We consider the effect of constraints on the number of nonnegative integer solutions of $x + y + z = n$, relating the number of solutions to linear combinations of triangular numbers. Our approach is geometric and may be viewed as an introduction to proofs without words. We use this geometrical perspective to prove identities by counting the number of solutions in two different ways, thereby combining combinatorial proofs and proofs without words.

This will be an interactive talk where those in attendance will get to use triangular graph paper to construct proofs of some of the results.

This talk is based on a paper of the same name that is co-authored with Matt Haines and Ryan Huddy.