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Polyhedra construction (Posted on 2006-07-17) Difficulty: 3 of 5
A certain polyhedron is constructed such that each vertex is the intersection of five triangles. How many vertices are there?

Was that easy? Try these.

...such that each vertex is the intersection of...
1. three squares and a triangle
2. three triangles and a square
3. four triangles and a square
4. a triangle, square, pentagon, and square in that order
5. a decagon, hexagon, and square

Notice any patterns?

See The Solution Submitted by Tristan    
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Solution Solution | Comment 1 of 4

Let
     v = number of polyhedron vertices
     f = number of polyhedron faces
     e = number of polyhedron edges
   f_n = number of polyhedron faces 
         that are n-gons
   m_n = number of n-gons at a
         polyhedron vertex
Then the following apply to each case:
   1) Euler's polyhedron theorem: v + f - e = 2
   2) m_n*v = n*f_n for each n
   3) f = sum of f_n
   4) 2*e = sum of n*f_n
Case 0: five triangles -
        3*f_3
   v = -------
          5
   f = f_3
        3*f_3
   e = -------
          2
   ===> v = 12
Case 1: three squares and a triangle -
        3*f_3     4*f_4
   v = ------- = -------
          1         3
   f = f_3 + f_4
        3*f_3 + 4*f_4
   e = ---------------
              2
   ===> v = 24
Case 2: three triangles and a square -
        3*f_3     4*f_4
   v = ------- = -------
          3         1
   f = f_3 + f_4
        3*f_3 + 4*f_4
   e = ---------------
              2
   ===> v = 8
Case 3: four triangles and a square -
        3*f_3     4*f_4
   v = ------- = -------
          4         1
   f = f_3 + f_4
        3*f_3 + 4*f_4
   e = ---------------
              2
   ===> v = 24
Case 4: a triangle, square, pentagon, and
        square in that order -
        3*f_3     4*f_4     5*f_5
   v = ------- = ------- = -------
          1         2         1
   f = f_3 + f_4 + f_5
        3*f_3 + 4*f_4 + 5*f_5
   e = -----------------------
                  2
   ===> v = 60
Case 5: a decagon, hexagon, and square -
        4*f_4     6*f_6     10*f_10
   v = ------- = ------- = ---------
          1         1          1
   f = f_4 + f_6 + f_10
        4*f_4 + 6*f_6 + 10*f_10
   e = -------------------------
                  2
   ===> v = 120
 

Edited on July 17, 2006, 12:23 pm
  Posted by Bractals on 2006-07-17 12:21:44

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