Given a grid strip (one cell wide), infinite in both directions. Two cells of the strip are traps, with $-N$ cells between them, one of which is occupied by a grasshopper. On each move, we call out a natural number, after which the grasshopper jumps that number of cells to the left or right (at its choice). For which $N$ can we call out numbers such that we can guarantee to drive the grasshopper into one of the traps, no matter where it initially is between the traps and how it chooses the directions of its jumps? (We always see where the grasshopper is sitting.)
Number the cells in order so that the traps get the numbers 0 and $N+1$.

Let $N+1=2^{k}$. We will call the distance $d$ to the nearest trap each time. The highest power of two that divides $d$ is less than $k$. If the grasshopper jumps in the opposite direction and does not fall into a trap, it will remain between the traps, and the distance to one of the traps will become $2 d$, and to the other $2^{k}-2 d$.

Both of these distances are divisible by a higher power of two. Therefore, sooner or later, the distances will be divided by $2^{k}$, which means the grasshopper will fall into a trap.

Let $N+1=2^{k} t$, where $t$ is odd and greater than 1. Declare all cells with numbers divisible by $t$ as traps. Suppose the grasshopper is not in a trap. Note that it is not standing exactly halfway between two traps, since all numbers of the form $1 / 2(a t+b t)$ are either non-integers or multiples of $t$. Therefore, for any number called, the grasshopper has a jump that does not lead to a trap.

## Answer

For $N=2^{k}-1$, where $k$ is a natural number.