AMC 8 · 2020 · #9

Grade 3 geometry-3dcounting
Archetype Casework by Position / Vertex Type └ Sub-archetype Grid / Solid Cell Position
spatial-visualizationsystematic-enumerationcombinations-basic caseworksystematic-enumeration ↑ Prerequisites: spatial-visualization
📏 Medium solution 💡 3 insights 📊 Diagram
📘 View easy version →
Problem
Akash's birthday cake is a 4 × 4 × 4 inch cube with icing on the top face and all 4 side faces but no icing on the bottom. The cake is sliced into 64 unit cubes (1 × 1 × 1 inch each). How many of these small cubes end up with icing on exactly 2 of their faces?

Pick an answer.

(A)
12
(B)
16
(C)
18
(D)
20
(E)
24

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Create a Physical Representation

Because the icing covers 5 of the 6 faces, the usual symmetric "corners have 3, edges have 2, faces have 1" rule does not apply — the missing bottom flips some categories. Tool #10 (build a physical cube of 4 × 4 × 4 unit cubes, or sketch it) makes it visible which positions touch exactly two iced faces. Tool #7 (Identify Subproblems) splits the count into three location types — top edges, vertical edges, and bottom corners — that each contribute to the "exactly two iced" count. Tool #2 (Systematic List) then counts each type without missing or double-counting any cube.

1STEP 1

Sort the 64 pieces by position — corners, edge-middles, face-centers, interior. A piece is iced only where it touches a frosted outer face.

corners=8, edge middles=12× 2 = 24, face centers=6× 4=24, interior=2× 2× 2=8
2STEP 2

Split the count into the three sub-cases that can give exactly two iced faces: the top edges, the vertical edges, and the bottom corners.

cases = {top edges, vertical edges, bottom corners}
3STEP 3

Top edges: each of the 4 top edges has 2 middle pieces touching the iced top plus one iced side, so exactly two faces are iced — 8 pieces.

4 top edges × 2 middle cubes = 8 cubes
4STEP 4

Vertical edges: each of the 4 vertical edges has 2 middle pieces touching two iced sides, exactly two faces again — 8 pieces.

4 vertical edges × 2 middle cubes = 8 cubes
5STEP 5

Bottom corners: the 4 bottom corners each touch two iced sides and the bare bottom, giving exactly two iced faces — 4 pieces.

4 bottom corners × 1 = 4 cubes
6STEP 6

Add the three groups to reach the total number of pieces frosted on exactly two faces.

8 + 8 + 4 = 20 → (D)
Answer
20
Sanity check the parts. The full big cube has 4 · 4 · 4 = 64 unit cubes. Of these, 8 are deep inside (no icing), 4 · 4 = 16 sit in the centers of the 4 side faces (each iced once), 4 centers of the un-iced bottom have 0 iced faces, 4 centers of the top have 1 iced face, and the 20 we counted have 2 iced faces, while the 4 top corners have 3 iced faces. The remaining bottom-edge middles (4 · 2 = 8 cubes) have 1 iced face. Tallying: 8 + 16 + 4 + 4 + 20 + 4 + 8 = 64. The bookkeeping closes, so 20 is consistent. Also 20 is choice (D), which is one of the offered answers.
💡Key takeaway

This AMC 8 problem only needs Grade 3 multiplication and addition you already know — count edges and corners, group them with ×, then add!

More like this

Same archetype — closest grade level first.