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”.
It’s Sinterklaas season in the Dutch-speaking world and, of course, as we have started baking the traditional spiced cookies called kruidnoten (“gingerbread buttons”) Simon didn’t want to miss an opportunity to play a version of peg solitaire with eatable pieces!
Such a pleasant play date last week with another eager learner. Simon shared his GeoGebra skills and some geometrical paper tricks, among other things. It’s heartwarming to see Simon blossom socially, he is growingly attentive to younger kids and generally engaging with people of various ages, as long as they show interest in anything Simon has an understanding of.
During Chinese lesson yesterday, Simon came up with what he calls his “Cycle formula” to calculate all the permutations of placing n numbers in a cyclical order (like on a clock face). He also proved the formula. Wait, Chinese lesson? Yes, I know, this guy manages to squeeze some math everywhere. His Chinese tutor loved it by the way. “Well, we’ve both learned something!” Simon exclaimed delightfully.
Simon has been enjoying Stephen Wolfram’s huge volume called A New Kind of Scienceand is generally growingly fascinated with Wolfram’s visionary ideas about the computational universe. We have been reading the 1500-page A New Kind of Science every night for several weeks now, Simon voraciously soaking up the behaviour of hundreds of simple programs like cellular automata.
Wolfram’s main message is that, contrary to our intuition, simple rules can result in complex and often seemingly random behaviour and since humanity now has the computer as a tool to study and simulate that behaviour, it could open a beautiful new alternative to the existing models used in science. According to Wolfram, we may soon realise that the mathematical models we are currently using, based on equations and constraints instead of simple rules, are merely a historical artefact. I’m amazed at how much this is in line with Simon’s own tentative thoughts he was sharing with me earlier this year, about how maths will be taken over by computer science and how algorithms are a more powerful tool than equations. When he came up with those ideas he hadn’t discovered Wolfram’s research and philosophy yet, he used to only know Wolfram as the creator of Wolfram Mathematica and the Wolfram language, both of which Simon greatly admires for being so advanced.
Last night, Simon was watching a TED talk Stephen Wolfram gave in 2010 about the possibilities of computing the much aspired theory of everything, but not in the traditional mathematical way. “It’s about the universe!” Simon whispered to me wide-eyed, when I came to the living room to fetch him. “Mom, and you know who was in the audience there? Benoit Mandelbrot!” (Simon knows Mandelbrot died the same year, he is intrigued by the fact that his and Mandelbrot’s lifetimes have actually overlapped by one year).
We have been informed by the World Science Scholars program that Stephen Wolfram will be one of the professors preparing a course for this year’s scholars cohort, so Simon will have the unique experience of taking that course and engaging in a live session with Stephen Wolfram. It is breathtaking, a chance to connect with someone who is much older, renowned and accomplished, and at the same time so like-minded, a soulmate.
Inspired by reading Stephen Wolfram, Simon has revisited the world of cellular automata and Turing machines, and created a few beautiful Langton’s Ants: