Problem 8 · 2009 Math Kangaroo
Medium
Algebra & Patterns
sum-constraintcasework
The diagram on the right shows a solid made up of 6 triangles. Each vertex is assigned a number, two of which are shown. The total of the three numbers on each triangular face is the same. What is the total of all five numbers?
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Answer: C — 17
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Hint 1 of 2
The solid is a triangular bipyramid: two apexes and a middle triangle of three vertices.
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Hint 2 of 2
Equal face sums force the three middle vertices to be equal and the two apexes to be equal.
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Approach: use equal triangle sums to pin down the vertex values
- Label the two tips t and the three middle vertices m₁,m₂,m₃.
- Each face uses one tip and two middle vertices; equal sums force m₁=m₂=m₃=m and both tips equal to t.
- The two given numbers 1 and 5 must be t and m; taking t=1, m=5 gives matching face sums.
- Total of all five = 2t + 3m = 2(1) + 3(5) = 17.
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