There's an advert on the underground at the moment that says "Listerine users are more likely to talk to a stranger on the tube." I have a couple of issues with this advert: one ethical and one mathematical.

First the ethical. It sounds like they're suggesting that that's a good thing. The guidance on talking to strangers on public transport is pretty clear, particularly if you're a man about to approach a woman: just don't. If she'd wanted to talk to you, she would be by now.

Okay. The mathematical issue is a little more subtle.

I'm usually a big fan of Numberphile but their most recent video seems to me to, at best, miss the point, and at worst, trade on dishonesty. If you haven't seen it, it's titled "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12". You should definitely watch it, because it's really interesting. What's the problem?

The thing is, there are lots of counterintuitive results in maths and physics. There are lots of moments of wonder that our students should experience and enjoy and understand. This is potentially one of them. But the wow in this video pivots on a deceipt that is never addressed: that there is a consistent meaning of sum for a divergent sequence which corresponds either to our intuition or to our regular definition of adding.

After discussing the possibility of finding any piece of information encoded somewhere in the digits of $\pi$, one of my year 12 students posed the following problem:

''Assuming the digits of $\pi$ form an infinitely long, essentially random sequence with no recurrence, is there definitely at least an initial repeat, in the sense that the first $n$ digits could repeat once before the rest continue 'randomly'?''

In other words, and more generally, is any irrational certainly in one of the forms $aabcdef...\,$, $ababcdef...\,$, $abcabcdef...\,$ and so on? (If you're worried about independence in subsequent digits then we can easily reformulate the problem to consider an infinite sequence of die rolls or coin flips or whatever.)

My first instinct was 'yes', perhaps because I'm pretty used to saying anything is possible given infinitely long. However, the anchoring at the start makes this a rather different and interesting problem. The probability of such a repeat seems to shrink much faster than the linear rate at which the digits accumulate. So, no then?

Well, it seems to be a bit more complicated than that...