Colliders use final focus systems to reduce the transverse beam sizes at the interaction point in order to increase collision event rates. The maximum focal strength (gradient) of the quadrupoles, and the maximum beam size in them, together limit the beam size reduction that is possible. The goal of a final focus system design is to find the best compromise between quadrupole aperture and quadrupole gradient, for the magnet technology that is used. This paper develops a design method that identifies the intrinsic limitations of a final focus system, validates the results of the method against realistic designs, and reports its application to the upgrade of the Large Hadron Collider final focus. © 2008 The American Physical Society.