  Ordering and classifying multipartite quantum states by their entanglement
content remains an open problem. One class of highly entangled states, useful
in quantum information protocols, the absolutely maximally entangled (AME)
ones, are specially hard to compare as all their subsystems are maximally
random. While, it is well-known that there is no AME state of four qubits, many
analytical examples and numerically generated ensembles of four qutrit AME
states are known. However, we prove the surprising result that there is truly
only {\em one} AME state of four qutrits up to local unitary equivalence. In
contrast, for larger local dimensions, the number of local unitary classes of
AME states is shown to be infinite. Of special interest is the case of local
dimension 6 where it was established recently that a four-party AME state does
exist, providing a quantum solution to the classically impossible Euler problem
of 36 officers. Based on this, an infinity of quantum solutions are constructed
and we prove that these are not equivalent. The methods developed can be
usefully generalized to multipartite states of any number of particles.
