Number of ways a laser will bounce on a circular reflective material in 4 times Is there a formula for getting the number of ways a laser will bounce on a circular reflective material in 4 times, returning to its original position?
So the correct answer is 4, but is there a formula?
Here's a picture showing the 4 ways:


The points where the beam first hits the circle are at angles $\frac{2k\pi}{n+1}$ from the initial point.  If $\gcd(k,n+1)\gt1$, then the beam will return to the start point after $\frac{n+1}{\gcd(k,n+1)}-1$ bounces, so we only want to count the number of $1\le k\le n$ so that $\gcd(k,n+1)=1$. This is $\phi(n+1)$ where $\phi$ is the Euler Totient Function.
Since $\phi(4+1)=4$, we get $4$ ways.
