Problem 5 · 2009 Math Kangaroo
Easy
Algebra & Patterns
substitutionsum-constraint
The diagram shows a solid made up of 6 triangles. Each vertex is assigned a number, two of which are indicated. 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
Adding the sums of all six faces counts every vertex the same number of times.
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Hint 2 of 2
Equal face sums force the three ‘waist’ vertices to be equal and the two tips to be equal.
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Approach: use equal face sums to collapse the unknowns
- Each triangular face joins a tip vertex to two neighbouring waist vertices.
- Equal sums make all three waist vertices equal and both tip vertices equal.
- The figure marks a tip as 1 and a waist vertex as 5, so the five numbers are 1, 1, 5, 5, 5, totalling 17.
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