IR-TST 2007 Problem 12

Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$ . We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$ . After each contact of light ray with a horizontal side, we put $0$ , and after each contact with a vertical side, we put $1$ . For each $n\geq 1$ , let $B_{n}$ be set of all blocks of length $n$ , in this sequence. a) Prove that $B_{n}$ does not depend on location of $P$ . b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$ .