AMC 8 · 2003 · #1

Grade 2 geometry-3d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

spatial-visualizationmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

Three kids look at a single cube. Jamie counts its edges, Jimmy counts its corners (vertices), and Judy counts its faces. Add the three counts together. What is the total?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Break into Subproblems

The question asks for one number, but that number is built from three independent counts. Tool #7 (Break into Subproblems) splits the task into three small jobs — count edges, count corners, count faces — and the final step just adds them. Tool #2 (Make an Organized List) helps each sub-count: list the faces by direction (top, bottom, and four sides), the corners by floor and ceiling, and the edges by which group of parallel lines they belong to. Listing in groups makes sure no part of the cube is missed or double-counted.

1STEP 1

Group the cube's faces by direction — top, bottom, and four sides — which totals 6.

faces = 1 (top) + 1 (bottom) + 4 (sides) = 6

2STEP 2

Corners sit at the four floor corners and the four ceiling corners, giving 8.

corners = 4 (floor) + 4 (ceiling) = 8

3STEP 3

Edges split into three families of parallel segments — bottom, top, and vertical — for 12.

edges = 4 (bottom) + 4 (top) + 4 (vertical) = 12

4STEP 4

Add the three subtotals — faces, corners, and edges — to reach 26.

12 + 8 + 6 = 26 → (E)

Answer

Quick check with Euler's relation for any convex polyhedron: V - E + F = 2. Plugging in V = 8, E = 12, F = 6 gives 8 - 12 + 6 = 2, which is correct, so the three counts are consistent. The sum V + E + F = 8 + 12 + 6 = 26 matches answer (E). Also, 26 is the largest choice, which fits because we are adding three positive whole numbers, not subtracting.

💡Key takeaway

Big questions about a 3D shape become easy when you split them up: count faces, count corners, count edges, then add. A cube always gives 6, 8, 12 — and they add to 26.

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