  The negative solution to the famous problem of $36$ officers of Euler implies
that there are no two orthogonal Latin squares of order six. We show that the
problem has a solution, provided the officers are entangled, and construct
orthogonal quantum Latin squares of this size. As a consequence, we find an
example of the long-elusive Absolutely Maximally Entangled state AME$(4,6)$ of
four subsystems with six levels each, equivalently a $2$-unitary matrix of size
$36$, which maximizes the entangling power among all bipartite unitary gates of
this dimension, or a perfect tensor with four indices, each running from one to
six. This special state deserves the appellation golden AME state as the golden
ratio appears prominently in its elements. This result allows us to construct a
pure nonadditive quhex quantum error detection code $(\!(3,6,2)\!)_6$, which
saturates the Singleton bound and allows one to encode a $6$-level state into a
triplet of such states.
