Question 125
A hunter and an invisible rabbit play on an infinite square-mesh grid, that is, one in which each cell has four neighbours: to the left, to the right, above and below. First, the hunter colours every cell of the grid, but may use only finitely many colours. The rabbit then chooses a cell of the grid, which will be its starting point. It then starts moving: every minute, it tells the hunter the colour of the cell it is on, and then moves to one of the four adjacent cells. Of course, since it is invisible, the only information the hunter has access to is the colours the rabbit announces each minute just before moving.
The hunter wins if, after a finite amount of time,
$\triangleright$ he can identify the cell the rabbit initially chose, or
$\triangleright$ the rabbit returns to a cell where it has already been before.
Does the hunter have a winning strategy?
The hunter wins if, after a finite amount of time,
$\triangleright$ he can identify the cell the rabbit initially chose, or
$\triangleright$ the rabbit returns to a cell where it has already been before.
Does the hunter have a winning strategy?