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.

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3 Responses to Predicting Pi

  1. Nick Johnson says:

    Well, the obvious thing one can learn given just |(10^n)r| is the first n digits of the decimal expansion of r. I presume there’s supposed to be more to learn?

  2. Brent says:

    Well done! But you can actually learn the decimal expansion of r faster than that.

  3. Pingback: Predicting pi: pretty graphs and convergents « The Math Less Traveled

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