Difference between revisions of "1988 AIME Problems/Problem 10"

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By the [[Euler characteristic]], we have that <math>V - E + F = 2</math>. The number of faces, <math>F</math>, is <math>12 + 8 + 6 = 26</math>. Since every point lies on exactly one vertex of a square/hexagon/octagon, we have that <math>V = 12 \cdot 4 = 8 \cdot 6 = 6 \cdot 8 = 48</math>. Substituting gives us <math>E = 72</math>.
 
By the [[Euler characteristic]], we have that <math>V - E + F = 2</math>. The number of faces, <math>F</math>, is <math>12 + 8 + 6 = 26</math>. Since every point lies on exactly one vertex of a square/hexagon/octagon, we have that <math>V = 12 \cdot 4 = 8 \cdot 6 = 6 \cdot 8 = 48</math>. Substituting gives us <math>E = 72</math>.
  
The number of segments joining the vertices of the polyhedron is <math>{48\choose2} = 1128</math>. Of these segments, <math>72</math> are edges. The number of diagonals of a square is <math>\frac{n(n+3)}{2} = 2</math>, of a hexagon is <math>9</math>, and of an octagon <math>20</math>. Hence the number of face diagonals is <math>2 \cdot 12 + 9 \cdot 8 + 20 \cdot 6 = 216</math>.
+
The number of segments joining the vertices of the polyhedron is <math>{48\choose2} = 1128</math>. Of these segments, <math>72</math> are edges. The number of diagonals of a square is <math>\frac{n(n-3)}{2} = 2</math>, of a hexagon is <math>9</math>, and of an octagon <math>20</math>. Hence the number of face diagonals is <math>2 \cdot 12 + 9 \cdot 8 + 20 \cdot 6 = 216</math>.
  
 
Subtracting, we get that the number of space diagonals is <math>1128 - 72 - 216 = 840</math>.
 
Subtracting, we get that the number of space diagonals is <math>1128 - 72 - 216 = 840</math>.

Revision as of 18:55, 19 June 2009

Problem

A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?

Solution

By the Euler characteristic, we have that $V - E + F = 2$. The number of faces, $F$, is $12 + 8 + 6 = 26$. Since every point lies on exactly one vertex of a square/hexagon/octagon, we have that $V = 12 \cdot 4 = 8 \cdot 6 = 6 \cdot 8 = 48$. Substituting gives us $E = 72$.

The number of segments joining the vertices of the polyhedron is ${48\choose2} = 1128$. Of these segments, $72$ are edges. The number of diagonals of a square is $\frac{n(n-3)}{2} = 2$, of a hexagon is $9$, and of an octagon $20$. Hence the number of face diagonals is $2 \cdot 12 + 9 \cdot 8 + 20 \cdot 6 = 216$.

Subtracting, we get that the number of space diagonals is $1128 - 72 - 216 = 840$.

See also

1988 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
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