Problem 8 · 1999 AMC 8
Medium
Geometry & Measurement
spatial-reasoningnet-folding
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Answer: A — Blue.
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Hint 1 of 2
On a cube, two faces are opposite exactly when they are NOT neighbors — they never share an edge or a fold-corner in the net. Pick the white square and rule out everything it touches as it folds.
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Hint 2 of 2
Faces that are edge-adjacent in the net, OR sit in an L (one step over, one step up), end up next to each other. The one square that can't reach white either way is its opposite.
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Approach: opposite = the face white can never sit beside
- Anchor the yellow square as the bottom and fold the rest up around it. White (directly below yellow) folds to become one side wall.
- Now eliminate white's neighbors: white shares the fold with yellow, and yellow's straight-strip partner is orange — those wrap around white's column. Red and green attach along the other edges. The only face left, sitting across the cube from white, is blue.
- The rule to keep: on any cube net, opposite faces are the pair that are neither edge-adjacent nor one-corner-apart. A quick test: in a straight run of three squares the two ends are opposite — here yellow's straight neighbors pin down what's beside white, leaving blue as the lone face across from it.
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