Question 81
A hunter and an invisible rabbit play a game on the integer lattice $\mathbb{Z}^2 := \{(x, y) : x, y \in \mathbb{Z}\}$ ($\mathbb{Z}$ is the set of all integers). Before the game starts, the hunter colors every point of $\mathbb{Z}^2$ with exactly one of finitely many colors; then the rabbit, having seen the hunter's coloring, secretly chooses a point as its starting point. In each subsequent minute, the rabbit first tells the hunter the color of the point it is on, and then secretly chooses one of the four neighboring points (up, down, left, right) that it has never visited before and moves to it.
The hunter wins if, within finite time, the rabbit can no longer move, or the hunter can determine with certainty the starting point chosen by the rabbit at the beginning of the game. Does the hunter have a strategy that guarantees a win in finite time?
The hunter wins if, within finite time, the rabbit can no longer move, or the hunter can determine with certainty the starting point chosen by the rabbit at the beginning of the game. Does the hunter have a strategy that guarantees a win in finite time?