A geometric perspective on counting non-negative integer solutions and combinatorial identities

We consider the effect of constraints on the number of non-negative 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.