Question

A hunter and a rabbit play a game in the euclidean plane. The rabbit starts at the origin (0,0). Every second, the hunter chooses a direction and moves the rabbit an integer length in that direction, and then the rabbit rotates around its previous position however much it chooses to. The winner of the game is the one who first moves the rabbit within k distance of the origin once 2 seconds have passed. For what k does the hunter win?

Answer 1

Trivial. For \(k\geq\frac12\) the hunter picks \(k+10\), then \(1\), then moves the rabbit to a place within \(\frac12\) from the origin. Otherwise, the rabbit can always move to a place that is integer+\(\frac12\) away from the origin after every move.

Answer 2

Ok, but can the rabbit win.

Answer 3

The hunter can always prevent the rabbit from winning.

Answer 4

ok actually it should be \(\lfloor k+10\rfloor\) for it to be an integer