UREM-ULB :
Unité de Recherche sur l’Enseignement des Mathématiques

Timothy Gowers, médaillé Fields, Professeur à l’Université de Cambridge, bientôt professeur au Collège de France a mis a profit son confinement avec ses enfants.

Devenu très actif sur Twitter, il a récemment posté sur Twitter une série de 14 tweets :

https://twitter.com/wtgowers/status/1264672918983802882?s=20

It is often suggested that one should teach Euclidean geometry in order to introduce children to the notion of proof. Today my children learnt a huge amount about proof in a completely different way when, as an experiment, I tried the "miracle Sudoku" with them. 1/

— Timothy Gowers (@wtgowers) May 24, 2020

In case you've been hiding under a rock recently, this delightfully compelling video can tell you what I'm talking about. 2/

— Timothy Gowers (@wtgowers) May 24, 2020

I had watched the video, which gave me a small advantage : I couldn't remember where anything went, but I had some idea of the style of argument required. Also, @RichardGowers had told me there was another puzzle of a similar kind, so I thought ... 3/

— Timothy Gowers (@wtgowers) May 24, 2020

we could do the first one together and then the second one individually. It's all made much less painful by a website that enables you to solve it online, but also to put in small numbers (e.g. to indicate the options not ruled out for a given number), ... 4/

— Timothy Gowers (@wtgowers) May 24, 2020

or colours (to indicate whatever you want them to indicate).

What's great about this is that it allows you to develop arguments that would, if written down, be proofs that were several lines long, without having to hold a lot in your head. Things like, "If I put a 2 ... 5/

— Timothy Gowers (@wtgowers) May 24, 2020

there, then none of those squares can be a 2, which means that there must be a 2 in that row, but that means that there can't be a 2 there, so there must be one there. But that's impossible, so I can't have put in the first 2."

What's also great is that it turns ... 6/

— Timothy Gowers (@wtgowers) May 24, 2020

very naturally into a cooperative exercise — it's easy for everyone to spot little arguments that other people haven't yet noticed. So it really felt as though I was collaborating on roughly equal terms with my 12yo son and 9yo daughter. 7/

— Timothy Gowers (@wtgowers) May 24, 2020

The first puzzle can be found here. 8/

https://cracking-the-cryptic.web.app/sudoku/tjN9LtrrTL

— Timothy Gowers (@wtgowers) May 24, 2020

A sign of the success of this, other than the fact that we were all hooked on it for over an hour, is that my children were desperate to do the other one. So we all got set up. And then something else interesting happened. We all got pretty bogged down, in my case ... 9/

— Timothy Gowers (@wtgowers) May 24, 2020

partly because I misunderstood the "check" function. At some point I thought I might have made a mistake, so I pressed "check" and it told me that it didn't look right, so I deleted a lot, before realizing that it was merely telling me I hadn't finished. (But actually ... 10/

— Timothy Gowers (@wtgowers) May 24, 2020

I think I *had* made a mistake.)

The interesting thing wasn't that, however. It was that we decided it would be more fun to collaborate again, and, just as is so often the case in serious mathematical research, we were far more efficient as a team than as individuals. 11/

— Timothy Gowers (@wtgowers) May 24, 2020

So they got a remarkable glimpse of what it's actually like to do mathematical research, with the way it can grab you, its dead ends, the getting stuck and unstuck, how tiny progress can unlock the door to much larger progress, how proofs can split up into lemmas, how ... 12/

— Timothy Gowers (@wtgowers) May 24, 2020

you sometimes need to consider several cases, the use of proof by negation, how a problem that initially looks impossible can sometimes be cracked, and the efficiency gains that can come from joint work.

Did I mention that we probably spent about three hours on it, all told ? 13/

— Timothy Gowers (@wtgowers) May 24, 2020

I'd do the same again, and so would they.

Here's puzzle 2. 14/

https://t.co/Y0JHt1tOke

— Timothy Gowers (@wtgowers) May 24, 2020