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 4 of 9: The hunter cannot identify the side

Previous step Next step

H^*\in r,\qquad |HH^*|=200,\qquad |H^*R|=200-d

Detailed analysis

Let $H^*$ be the point on $r$ reached by going $200$ units from $H$ through $R$ . Every point $Q$ reachable by the hunter after $200$ unit moves has projection on $r$ no farther in the direction from $H$ through $R$ than $H^*$ . Since $d\ge1$ while the small offset of $Y_i$ from the point $200$ units beyond $R$ is less than $1/400$ , both destinations lie beyond $H^*$ . The reports are identical for the two routes, so the hunter follows the same path in both cases. Whichever side of $r$ the final $Q$ lies on, the destination on the opposite side is at least as far from $Q$ as $H^*Y_i$ . If $Q$ lies on $r$ , either destination has that lower bound. Thus the rabbit can choose a route with final distance at least $y:=H^*Y_1=H^*Y_2$ .