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 2 of 9: Set up one amplification block

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1\le d\le100,\qquad d=|HR|

Detailed analysis

Consider a block in which the hunter is at $H$ , the rabbit is at $R$ , and $d=|HR|$ satisfies $1\le d\le100$ . Let $r$ be the line $HR$ . We may even give the hunter the exact current position $R$ at the beginning of the block; a strategy that defeats an informed hunter also defeats the actual hunter, who has no more information.