What’s the 99th digit to the right of the decimal point in the decimal expansion of ?
Of course, it’s simple enough to use a computer to find the answer; any language or software system that can compute with arbitrary-precision real numbers can find the correct answer in a fraction of a second. But that’s obviously not the point! Can we use logical reasoning to deduce or prove the correct answer, without doing lots of computation? Even if we find the answer computationally, can we explain why it is the right answer? Solving this puzzle took me down a fascinating rabbit hole that I’d like to share with you over the next post or three or eight.
For the moment I’ll just let you think about the puzzle. Although using a computer to simply compute the answer is cheating, I do encourage the use of a computer or calculator to try smaller examples and look for patterns. It is not too hard to see a pattern and conjecture the right answer; the interesting part, of course, is to figure out why this pattern happens, and to prove that it continues.