Purely combinatorial proofs are given for the sum of squares formula, 1[superscript 2] + 2[superscript 2] + ... + n[superscript 2] = n(n + 1) (2n + 1) / 6, and the sum of sums of squares formula, 1[superscript 2] + (1[superscript 2] + 2[superscript 2]) + ... + (1[superscript 2] + 2[superscript 2] + ... + n[superscript 2]) = n(n + 1)[superscript 2] (n + 2) / 12. More precisely, the following algebraic equivalents are derived, [image omitted]. The proofs obtained literally count squares, namely lattice squares whose vertices are in an "n" x "n" grid. For the first formula, only lattice squares aligned to the grid are counted; for the second formula, aligned "and" tilted squares are counted.