The Five-Button Door Lock – Experiment and Discovery in Mathematics

Experimentation, gathering data, and computation are an integral part of the process whereby mathematicians discover theorems and their proofs. Mathematics may be a science of exact proof, but the data and the process used in the discovery of certain proofs adds an experimental component. And, computers can help by creating data that reveal patterns. This productive methodology is one worth emphasizing at every level of mathematical instruction, but especially in primary, middle and high school curricula, where the emphasis on algorithms, methods, technique, and vocabulary leaves experiment and discovery as an afterthought. Inspired by a question posted by Marc Dostie on the Rediscovering Mathematics Facebook page 1, we consider a number of problems related to the following door lock. We offer this exploration as an example of how to incorporate discovery, experiment, and calculation into mathematics and pedagogy. An early version of this work was presented in an invited lecture at the Mathematics and Computer Science Colloquium at Providence College in 2015.2 Programmable door locks such as the one in the figure, commonly found in schools, hospitals, and office buildings, provide a flexible way to maintain selective security and entry to different rooms and areas of buildings. To enter a room, a person presses certain buttons, then enter, and turns the handle. In this particular model of the lock, once a button is pressed, it cannot be pressed again, however, buttons can be pressed simultaneously, and the order in which the presses occur is significant.