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 3 of 9: Two symmetric hidden routes

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|RY_1|=|RY_2|=200,\qquad \operatorname{dist}(Y_i,r)=1,\qquad |Y_1Y_2|=2

Detailed analysis

Choose points $Y_1$ and $Y_2$ on opposite sides of $r$ , each at distance $1$ from $r$ and distance $200$ from $R$ . They are symmetric about $r$ , so $|Y_1Y_2|=2$ . The rabbit chooses either endpoint and follows the straight segment from $R$ to it in $200$ unit hops. Every intermediate point is at distance at most $1$ from $r$ , so the device may report its orthogonal projection onto $r$ . The reports are then identical for the two possible routes and are always legal.