Problem 18 · 2006 AMC 8
Medium
Geometry & Measurement
corner-cubessurface-fraction
A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?
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Answer: D — 5/9.
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Hint 1 of 2
Don't think about all 6 faces and 8 cubes at once. By symmetry every face looks identical, so the whole-cube fraction equals the fraction on a SINGLE face.
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Hint 2 of 2
On one 3×3 face, ask: which of the 9 little squares are black? A corner cube of the big cube touches each face it's on at exactly that face's corner square.
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Approach: symmetry — just analyze one face
- Every face is identical, so the surface fraction for the whole cube equals the fraction on one face. Look at a single 3 × 3 face: 9 unit squares.
- The black cubes sit at the big cube's 8 corners. On any one face, the four corner unit squares are exactly those corner cubes — so 4 squares are black, leaving 9 − 4 = 5 white.
- White fraction = 5/9.
- Why one face is enough: the cube's symmetry makes every face the same, so a count on one face IS the answer — no need to total all 54 surface squares. Spotting symmetry to shrink the work is the whole game here.
- Trap to dodge: the choice 19/27 counts white CUBES out of all cubes, but the question asks about surface AREA — the buried center cube and hidden faces don't show, so don't mix the two up.
Another way — total surface area, counting black squares directly:
- The big cube has 6 faces × 9 = 54 unit squares of surface. Each of the 8 corner (black) cubes shows on 3 faces, contributing 3 black squares, for 8 × 3 = 24 black squares.
- White squares = 54 − 24 = 30, so white fraction = 30/54 = 5/9 — matches the one-face shortcut.
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