  The classical combinatorial problem of $36$ officers has no solution, as
there are no Graeco-Latin squares of order six. The situation changes if one
works in a quantum setup and allows for superpositions of classical objects and
admits entangled states. We analyze the recently found solution to the quantum
version of the Euler's problem from a geometric point of view. The notion of a
non-displaceable manifold embedded in a larger space is recalled. This property
implies that any two copies of such a manifold, like two great circles on a
sphere, do intersect. Existence of a quantum Graeco-Latin square of size six,
equivalent to a maximally entangled state of four subsystems with d=6 levels
each, implies that three copies of the manifold U(36)/U(1) of maximally
entangled states of the $36\times 36$ system, embedded in the complex
projective space ${C}P^{36\times 36 -1}$, do intersect simultaneously at a
certain point.
