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The classical combinatorial problem of $36$ officers has no solution, asthere are no Graeco-Latin squares of order six. The situation changes if oneworks in a quantum setup and allows for superpositions of classical objects andadmits entangled states. We analyze the recently found solution to the quantumversion of the Euler's problem from a geometric point of view. The notion of anon-displaceable manifold embedded in a larger space is recalled. This propertyimplies that any two copies of such a manifold, like two great circles on asphere, do intersect. Existence of a quantum Graeco-Latin square of size six,equivalent to a maximally entangled state of four subsystems with d=6 levelseach, implies that three copies of the manifold U(36)/U(1) of maximallyentangled states of the $36\times 36$ system, embedded in the complexprojective space ${C}P^{36\times 36 -1}$, do intersect simultaneously at acertain point.

Quantum version of the Euler's problem: a geometric perspective