Solution
= Solution
For $J>0$ each bond minimizes its <energy> when its two <Ising spins> are opposite, giving bond <energy> $-J$. An even ring allows every bond to do this simultaneously, including the closing bond required by the <periodic boundary conditions>. Its two <ground states> are
$$
\boxed{S_j=\sigma(-1)^{j-1},\quad\sigma=\pm1;
\qquad E_0=-NJ,\quad g_0=2.}
$$
The two configurations are related by <spin inversion symmetry>. There is no <geometrical frustration>.