Just found this charming student video proof without words of the value of two Ramsey numbers: R(3,3)=6 and R(3,3,3)=17. Probably hard to follow if you don't already know the proofs anyway, but for those who do, it's very relaxing to watch! www.youtube.com/watc…

Another surprising open problem: Is the sum of 1/(n!-1) for n >1 (so 1+1/5+1/23+...) irrational? The sum of 1/n! (1 + 1/2 + 1/6 + 1/24 +...) is e = 2.71728... which Euler showed almost 300 years ago is irrational. But change the series just by -1 and it's a mystery.

Let x be a real number such that 2^x and 3^x are both integers. Must x be an integer? Surely yes, but this is not known! (It follows from Schanuel's conjecture, and the weaker four exponentials conjecture, but both of these are also very open.)

This is a famous picture of Erdős and a young Terence Tao in 1985 discussing mathematics. I recently learnt what they were talking about.