  It is known from clever mathematical examples \cite{Ca10} that the Monge
ansatz may fail in continuous two-marginal optimal transport (alias optimal
coupling alias optimal assignment) problems. Here we show that this effect
already occurs for finite assignment problems with $N=3$ marginals, $\ell=3$
'sites', and symmetric pairwise costs, with the values for $N$ and $\ell$ both
being optimal. Our counterexample is a transparent consequence of the convex
geometry of the set of symmetric Kantorovich plans for $N=\ell=3$, which -- as
we show -- possess 22 extreme points, only 7 of which are Monge. These extreme
points have a simple physical meaning as irreducible molecular packings, and
the example corresponds to finding the minimum energy packing for
Frenkel-Kontorova interactions. Our finite example naturally gives rise, by
superposition, to a continuous one, where failure of the Monge ansatz manifests
itself as nonattainment and formation of 'microstructure'.
