  We investigate the maximum number of embedded patterns in the two-dimensional
Hopfield model. The grand state energies of two specific network states,
namely, the energies of the pure-ferromagnetic state and the state of specific
one stored pattern are calculated exactly in terms of the correlation function
of the ferromagnetic Ising model. We also investigate the energy landscape
around them by computer simulations. Taking into account the qualitative
features of the phase diagrams obtained by Nishimori, Whyte and Sherrington
[Phys. Rev. E {\bf 51}, 3628 (1995)], we conclude that the network cannot
retrieve more than three patterns.
