Problem 27 · 2019 Math Kangaroo
Stretch
Spatial & Visual Reasoning
cube-viewscareful-counting
How many planes exist that go through exactly three vertices of a given cube?
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Answer: D — 8
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Hint 1 of 2
Three vertices fix a plane; avoid sets where a fourth vertex lies on it (a face or a diagonal rectangle).
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Hint 2 of 2
Each plane through exactly three vertices slices off one corner of the cube.
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Approach: count corner-cutting triangles
- A plane through exactly three vertices uses the three vertices around a single corner.
- The cube has 8 corners, each giving one such triangular plane.
- So there are 8 planes.
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