Nuclear fission of heavy (actinide) nuclei results predominantly in asymmetric mass splits(1). Without quantum shell effects, which can give extra binding energy to their mass-asymmetric shapes, these nuclei would fission symmetrically. The strongest shell effects appear in spherical nuclei, such as the spherical 'doubly magic' (that is, both its atomic and neutron numbers are 'magic' numbers) nucleus Sn-132, which contains 50 protons and 82 neutrons. However, a systematic study of fission(2) has shown that heavy fission fragments have atomic numbers distributed around Z = 52 to Z = 56, indicating that the strong shell effects in Sn-132 are not the only factor affecting actinide fission. Reconciling the strong spherical shell effects at Z = 50 with the different Z values of fission fragments observed in nature has been a longstanding puzzle(3). Here we show that the final mass asymmetry of the fragments is also determined by the extra stability provided by octupole (pear-shaped) deformations, which have been recently confirmed experimentally around Ba-144 (Z = 56)(4,5), one of very few nuclei with shell-stabilized octupole deformation(6). Using a quantum many-body model of superfluid fission dynamics(7), we find that heavy fission fragments are produced predominantly with 52 to 56 protons, which is associated with substantial octupole deformation acquired on the way to fission. These octupole shapes, which favour asymmetric fission, are induced by deformed shells at Z = 52 and Z = 56. By contrast, spherical magic nuclei are very resistant to octupole deformation, which hinders their production as fission fragments. These findings may explain surprising observations of asymmetric fission in nuclei lighter than lead(8).