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Each of the 24 vertices has 23 other vertices to go to. Of these, there are 7 that it shares with each of the two octagonal faces on which it's located, but one of these vertices is on both these faces, so 2 x 7 - 1 = 13 are accounted for. The two vertices that share its triangular face are also among the 13 already counted, and so do not count again.
That means there are 23 - 13 = 10 destination vertices that require going through the body of the solid.
There are 24 vertices that each have 10 other vertices to go to through the body of the solid, making 240 routes. However each of these segments has now been counted twice--once in each direction. So the answer is half of this, or 120 segments. |
