Question 120
Let $k$ be a nonnegative integer. There is a pile of $2^{2^k}$ coins, exactly one of which is counterfeit. We have an unlimited number of police dogs, exactly one of which is sick, but we do not know which one. A test of the coins consists of the following three steps: choose some of the coins; pick a police dog; let this dog sniff the chosen coins simultaneously. If the counterfeit coin is among the chosen coins, a healthy dog will certainly bark after sniffing; otherwise it certainly will not bark. Whether the sick dog barks or not is random. Design a procedure using at most $2^k + k + 2$ tests that identifies the counterfeit coin, and prove that the procedure works.