Problem 6

A robot sits at zero on a number line. Each second the robot chooses a direction, left or right, and at the $s$-th second the robot moves $2^{s-1}$ units in that direction on the number line.

For which integers $n$ are there infinitely many routes the robot can take to reach $n$?

(You may use the fact that every positive integer can be written as a sum of different powers of 2. For example, $19 = 2^0 + 2^1 + 2^4$)