Problem 23 · 1989 AJHSME
Stretch
Geometry & Measurement
surface-areaexposed-faces
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Answer: C — 33.
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Hint 1 of 3
Painted area = number of EXPOSED unit faces (each 1 m²). A face is unpainted only if it touches the ground or is pressed against a neighboring cube.
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Hint 2 of 3
Counting one cube at a time invites double-counting. Instead sweep the whole sculpture by viewing direction — top, then each side — and tally the faces you can actually see from each.
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Hint 3 of 3
The bottom faces sit on the ground (never painted), so the work is: all top-visible faces plus the exposed faces on the front, back, and two sides.
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Approach: tally exposed faces by viewing direction
- Only faces touching nothing get paint — faces on the ground or glued to a neighbor are skipped. So organize the count by direction instead of cube-by-cube, which avoids missing or repeating faces.
- Looking straight down, 10 cube-tops are uncovered → 10. The front shows 6 exposed faces and the back another 6. The two side walls together expose 11 more.
- Total painted = 10 + 6 + 6 + 11 = 33 m².
- Why this transfers: for any blocky solid, painted surface = total faces (cubes × 6) minus the hidden ones (on the ground or shared between touching cubes). Counting by direction is just an organized way to do exactly that without losing track.
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