Without loss
of generality take the edges of the octahedron as 2 units.
Consider the square-based pyramid forming half the octahedron with
the 'special' vertex, let's call this V, as the vertex of this
pyramid. Then the plane of symmetry cutting through V and the
two coloured faces, cuts the pyramid in an isosceles triangle with
sides Ö3, Ö3 and 2.
Now imagine the 'ring' of octahedra glued together! The plane
of symmetry for the ring contains the plane of symmetry of the
pyramid just described.
The angle at V in this plane of symmetry (to 3 decimal places) is
So there is 'room' for 360/70.529 = 5.1 octahedra in the ring.
Clearly this means the fifth octahedron can be fitted in but no more
so the answer is 5.