## Predicting Pi

Inspired by a recent post over at Foxmaths!, here’s an interesting challenge question for you to think about:

Suppose I give you the sequence of integers $\lfloor \pi \rfloor, \lfloor 2 \pi \rfloor, \lfloor 3 \pi \rfloor$, and so on, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to x—in other words, it means to round down. So the first number in the sequence would be $\lfloor \pi \rfloor = 3$, the next number would be $\lfloor 2 \pi \rfloor = 6$, and so on. Given this sequence, what can you learn about $\pi$ (assuming that you didn’t already know anything about it)?

A more general question: given the sequence of integers $\lfloor k r \rfloor$ for k = 1,2,3…, what can you learn about r?

The answer has many interesting connections to the theory of irrational numbers and continued fractions.