Consider a unconventional billiard board in the shape of an equilateral triangle (depicted below). An incredibly small ball (size in picture is increased for the *sake of visibility* on your screen) is put in the A corner.

[![Ett javla fint biljardbord, eller hur?][1]][1]

> How many paths for the ball are there such that the ball bounces off the sides exactly 20160127 times, starting and ending in the A corner?

**Note**: the question is heavily inspired by a question on the [Project Euler](https://projecteuler.net/) website, which I enjoyed very much solving.

Hint:
>! Case: 11 bounces, 2 paths; Case: 10001 bounces, 800 paths; Case: 1000001 bounces, 80840 paths.

**Additional information**: There is also a [paper][2] on the matter if you are intrested to read more.


  [1]: https://i.sstatic.net/Jy0nY.png
  [2]: https://arxiv.org/PS_cache/math/pdf/0509/0509292v7.pdf