Question 47
A flea is initially at the point $(0,0)$ of the Cartesian plane. It then makes $n$ jumps. Each jump is made in one of the four cardinal directions, chosen freely. The first jump has length $1$, the second length $2$, the third length $4$, and so on, up to the $n$-th jump, which has length $2^{n-1}$. Prove that, if the final position of the flea is known, then its position after each of the $n$ jumps can be determined uniquely.