A fly flying to $300$ points, but each time it increases its flying distance by $1.$

Question

There are 300 points around circle. A fly sits in one of the points. It starts moving clockwise. At first the fly moves to the next point. Then it skips one point. Then it flies skipping 2 points, then 3 points and so on. Will the fly eventually visit all the points?
I am not sure where to start. I have tried to shown it for a smaller case with the number of points being 6, but it doesn't seem to work. How can I generalize it?

Answer 1

If the starting point is point $0$, the the points are numbered clockwise, then the fly does not get to any of the points $2,5,8,11,$ etc. i.e. any point $n$ for which $n \equiv 2 (mod \ 3)$

This is because if you look at the numbers it does visit: $0,1,3,6,10,15,21,...$ you will find that they are $0,1,0,0,1,0,0,... (mod \ 3)$

For a rigorous proof:

Let's prove by induction that:

$$\sum_{n=0}^k n \equiv \pmod 3 \begin{cases} 0 & \text{if $k \equiv 0 \pmod 3$} \\ 1 & \text{if $k \equiv 1 \pmod 3$}\\ 0 & \text{if $k \equiv 2 \pmod 3$} \end{cases}$$

Base:

$$\sum_{n=0}^0 n = 0 \equiv 0 \pmod 3$$

$$\sum_{n=0}^1 n = 1 \equiv 1 \pmod 3$$

$$\sum_{n=0}^2 n = 3 \equiv 0 \pmod 3$$

Step:

$$\sum_{n=0}^{3(k+1)} n = \sum_{n=0}^{3k} n + (3k+1) + (3k+2) + (3k+3) = \sum_{n=0}^{3k} n + 9k + 6 \overset{Ind.Hyp.}{\equiv} 0 \pmod 3$$

$$\sum_{n=0}^{3(k+1)+1} n = \sum_{n=0}^{3k+1} n + (3k+2) + (3k+3) + (3k+4) = \sum_{n=0}^{3k+1} n + 9k + 9 \overset{Ind.Hyp.}{\equiv} 1 \pmod 3$$

$$\sum_{n=0}^{3(k+1)+2} n = \sum_{n=0}^{3k+2} n + (3k+3) + (3k+4) + (3k+5) = \sum_{n=0}^{3k+2} n + 9k + 12 \overset{Ind.Hyp.}{\equiv} 0 \pmod 3$$

Also, since $300$ is a multiple of $3$, these numbers won't change no matter how many times the fly goes around.

This is why the fly also didn't visit all points when you tried it for $n=6$, and in fact the fly won't even visit all points when there are just $3$ (try it!)