Paul Cahen

Paul Cahen

PhD Student in Mathematics

Mathematics toward a broader audience

I discovered juggling at 17 (actually thanks to other young persons into mathematics) and since then I practiced thoroughly or not-so-thoroughly depending on the moments. Anyway, juggling has much to do with mathematics and I used the mathematization of juggling, famously known in the juggling community as the siteswap notation, to show how mathematical reasonments could be useful in practice. Here are the slides of a talk that I gave in front of high school students on June 15, 2026 at the ICJ (my lab in Lyon). I hope to give more talks about that in the future.

This olympic problem, who was the last problem at IMO 2015, has much to do with this mathematical theory of siteswaps.

Levine hat puzzle has $n\ge 2$ players given each of them an infinite stack hats, each of them colored at random either black or white, the colors being chosen i.i.d. flipping a fair coin. They are allowed to communicate before the random colors are sampled but not after. The players see everyone’s hats but theirs, and they simultaneously try to pick a black hat of their own stack. They win precisely when all the players get it right. Let $V_n$ be the supremum of their success probability over all possible strategies. Levine’s conjecture states that $V_n\to 0$ as $n$ goes to infinity. Even the value $V_2$ is unknown.

I gave a talk about this fascinating problem at the Séminaire Pizza de l’ICJ in front of the interested researchers of all the lab. I mostly based my talk on the paper On Levine’s notorious hat puzzle, but many other papers brought new ideas to that subject: see e.g. this one, or that one.

As a high school student I took part in the Olympiads, and attended many courses organized by the POFM who were meant to prepare us. After starting university, I started to give myself exercice sessions in many occurences. Here is a short resume of my activity there: TBD

Since the beginning of my PhD I helped MathàLyon a few times. This is an association that regularly visits middle schools to run mathematical workshops with students and help them discover various ideas beyond school mathematics through their own exploration. Here is a document presenting the activities of MathàLyon.