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

TL;DR

This paper investigates the average number of trials needed to open multiple locks with a larger set of keys, comparing three strategies through theoretical analysis and implementation examples.

Contribution

It introduces and compares three strategies for unlocking multiple locks with more keys than locks, providing both theoretical insights and practical implementations.

Findings

Strategy efficiency varies with lock and key configurations

Theoretical analysis of expected trials for each strategy

Practical implementation examples illustrate the strategies

Abstract

This short note is the result of a French "Hippocampe internship" 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 $N \geq n$ 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.