[ Game Theory_(miscellaneous) ] [ Parity and Imparity ]

On a number line, a point grasshopper sits at point $P$. Points 0 and 1 are traps. On each move, we call out any positive number, after which the grasshopper jumps either to the left or to the right (at its choice) by a distance equal to this number. For which $P$ can we call out numbers in such a way as to guarantee driving the grasshopper into one of the traps? (We always see where the grasshopper is sitting.)
Let $P$ be a number of the form given in the answer, and the fraction is irreducible (i.e., $a$ is odd). Then the number $1-P$ also has the same form (with the same denominator). We will call the smallest of the numbers $P$ and $1-P$. If the grasshopper jumps and does not fall into a trap, it will remain between the traps, and the distance to one of the traps will double. As a result, the denominator of the grasshopper's coordinate will be halved. After $k$ jumps, the coordinate will become an integer, which means falling into a trap.

Let $P$ be a non-binary rational number. We will declare all binary rational numbers as traps. We will show that if the grasshopper is not in a trap, it can always jump to a non-trap. Note that the midpoint of two traps is again a trap. Let the number $d$ be named. Since $P = \frac{1}{2}((P+d) + (P-d))$, and $P$ is not a trap, then at least one of the points $P+d$ and $P-d$ is also not a trap. We will jump there.

Let $P$ be a binary rational number not lying in the interval [0, 1]. Then the grasshopper can move away from this interval with each jump.

## Answer

For $P = a \cdot 2^{-k}$, where $a$ and $k$ are natural numbers, $0 < a < 2^k$.