Problem 25 · 2021 Math Kangaroo
Stretch
Spatial & Visual Reasoning
careful-counting
The solid shown in the diagram has 12 regular pentagonal faces, the other faces being either equilateral triangles or squares. Each pentagonal face is surrounded by 5 square faces and each triangular face is surrounded by 3 square faces. John writes 1 on each triangular face, 5 on each pentagonal face and \(-1\) on each square. What is the total of the numbers written on the solid?
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Answer: B — 50
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Hint 1 of 2
Each pentagon is ringed by 5 squares and each triangle by 3 squares — use this to count the faces of each type.
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Hint 2 of 2
Multiply each face count by its written number and add.
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Approach: count each face type, then total the labels
- The solid is the rhombicosidodecahedron: 12 pentagons, 20 triangles and 30 squares.
- John's total is 12·5 + 20·1 + 30·(−1) = 60 + 20 − 30.
- That equals 50.
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