The energetic cost of introducing square faces to fullerenes with adjacent pentagons is investigated theoretically. Relative energies of all 1735 hypothetical C-40 cages that can be assembled from square, pentagonal, and hexagonal faces are calculated within two independent semiempirical models. All isomers are found to lie in local minima on the potential surface. The QCFF/PI (quantum consistent force field/pi) and DFTB (density functional tight binding) approaches agree in predicting that no cage with one or more squares is of lower energy than the best classical C-40 fullerene but that many such cages are more stable than many C-40 fullerenes. Energy penalties of 160-200 kJ mol(-1) per square are suggested by the DFTB calculations, and penalties of about twice this size by the QCFF/PI model. The energy variation across the range of fullerenes and pseudofullerenes is steric in origin and correlates well with the normalized second moment of the hexagon neighbor signature : aggregation of hexagons in one part of the cage surface is incompatible with even distribution of curvature and implies crowding of defects elsewhere. QCFF/PI calculations for selected isomers of C-62 to C-68 also show that though cages with squares may again be more stable than some fullerenes, they are all bettered in energy by the best classical fullerene at each nuclearity.