  We investigate the two well-known ground states of rings of $N$ classical
magnetic dipoles that are given by clockwise or anti-clockwise spin
orientations tangent to the circle encompassing the dipole ring. In particular,
we formulate a rigorous proof of the ground state property of the states in
question. The problem can be reduced to the determination of the lowest
eigenvalue of a $3N\times 3N$ matrix ${\mathbf J}$. We show that all
eigenvalues of ${\mathbf J}$ can be analytically calculated and, at least for
$N=3,\ldots,8$, the lowest one can be directly determined. The main part of the
paper is devoted to the completion of the proof for $N\ge 9$ based on various
estimates and case distinctions. We also discuss the question to what extent
computer-algebraic results should be allowed to contribute to a mathematical
proof.
