AMC 8 · 2014 · #19

Grade 6 geometry-3d

Archetype Casework by Position / Vertex Type └ Sub-archetype Grid / Solid Cell Position

surface-areaspatial-visualizationoptimization-counting caseworkoptimization-counting ↑ Prerequisites: surface-areaspatial-visualization

📏 Medium solution 💡 4 insights

Problem

A 3-inch cube is built from twenty-seven 1-inch unit cubes: 21 red and 6 white. Place the white cubes so that the white area showing on the outside is as small as possible. What fraction of the 3-inch cube's surface area is white?

Pick an answer.

(A)

$frac{5}{54}$

(B)

$frac{1}{9}$

(C)

$frac{5}{27}$

(D)

$frac{2}{9}$

(E)

$frac{1}{3}$

AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Create a Physical Representation

This is a 3D placement problem, exactly the trigger for Tool #10 — picture (or build with 27 blocks) a 3 × 3 × 3 cube and notice that not every slot exposes the same number of faces. Tool #7 (Identify Subproblems) splits the task into three clean pieces: (i) find the total surface area, (ii) classify the 27 slots by how many faces they expose, (iii) place the 6 white cubes greedily in the most-hidden slots. Tool #2 (Systematic List) handles step (ii): list the slot types in order from least exposed to most exposed and count each.

1STEP 1

A cube has 6 square faces, each 3 × 3 = 9 sq in, so the total surface area is 54 square inches.

Total surface area = 6 × 9 = 54 sq in

2STEP 2

Sort the 27 slots by exposed faces: a corner shows 3, an edge 2, a face-center 1, and the dead center shows 0.

corners: 8, edges: 12, face-centers: 6, center: 1

3STEP 3

List the types by exposure and add the counts — they total 27, so no slot is missed.

1 + 6 + 12 + 8 = 27 ✓

4STEP 4

Fill the most-hidden slots first: 1 white cube in the dead center (0 faces), the other 5 in face-centers (1 face each).

1 center + 5 face-centers = 6 white cubes placed

5STEP 5

The center cube shows 0, and each of the 5 face-center cubes shows a 1 × 1 square, so the white area is 5 sq in.

White area = 1 × 0 + 5 × 1 = 5 sq in

6STEP 6

Put white area over total to get $\frac{5}{54}$ , already in lowest terms since gcd(5, 54) = 1.

white/total =

\frac{5}{54}

→ (A)

Answer

\frac{5}{54}

Sanity check the placement: any other placement is worse. If we instead put a white cube on an edge slot, it would show 2 faces, raising the white area by at least 1 square inch above 5; on a corner slot it would show 3 faces. So 5 sq in is genuinely the minimum, and

\frac{5}{54}

≈ 9.3% — small, as expected when we hide white cubes inside. Choice (A)

\frac{5}{54}

matches.

💡Key takeaway

Hide one white cube in the very middle, and the other five in face-centers so each shows just one square — that is the smallest white surface possible.

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