# Problem 202: Laserbeam

![graphic](img202.gif)

Three mirrors are arranged in the shape of an equilateral triangle, with
their reflective surfaces pointing inwards. There is an infinitesimal
gap at each vertex of the triangle through which a laser beam may pass.
Label the vertices A, B and C. There are 2 ways in which a laser beam
may enter vertex C, bounce off 11 surfaces, then exit through the same
vertex: one way is shown below; the other is the reverse of that. There
are 80840 ways in which a laser beam may enter vertex C, bounce off
1000001 surfaces, then exit through the same vertex. In how many ways
can a laser beam enter at vertex C, bounce off 12017639147 surfaces,
then exit through the same vertex?