This blog is about Simon, a young gifted mathematician and programmer, who had to move from Amsterdam to Antwerp to be able to study at the level that fits his talent, i.e. homeschool. Visit https://simontiger.com
Today, Simon returned to a problem he first encountered at a MathsJam in summer: “Pick random numbers between 0 and 1, until the sum exceeds 1. What is the expected number of numbers you’ll pick?” Back in June, Simon already knew the answer was e, but his attempt to prove it didn’t quite work back then. Today, he managed to prove his answer!
The same proof in a more concise way:
At MathsJam last night, Simon was really eager to show his proof to Rudi Penne, a professor from the University of Antwerp who was sitting next to Simon last time he gave it a go back in June. Rudi kept Simon’s notes and told me he really admired the way Simon’s reasoning spans borders between subjects (the way Simon can start with combinatorics and jump to geometry), something that many students nurtured within the structured subject system are incapable of doing, Rudi said. Who needs borders?
Later the same evening, Simon had a blast demonstrating the proof to a similar problem to a larger grateful and patient audience, including Professor David Eelbode. The first proof was Simon’s own, the second problem (puzzle with a shrinking bullseye) and proof came from Grant Sanderson (3Blue1Brown) on Numberphile.
“Don’t allow any constraints to dull his excitement and motivation!” Rudi told me as Simon was waiting for us to leave. “That’s a huge responsibility you’ve got there, in front of the world”.
Take any real number and call it x. Then plug it into the equation f(x) = 1 + 1/x and keep doing it many times in a row, plugging the result back into the equation.
At some point you will see that you arrive at a value that will become stable and not change anymore. And that value will be… φ, the golden ratio!
But this equation also has another answer, -1/φ. If you plug that value into the equation, it will be the same, too. The real magic happens once you have rounded the -1/φ down (or up), i.e. once what you plug into the equation is no longer exactly -1/φ. What happens is that, if you keep going, you will eventually reach… φ as your answer!
Simon believes that he has found a mistake in one of the installations at the Technopolis science museum. Or at least that the background description of the exhibit lacks a crucial piece of info. The exhibit that allows to simultaneously roll three equal-weight balls down three differently shaped tracks, with the start and the end at identical height in all the three tracks, supposes that the ball in the steepest track reaches the end the quickest. The explanation on the exhibit says that it is because that ball accelerates the most. Simon has noticed, however, that the middle track highly resembles a cycloid and says a cycloid is known to be the fastest descent, also called the Brachistochrone Curve in mathematics and physics.
In Simon’s own words:
You need the track to be steep, because then it will accelerate more – that’s right. But it also has to be quite a short track, otherwise it takes long to get from A to B – which is not in the explanation. It’s not the steepest track, it’s the balance between the shortest track and the steepest track.
Galileo Galilei thought that it is the arc of a circle. But then, Johan Bernoulli took over, and proved that the cycloid is the fastest.
We’ve also made some slow motion footage of us using the exhibit (you can see that the cycloid is slightly faster, but as far as I can tell, it’s not precision-made, so it wasn’t the fastest track every time): https://www.youtube.com/watch?v=5Brub0FnpmQ
I hope that you could mention the brachistochrone/ cycloid in your exhibit explanation. I don’t think you can include the proof, because for such a general audience, it can’t fit on a single postcard!
Simon loves looking at things geometrically. Even when solving word problems, he tends to see them as a graph. And naturally, since he started doing more math related to machine learning, graphs have occupied an even larger portion of his brain! Below are his notes in Microsoft Paint today (from memory):
Slope of Line:
Steepness of Curve:
An awesome calculator Simon discovered online at desmos.com/calculator that allows you to make mobile and static graphs:
Simon programmed a presentation to explain why 28×28 is not the same as 20×20 + 8×8 geometrically. The code is quite complicated and involves some trigonometry and conditional statements: the grid is divided into different parts every time Simon clicks and depending on how many times he has already clicked. This is typical Simon – coming up with an inherently arduous and complex system to visualise the beauty of the world around him, even of the seemingly trivial things. By the way, the inspiration for the 28×28 grid came from Simon’s favourite math channel, 3Blue1Brown and its latest video on Neural Networks (the grid was used to explain computer vision).
Simon is doing quite a lot of sums in his head nowadays, looks like it’s a new trend. Today, while bathing in the fountain outside, he was calculating how long 1/16th of a minute lasted. And a couple days ago, while waiting for his appointment at the hospital, he calculated how long it would take someone to read a whole page of random numbers, taking an educated guess that one takes 4 seconds to read out one number and remembering Daniel Shiffman mentioned there were 100×5 numbers per page in his book.
Simon is a fan of the 3Blue1Brown channel and absolutely loved their video on solving the Towers of Hanoi puzzle with binary and ternary numbers. He practiced a lot with both. Eventually he developed his own, rhythmic, way to solve the puzzle:
Applying ternary numbers (solving the puzzle in 80 steps):
The video on 3Blue1Brown channel:
Applying binary numbers (solving the puzzle in 15 steps):