Locks and keys: How fast can you open several locks with too many keys?

This short note is the result of a French " Hippocampe internship " 2 that aims at introducing the world of research to young undergraduate French students. The problem studied is the following: imagine yourself locked in a cage barred with n different locks. You are given a keyring with N ≥ n keys containing the n keys that open the locks. In average, how many trials are required to open all locks and get out? The article studies 3 different strategies and compare them. Implementation of the strategies are also proposed as illustrations of the theoretical results. The precise statement of the problem and the various assumptions are the following: Statement of the problem: We need to open n locks with a keyring containing N ≥ n keys. We are interested in determining the average and the standard deviation of the number of trials to open all locks. The special case when the number of keys equals the number of locks (n = N) is included. Assumptions: The following assumptions about locks and keys are made: • All locks and keys are assumed to be indistinguishable. • All locks are different and require a specific key to be opened. • A lock can always be opened, i.e. the key required to open it belongs to the keyring. • A key cannot open more than one lock. Consequently, the keyring contains the n different keys corresponding to the locks as well as additional keys that do not open anything when N > n. The keys are randomly placed in the keyring. We denote by T n,N the number of trials to open all locks. We are interested in gathering as much information as possible on T n,N especially E(T n,N) and Var(T n,N). 2 Strategy 1: The totally random strategy The first strategy one can think of is to select randomly one of the remaining keys and try it randomly into one the remaining locks to open. In particular we do not keep track about the previous tries except for the fact that we discard locks that are already opened and their corresponding 2 " Hippocampe " (sea-horse or hippocampus in French) internships are three days sessions during which small groups of students specialized in mathematics and physics are presented with a problem that they have to solve under the supervision of an academic professor.