Problem 3

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point,

A_0

, and the hunter's starting point,

B_0

, are the same. After

n-1

rounds of the game, the rabbit is at

A_{n-1}

and the hunter is at

B_{n-1}

. In the

n

th round, three things occur in order: (i) the rabbit moves invisibly to a point

A_n

such that the distance between

A_{n-1}

and

A_n

is exactly

1

; (ii) a tracking device reports a point

P_n

to the hunter, with the only guarantee that the distance between

P_n

and

A_n

is at most

1

; (iii) the hunter moves visibly to a point

B_n

such that the distance between

B_{n-1}

and

B_n

is exactly

1

. Is it always possible, no matter how the rabbit moves and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after

10^9

rounds she can ensure that the distance between her and the rabbit is at most

100

Step 1 of 9: The rabbit can create an initial separation

Previous step Next step

P_1=A_0,\qquad d_1=|A_1B_1|\ge 1

Detailed analysis

For the first report choose $P_1=A_0$ , which is legal for every unit move of the rabbit. The hunter's response $B_1$ to this report is therefore fixed by her strategy and does not depend on the rabbit's direction. The rabbit can plan its unit move $A_1$ opposite to that predetermined $B_1$ ; consequently $d_1=|A_1B_1|\ge1$ . This only uses the hunter's strategy and is a valid first-round choice made before the report.