Competition · AMC preparation · step 4 of 4

AMC 8 · 2006 · #18

Grade 6 geometry-3d
Archetype Casework by Position / Vertex Type └ Sub-archetype Grid / Solid Cell Position
surface-areaspatial-visualizationfraction-arithmeticcasework caseworkidentify-subproblemscomplementary-counting ↑ Prerequisites: surface-areaarea-rectangles
📏 Short solution 💡 3 insights
Problem
A 3 × 3 × 3 cube is built from 27 unit cubes. The 8 corner unit cubes are black; the other 19 are white. What fraction of the big cube's outer surface is white?

Pick an answer.

(A)
$\frac{1}{9}$
(B)
$\frac{1}{4}$
(C)
$\frac{4}{9}$
(D)
$\frac{5}{9}$
(E)
$\frac{19}{27}$

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

Counting white unit-squares directly is messy — the edge and center cubes contribute different numbers of faces. Tool #16 (Count the Complement) flips the problem: count the black unit-squares instead. Black cubes sit only at the 8 corners, and Tool #17 (Visualize Spatially) tells us each corner cube shows exactly 3 faces on the surface. That makes the black count a single multiplication. Tool #7 (Identify Subproblems) keeps the work organized: (a) find the total surface, (b) find the black surface, (c) subtract, (d) form the fraction.

1STEP 1

Count all surface squares

The big cube has 6 faces, each a 3×3 grid of 9 squares, giving 54 surface squares in all.

Total surface squares = 6 × 9 = 54
2STEP 2

Count the black squares

The 8 black cubes sit at corners, and each corner shows exactly 3 faces, so there are 24 black squares.

Black surface squares = 8 × 3 = 24
3STEP 3

Subtract for the white count

Every surface square is black or white, so subtract: 54 minus 24 leaves 30 white squares.

White surface squares = 54 - 24 = 30
4STEP 4

Simplify the white fraction

Write the white fraction 3054\frac{30}{54} and divide top and bottom by 6 to get 59\frac{5}{9}.

30/54 = 5/9 → (D)
Answer
5/9
Cross-check by counting white directly using the symmetry of one face. One face of the big cube is a 3 × 3 grid. The 4 corners of that grid are corners of the big cube and are black; the 4 edge squares and the 1 center square are white. So each face shows 5 white and 4 black unit squares, giving 59\frac{5}{9} per face — and therefore 59\frac{5}{9} overall. This matches answer (D). Magnitude is sensible: 19 of 27 cubes are white but they sit mostly inside, so the white surface fraction 59\frac{5}{9} is much smaller than the white cube fraction 1927\frac{19}{27}, which rules out (E).
💡Key takeaway

When most of the surface is one color, count the other color instead — each of the 8 black corner cubes shows just 3 faces, so 24 black squares out of 54 leaves 59\frac{5}{9} white.

  • Count all surface squares
  • Count the black squares
  • Subtract for the white count
  • Simplify the white fraction

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