Question 90
Alice and Bob are playing hide and seek. Initially, Bob chooses a secret fixed point $B$ in the unit square. Then Alice chooses a sequence of points $P_0, P_1, \ldots, P_N$ in the plane. After choosing $P_k$ (but before choosing $P_{k+1}$) for $k \ge 1$, Bob tells "warmer" if $P_k$ is closer to $B$ than $P_{k-1}$, otherwise he says "colder". After Alice has chosen $P_N$ and heard Bob’s answer, Alice chooses a final point $A$. Alice wins if the distance $AB$ is at most $1/2020$, otherwise Bob wins. Show that if $N = 18$, Alice cannot guarantee a win.