Problem 21 · 1997 AJHSME
Stretch
Geometry & Measurement
surface-areainvariance
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Answer: D — 54 sq cm.
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Hint 1 of 2
Before computing anything, ask what a single corner-cut actually DOES to the surface. A corner cube shows 3 faces to the outside; pull it out and you expose 3 fresh faces from the notch. Notice anything?
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Hint 2 of 2
Look for an invariant: if every move removes exactly as much surface as it adds, the total can't change — so you only need the original.
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Approach: spot the surface-area invariant
- A corner cube touches 3 of the big cube's outer faces. Removing it deletes those 3 unit squares but uncovers 3 new unit squares (the inside walls of the notch) — a perfect trade, so surface area is unchanged.
- All 8 corners are separated by a middle cube, so the cuts don't interfere; the area still equals the original cube's: 6 × 3² = 6 × 9 = 54 sq cm.
- Why this transfers: the elegant move is recognizing 'lose 3, gain 3 = no change' before crunching numbers. Hunting for an invariant turns a scary 3-D figure into a one-line answer.
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