I’ve just been working with my 12-year-old daughter on Cantor’s diagonal argument, and countable and uncountable sets.

Why? Because the maths department at her school is outrageously good, and set her the task of researching a mathematician, and understanding some of the maths they did – the real thing.

So what else could we have done – thinking that we know our multiplication tables and fractions, but aren’t yet completely confident with equations which have letters for unknown numbers?

I did think of proving that there are infinitely many primes – we can follow an argument – other suggestions welcome.

And incidentally, tell your local high school to do this …

**Answer**

Six people at a dinner party is sufficient to ensure that there are either three mutual strangers or three mutual acquaintances. In fact, six is the *smallest* number that ensures this phenomenon. This is the diagonal Ramsey number $R(3,3)$, and the proof can be demonstrated with a couple pictures and just a dash of the pigeonhole principle. There are lots of directions she could go after understanding $R(3,3)$ (though most of it is not due to Ramsey).

**Attribution***Source : Link , Question Author : Community , Answer Author :
Austin Mohr
*