Problem 24 · 2019 Math Kangaroo
Stretch
Geometry & Measurement
careful-countingcaseworkspatial-reasoning
Consider a cube. How many planes pass through at least three vertices of this cube?
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Answer: E — 20
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Hint 1 of 2
Planes through 3+ cube vertices split into types: the 6 faces, diagonal planes through opposite edges, and corner-cutting planes.
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Hint 2 of 2
Count each type: 6 faces, 6 diagonal rectangles, 8 corner triangles.
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Approach: classify the planes by how they cut the cube
- Face planes: 6.
- Diagonal planes through two opposite edges (rectangles): 6.
- Planes through three vertices around one corner: 8.
- Total 6 + 6 + 8 = 20 — answer (E).
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