I am also interested in the popularisation of Mathematics and enjoy showing that simple ideas when correctly formulated may lead to innovative solutions which solve complex problems.
In this context, I had the opportunity to work on a popularisation project with Thibault Defourneau and Valentin Montmirail which is described below.
The Lights Out is a game consisting of a 5 by 5 grid of lights. Given a pattern of switched lights at the beginning, the goal of the game is to switch all the lights off with the following rule: pressing a light toggles it and the adjacent ones. While being simple at first sight, it turns out to be hard to solve in practice without an efficient strategy.
Linear algebra offers actually a very elegant and intuitive framework to model the game. In particular abstract notions, such as the null space of a matrix, can be easily interpreted in terms of the game. An intuitive algorithm coming from this modelling permits to compute all the solutions of the game and to propose the best one, namely the solution with the fewest pressed buttons.
We have created workshops and talks based on the Lights Out for high-school students. The goal was to illustrate through a game how mathematics can be smartly used for the understanding and the resolution of complex problems. We also wrote an article (in French) detailing our workshop; see Publications.
We have implemented an algorithm in Java together with an interface permitting to play and to solve the Lights Out. It is available at https://github.com/Mystelven/LightsOuts-Math.