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All possible straight lines joining the vertices of a cube with mid-points of its edges are drawn. At how many points inside the cube do two or more of these lines meet?

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By symmetry, I can find 6 of these points inside a cube. The answer is 14. That means there are 8 more points? Where are these points?

Oziter
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  • I'm sorry, you came up with $6$? Are you talking about within the interior of the cube only? If we were to include points being on the exterior that would have included then the vertices which fell under your first and/or second cases and there being eight vertices would account for your missing eight. – JMoravitz Sep 14 '20 at 02:24
  • @JMoravitz if you include points on the boundary then there are many more than 8, since you can make a crossing at each edge of a face – Calvin Khor Sep 14 '20 at 02:27
  • Only the points inside the cube, not on the surface or edge. – Oziter Sep 14 '20 at 11:35

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Take a diagonal between vertices on the base, and a parallel diagonal between midpoints on the top face. That gives a trapezoid whose diagonals meet.
$(0,0,0)-(1,1,1/2)$ and $(0,1,1)-(1,1/2,0)$ meet at $(2/3,2/3,1/3)$

Empy2
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