AMC 8 · 2002 · #22

Grade 6 geometry-3d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border

surface-areaspatial-visualizationface-adjacency identify-subproblemscasework ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Six unit cubes (each 1 inch on every edge) are fastened together into the composite solid shown. Find its total surface area, in square inches, counting the top, the bottom, and every visible side.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

Counting visible faces one by one on a lumpy stack is error-prone. Tool #16 (Count the Complement) flips the job: start with all the faces the 6 cubes would have if they were separate (6 × 6 = 36), then subtract only the faces that got hidden by being pressed against another cube. Tool #1 (Draw a Diagram) makes the hidden faces easy to find — each place where two cubes touch is one contact, and one contact hides two unit faces. Tool #7 (Identify Subproblems) splits the count into two clean pieces: (a) total faces of separate cubes, (b) hidden faces from contacts.

1STEP 1

Pretend the 6 cubes are separate: each has 6 faces of area 1, so 6 × 6 = 36 unit faces in all.

Separate total = 6 cubes × 6 faces × 1 in² = 36 in²

2STEP 2

Trace the figure cube by cube: the pairs that share a whole face give 5 contacts.

Contacts = 5

3STEP 3

Each contact hides 2 faces (one per cube), so 5 × 2 = 10 unit faces disappear.

Hidden faces = 5 × 2 = 10, Hidden area = 10 × 1 = 10 in²

4STEP 4

Subtract hidden from the separated total: 36 - 10 = 26 in² → (C).

Surface area = 36 - 10 = 26 in² → (C)

Answer

Two quick checks. First, bounds: every contact hides 2 faces, so the surface area must drop by an even number from 36. The choices 18, 24, 26, 30, 36 are all even drops (18, 12, 10, 6, 0), and only 36 - 26 = 10 matches 5 contacts × 2 = 10 — so (C) is the only contact-count that is consistent with the figure. Second, sanity: a single cube has surface area 6; six cubes glued in a straight line would have 5 contacts too and give 36 - 10 = 26 — the same as this arrangement, which is a useful reminder that surface area depends on the number of contacts, not the exact shape.

💡Key takeaway

Imagine the cubes apart (36 faces), then erase 2 faces for every spot where two cubes touch (5 spots = 10 faces). What's left is the surface area: 36 - 10 = 26.

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