10 Think Outside the Box

We found that a very similar problem could be easily solved and that it agrees with the given problem to within . Since the answer is about , this is enough. The variation is to consider the region with no right boundary and ask for the probability that a particle hits the left before hitting the top or bottom. This turns out to have the simple and exact answer: . Doubling this, and doing some very rough estimation on the discrepancy between our variation and the given problem, leads to the slightly better formula , which gives the answer to the problem posed to 14 digits. The answer to 100 digits is:
0.00000003837587979251226103407133186204839100793005594072509569030022799173436606852743276500842845647269910.

So, given the mathematical work we did, one can easily get 14 digits as follows.

Or 13 digits from:

Because the argument is small, one gets the same digits even if the arctangent is ignored. We were lucky, since our method would fail if the rectangle was closer to a square!

Bornemann showed that the exact answer to the given question is , which is very similar to the steady-state series in #8. One term of this series gives 14 digits. Moreover, Bornemann observed that this series equals (the first term of which is our arctan approximation) and it is well known (Borwein & Borwein, Pi and the AGM, §3.2, exer. 12) that these series equal , where is the modular elliptic function. This formula is fast, taking only a small fraction of a second to get 500 digits.

Created by Mathematica  (June 27, 2004)