Unlocking Your Bike the Easy Way

Abstract: Combination locks are widely used to secure bicycles. We consider a combination lock consisting of adjacent rotating dials with the first nonnegative integers printed on each of them. Assuming that we know the correct combination and we start from an incorrect combination, what is the minimal number of steps to arrive at the correct combination if in each step we are allowed to turn an arbitrary number of adjacent dials once in a common direction? We answer this question using elementary methods and show how this is related to a variation of (multivariate) functions.