Summary: We consider two non-overlapping congruent squares \(q_1\), \(q_2\) and the homothetic congruent squares \(q^k_1\), \(q^k_2\) obtained from two similitudes centered at the centers of the squares. We study the supremum of the ratios of these similitudes for which \(q^k_1\), \(q^k_2\) are non-overlapping. This yields a function \(\psi= \psi(q_1, q_2)\) for which the squares \(q^\psi_1\), \(q^\psi_2\) are non-overlapping although their boundaries intersect. When the squares \(q_1\) and \(q_2\) are not parallel, we give a 8-step construction using straight edge and compass of the intersection \(q^\psi_1\cap q^\psi_2\) and we obtain two formulas for \(\psi\). We also give an angular characterization of a vertex which belongs to \(q^\psi_1\cap q^\psi_2\).