As a late adapter to smart phones, I just recently started idling away some time playing the popular app Flow Free (Big Duck Games). The goal is to connect pairs of dots of the same color with grid paths that do not intersect and cover the entire grid. It makes me think of the Gessel-Viennot lemma about a determinant counting nonintersecting sets of paths (although there often only moves right and up are allowed).
One could make a computer display where the pairs of dots have a unique set of nonintersecting paths. An initial step could lead visitors through the number of paths between two points being counted by binomial coefficients. Then finding a set of nonintersecting paths is similar to the game (same sort of touchscreen interface), with the bonus that your work shows that a particular determinant is 1 (without all the arithmetic and plus / minus signs).
The same interface could have an exploration of paths strictly below the diagonal that lead to Catalan numbers, which connects to a whole host of visually engaging things such as triangulating regular polygons and making "penny piles" (Richard Stanley is up to 202 things counted by these numbers -- that could be a whole special exhibit).