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Truncated Cube (Posted on 2008-02-14) Difficulty: 3 of 5
A cube has 8 vertices. If each vertex is cut off to form a triangle, the new solid will have 3 x 8 = 24 vertices. If each of these vertices is then connected directly to each of the others via a straight line segment, how many of these segments will go through the body of the solid, rather than along its surface?

  Submitted by Charlie    
Rating: 3.0000 (3 votes)
Solution: (Hide)
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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
answerK Sengupta2008-02-26 05:19:41
The source of the puzzleCharlie2008-02-15 11:56:14
SolutionOn second thought..FrankM2008-02-15 07:30:21
Some Thoughtsgot different answerPraneeth2008-02-15 04:04:48
re(2): SolutionDej Mar2008-02-14 23:10:58
SolutionBractals2008-02-14 22:00:42
Some Thoughtsre(2): SolutionBractals2008-02-14 18:25:18
re: SolutionStephanie2008-02-14 18:13:22
QuestionHmm, Can I have that again?FrankM2008-02-14 18:09:41
SolutionLightning guessFrankM2008-02-14 18:05:10
SolutionSolutionDej Mar2008-02-14 12:46:26
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