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Playing hide and seek on the unit square

Source: Baltic Way 2020, Problem 10

November 14, 2020
combinatoricscombinatorics proposed

Problem Statement

Alice and Bob are playing hide and seek. Initially, Bob chooses a secret fixed point BB in the unit square. Then Alice chooses a sequence of points P0,P1,…,PNP_0, P_1, \ldots, P_N in the plane. After choosing PkP_k (but before choosing Pk+1P_{k+1}) for k≥1k \geq 1, Bob tells "warmer'' if PkP_k is closer to BB than Pk−1P_{k-1}, otherwise he says "colder''. After Alice has chosen PNP_N and heard Bob's answer, Alice chooses a final point AA. Alice wins if the distance ABAB is at most 12020\frac 1 {2020}, otherwise Bob wins. Show that if N=18N=18, Alice cannot guarantee a win.
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