From A Golden March from the futility closet.
Draw a circle whose circumference is the golden mean. Choose a point and label it 1, then move clockwise around the circle in steps of arc length 1, labeling the points 2, 3, and so on. At each step, the difference between each pair of adjacent numbers on the circle is a Fibonacci number.
When would the first collision occur? I think this problem could be solved by some modular arithmetic; however, I only know how to use integers, not Fibonacci numbers when doing modular arithmetic.
There will never be a collision because the golden ratio is irrational. If there were a collision, the first would be some point $m$ landing on $1$. The distance traveled would be $m-1$. That would have to be a multiple of $\phi$, which is impossible.