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octahedron
Start with a thought experiment in which you take some regular octahedra and, on each one, colour two faces that meet at a vertex but not along an edge. Now imagine taking two octahedra and gluing them together with coloured faces in contact so that the vertices where the coloured faces meet coincide. Now glue another octahedron on, coloured face to coloured face, so that the three vertices where the coloured faces meet coincide. Continue in the same way as long as possible. Calculate how many octahedra can be joined together in this way.