Problem 14 · 2017 Math Kangaroo
Hard
Counting & Probability
careful-counting
Each face of the polyhedron shown is either a triangle or a square. Each square borders 4 triangles, and each triangle borders 3 squares. The polyhedron has 6 squares. How many triangles does it have?
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Answer: D — 8
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Hint 1 of 2
Count the square-touches-triangle contacts in two different ways.
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Hint 2 of 2
Each square contributes 4 such contacts; each triangle uses up 3 of them.
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Approach: double-count the square-triangle adjacencies
- There are 6 squares, each bordering 4 triangles, giving 6 x 4 = 24 square-triangle borders.
- Each triangle borders 3 squares, so it accounts for 3 of those borders.
- Number of triangles = 24 / 3 = 8.
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