Problem 13 · 2003 AMC 8
Hard
Geometry & Measurement
spatial-reasoning
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Answer: B — 6 cubes.
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Hint 1 of 2
"Painted faces" just means "exposed faces", so reword the question: which cubes have exactly four sides touching the outside air?
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Hint 2 of 2
Don't paint every cube — instead find the misfits. Sort the 14 cubes by exposure: the perched-on-top ones are too open, the buried bottom corners too hidden, and what's left must be the fours.
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Approach: count exposed faces per cube, then subtract the misfits
- Paint reaches a cube's face only if that face is exposed, and the bottom counts as exposed too. So "exactly four painted faces" means "exactly four exposed sides." Rather than tally all 14, find the cubes that are too exposed or too hidden and subtract.
- The 4 cubes perched on top of the base are open on all 4 sides plus the top — that's 5 painted faces, too many.
- The 4 corner cubes of the bottom layer touch neighbors on two sides, leaving only 3 faces (two sides + bottom) exposed — too few.
- Everything else has exactly four exposed sides. So 14 − 4 (tops) − 4 (bottom corners) = 6 cubes.
- You'll see this again: for "how many cubes have exactly k painted faces," count by a cube's position (corner / edge / face-center / hidden) instead of inspecting cubes one at a time — position fixes the exposure.
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