Problem 16 · 2009 Math Kangaroo
Hard
Algebra & Patterns
sum-constraintsubstitution
The diagram shows an object with 6 triangular faces. On each corner there is a number (two are shown). The sum of the numbers on the corners of each face is the same. What is the sum of all 5 numbers?
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Answer: C — 17
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Hint 1 of 2
Equal face-sums force just two values: the two tips are equal and the three middle corners are equal.
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Hint 2 of 2
Use the two shown numbers to tell which value is a tip and which is a middle corner.
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Approach: exploit equal face sums to find the two values
- The solid is a triangular bipyramid: two tips plus three middle corners. Equal face sums force all three middle corners equal and both tips equal.
- The shown 1 is a tip and the shown 5 is a middle corner, so the numbers are 1, 1 (tips) and 5, 5, 5 (middle).
- Their total is 1+1+5+5+5 = 17.
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