I will discuss how the numerical analysis of the game p-Laplacian, developed with Falcone, Finzi Vita and Giorgi has led to new insights into the behavior of that operator. Particular results obtained with collaborators including a mean value property for p-harmonic functions, a concrete realization of Wolff’s example used in the proof that p-harmonic measure is not additive, and a discussion of the local behavior of the Poisson problem for the game p-Laplacian in a corner.