  We study the variation of the hadron masses in the presence of external
magnetic fields of strength of the order of the masses themselves. We identify
the main factors affecting the lattice simulation results: - the boundary
discontinuities for $eB << 2\pi / L^2 a^2$. - the SU(6) choice of the hadron
wave-function. We confirm qualitatively the earlier theoretical ansatz on the
linear behaviour of the masses with the magnetic field and, as a by-product, we
improve the lattice measurements of the nucleon magnetic moments. However our
systematic and statistical errors preclude us from measuring the theoretically
predicted field strength at which the proton becomes heavier than the neutron.
