Problem 12 · 2019 AMC 8
Medium
Geometry & Measurement
spatial-reasoningcasework
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Answer: A — Red.
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Hint 1 of 2
Key fact about a cube: two faces are opposite exactly when they can never show up together in one view (you'd have to see through the cube). So the opposite of aqua is whichever color shares no view with it.
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Hint 2 of 2
Easier to chase the other way: list every color that does appear alongside red. Red touches four colors — the one it's missing must be its opposite.
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Approach: opposite faces never appear in the same view
- Two faces sharing a view are adjacent; opposite faces can never both be seen at once. Track what red is adjacent to across the three pictures.
- Red appears with brown and green in one view, with brown and white in another, and with purple and green in the third — so red sits next to brown, green, white, and purple.
- Red is adjacent to four of the five other colors; the one it never meets is aqua. So aqua and red are opposite, and the face opposite aqua is red.
- Why this transfers: for any "net or views of a cube" puzzle, find a face and rule out its four neighbors — the single color it never shares a view with is forced to be its opposite.
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