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...