When you meet someone and explain that you are a mathematician, the most common reactions are:

- Ah, I never understood math!

- Is there something more to discover in math?

- Math is very difficult …

Or, in recent years:

- There’s a mathematician you’ve seen all over the media for some time, do you know him?

But the other night at a birthday party, I was asked an unexpected question:

- Can we have fun doing math?

- Yes, of course, I replied.

But I had to, as a self-respecting mathematician, try to demonstrate it.

And you, how would you go about demonstrating that?

I decided to try to make my interlocutors feel the mathematical pleasure with a funny problem. I explained the problem to them on a paper towel, between a bowl of guacamole and another of crisps. So, to keep that messy party night style a bit, I’m also accompanying this post with sketches made on a napkin.

Imagine a room covered with a 5 × 5 tile, as in the following figure:

In each of the tiles — which we imagine large enough — there is a person, as in this drawing:

At a signal, each person must move to one of the neighboring tiles. The following drawing shows the possible choices for three different people

Now it is a question of proving that after that we end up with at least one tile containing several people. Or, which amounts to the same thing, with at least one empty tile.

-How do you expect me to demonstrate this, I haven’t done maths since high school.

-Well, I said, first of all, you have to forget what you’ve done or not done, and take to the game like a child. Handle the problem all over the place, until an idea pops up. For example, why not take a simpler case, with a 2 × 2 tiling? And there, look, people can move around and then find themselves alone again in their tiles:

- So it’s probably important that in our problem we have an odd number of tiles along each side!

This was the sentence that sprang from one of my interlocutors, to which I replied:

- You just got a really good idea. But how to make it grow? How to use the fact that 5 is odd?

- … (concentrated silence).

I resumed:

- I will now present to you a fundamental ingredient of mathematical pleasure: the dream. Once there, you have to start dreaming, in order to discover analogies with other objects or other thought experiences. If you look at the drawings we made, what do they remind you of?

- … to a chessboard?

- Perfect, excellent answer! And now I’m going to tell you one of the reasons why analogies are important in math. This is because they are not perfect! We identify an analogy by the dream — I don’t know if the term is so well chosen, but I want to indicate by this that it is a process which is not conscious — but once the analogy is discovered, we can call the conscience to the rescue. More precisely, one can wonder what differentiates the analogous cases. What is a chessboard here that cannot be found in our tiles?

- … its boxes are colored in black and white?

-Magnificent ! I cried, well, there you are very close to the solution of the problem. Let’s color our tiles like the squares on a chessboard … and think about our question again …

There were a few seconds of concentration, then one of the interlocutors said with the joy of understanding in his eyes:

- There are not the same number of black boxes and white boxes! I don’t know which ones are more numerous, you have to count, but it shows that after the movement of people there will necessarily be an empty square.

- You understand, bravo! But you didn’t say the essential fact that makes your argument work …

And my interlocutor replied, after a few more seconds of reflection:

- I see, a person located on a white tile passes over a black tile and vice versa …

- So there it is, I congratulate you! And there you have it, you have just seen some ingredients of mathematical pleasure: dreaming in order to discover analogies. Then work more consciously to detect the differences between analogous objects. Thanks to these differences, dress the initial context with a fabric that is suited to the problem. Finally, after a more or less long journey in dreaming and conscious work, feel the flash of understanding.

- Ah, why wasn’t math explained to me like that in high school ?! I was disgusted by the calculations of integrals, I never understood what it was for …

This last remark made me feel like I had succeeded in my demonstration. But that also posed a new problem for me: how to demonstrate mathematical pleasure by calculating integrals?

Do you have any idea to solve it?




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