Inspirational talk for final year high school pupils
There is a neat way to introduce zero-knowledge proofs to students, which I think is originally due to Oded Goldreich (please correct me if I'm wrong). 

You have a red ball and a green ball, which poor colorblind Charlie believes are the same color. You want to convince Charlie that you can tell the difference between the red ball and green ball, and you want to do this in a way that Charlie does not learn which is red and which is green. (You want to prove something is true, in such a way that no one else can turn around and claim a proof of that something as their own.) How can you do this? Or is it impossible? 

One protocol is the following. Charlie puts a ball in each hand, then chooses to either switch the two balls behind him, or not. Then he presents the two balls again. If you can always detect whether he switched the two balls or not, then Charlie is increasingly convinced that you can tell the difference between them. If Charlie does this shuffle at random and you really can't tell the difference between the colors, then you will only guess correctly with probability $1/2$. After $k$ trials, Charlie should be convinced that you can tell the difference with probability at least $1-1/2^k$.

Now while Charlie becomes increasingly convinced that you can tell the difference, he frustratingly never learns which ball is red and which one is green.