Problem 22 · 2002 AMC 8
Stretch
Geometry & Measurement
hide-faces-in-3dspatial-reasoning
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Answer: C — 26 square inches.
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Hint 1 of 2
Start from the easy over-count: 6 loose cubes show 6 × 6 = 36 little faces. Gluing them together can only *hide* faces, never add any.
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Hint 2 of 2
Look at the *glue joints*, not the cubes. Every place two cubes are pressed together buries exactly 2 faces (one off each cube). Count the joints, double it, subtract.
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Approach: total faces minus the hidden (touching) faces
- Six separate unit cubes would show 6 × 6 = 36 unit faces.
- Now subtract for contact. Each spot where two cubes touch hides 2 faces. Tracing the figure, the cubes are joined at 5 places, so 5 × 2 = 10 faces are buried.
- Exposed surface area = 36 − 10 = 26 square inches.
- *The reusable idea:* for any blob of glued cubes, surface area = 6·(number of cubes) − 2·(number of touching pairs). Counting joints is faster and less error-prone than checking each cube's hidden sides one at a time.
Another way — tally per cube:
- Go cube by cube and count buried faces: three cubes hide 1 face each, two hide 2, one hides 3 → 3 + 4 + 3 = 10 hidden.
- Exposed = 36 − 10 = 26.
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