Problem 15 · 1991 AJHSME
Hard
Geometry & Measurement
surface-areainvariance
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Answer: C — the same.
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Hint 1 of 3
Don't compute the whole surface area before and after — just track what CHANGES at the corner. Removing the little cube takes away some surface squares but opens up a notch with new walls. Compare those two amounts.
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Hint 2 of 3
The cut-out cube sat at a corner, so exactly 3 of its faces were on the outside (now gone). The notch it leaves behind exposes 3 fresh inside faces. Are those numbers the same?
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Hint 3 of 3
Picture biting one cube out of a corner: 3 squares disappear from the outside, 3 brand-new squares appear inside. The trade is even.
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Approach: compare only the squares removed vs. the squares newly exposed
- Total surface area barely matters — only the corner changes, so just weigh what's lost against what's gained there.
- The unit cube was at a corner, so 3 of its faces were part of the outer surface; removing it erases those 3 unit squares.
- But the empty notch now shows 3 new inside faces (the walls of the bite), adding 3 unit squares back.
- 3 removed, 3 revealed — they cancel exactly, so the surface area is the same.
- Why this transfers: for any "before vs. after" question, don't recompute the whole thing — track only the difference. And a corner notch always trades 3 outer faces for 3 inner faces (an edge notch trades 2 for 4, growing the area), so the geometry of where you cut decides the outcome.
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