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 8 of 9: Iterate until the distance exceeds 100

Previous step Next step

m=20000,\qquad d^2>1+\frac m2=10001

Detailed analysis

Start with $d_1\ge1$ . Apply the block strategy whenever the current distance is at most $100$ . If the distance never exceeded $100$ through $m=20000$ blocks, the preceding step would give $d^2>1+m/2=10001$ , a contradiction. Hence the rabbit reaches distance greater than $100$ during these blocks.