the way. Once we got home I sat down to listen to the elaborate proof he had coined for his hypothesis. Here is comes, in his own words:
Someday mathematics may become computer science because most of mathematics uses simple equations and stuff like that, but computer science uses algorithms instead. And of course, algorithms are more powerful than equations. Let me just give you an example.
There’s this set of numbers called algebraic numbers, and there’s this set of numbers called computable numbers. The algebraic numbers are everything you can make with simple equations (finite polynomials), so not like trig numbers, which are actually infinite polynomials, just simple finite equations with arithmetic and power. Computable numbers, however, are a set of numbers that you can actually make with a finite algorithm. It may not represent a finite equation, but the rules for the equation have to be finite. So the algorithm that generates that equation has to be finite. It’s pretty easy to see that every algebraic number is by definition computable. Because the algorithm would just basically be the equation itself.
Is every computable number algebraic? Well, we can easily disprove that. It took very long to prove that Pi is not algebraic, that it is transcendental, as it’s called. But Pi is computable, of course, because, well, that’s how we know what Pi is, to 26 trillion decimal places. So there you go. That’s a number that is computable but not algebraic. So the Euler diagram now looks like this:
Now we look back at the beginning and we see that algebraic numbers have to do with equations and computable numbers have to do with algorithms. And because the set of all algebraic numbers is in the set of all computable numbers as we’ve just proved, the set of computable numbers will have more numbers than algebraic numbers. We have given just one example of how algorithms are more powerful than equations.
What about the mathematics that deals with numbers that are incomputable? – I asked.
Well, that’s set theory, a different branch of mathematics. I meant applied mathematics, the mathematics that has application.
Here Simon explains one more effect he has played with at home, the Magnus effect.
Sunday at the beach, Simon was reenacting the 5 doors and a cat puzzle (he had learned this puzzle from the Mind Your Decisions channel). The puzzle is about guessing behind which door the cat is hiding in as few guesses as possible, while the cat is allowed to move one door further after every wrong guess.
“Here’s a fun fact!” Simon said all of a sudden. “If you add up all the grains of sand on all the beaches all over the world, you are going to get several quintillion sand grains or several times 10^18!” He then proceeded to try to calculate how many sand grains there might be at the beach around us…
In the evening, while having a meal by the sea, Simon challenged Dad with a Brilliant.org problem he particularly liked:
Simon’s explanation sheet (The general formulas are written by Simon, the numbers underneath the table are his Dad’s, who just couldn’t believe Simon’s counterintuitive solution at first and wanted check the concrete sums. He later accepted his defeat):
Simon is looking at his subscribers count on YouTube. We speculate if he gets to 1000 before the end of the academic year. Simon tells me that’s because subscriber count is just another example of Benford’s law in action. What is Benford’s law? – I ask.
“If you take some data that spans a few orders of magnitude and take the leading digits of all numbers, then you’re most likely not going to get a uniform distribution. Instead, 30% of the time, the numbers will start with a 1, a little bit less of the time – with a 2, even less – with a 3, and so on all the way to 9 (which has a low chance of occurring). For example, the populations of countries would follow that law. If something is not random enough, though (like human height in meters), then it wouldn’t follow that law. If something is too random, it also wouldn’t follow that law.”
Simon explains further: “Consider YouTube subscriber count over time. If you have 100 subscribers, then to get up to 200 is an increase of 100% (which is pretty big). But to get from 200 to 300 is only a 50% increase. From 900 to 1000 is just 11 %.”.
Then his dad asks: “What about going from 1000 subscribers to 1100 subscribers?”
“Well, Benford’s law only cares about the leading digit (and that’s what you want to increase as well). So you don’t want to increase from 1000 to 1100, you want to increase from 1000 to 2000! In other words, start a new Benford’s law ‘Epoch’.”