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 in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: Akash's birthday cake is a $4 \times 4 \times 4$ 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 ($1 \times 1 \times 1$ inch each). How many of these small cubes end up with icing on exactly $2$ of their faces?

Givens: Big cube edge length $= 4$ inches; sliced into $4 \times 4 \times 4 = 64$ unit cubes; Icing on the top face and all $4$ side faces ($5$ faces total); No icing on the bottom face; Answer choices: (A) $12$, (B) $16$, (C) $18$, (D) $20$, (E) $24$

Unknowns: The number of unit cubes that have icing on exactly $2$ of their $6$ faces

Understand

Plan

Primary tool: #10 Create a Physical Representation

Secondary: #7 Identify Subproblems, #2 Make a Systematic List

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 $4 \times 4 \times 4$ 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.

Execute — Answer: D

#10 Create a Physical Representation K.G.B.4 Step 1

Picture the cube and label its $6$ kinds of positions: $8$ corners, $12$ edges (each edge has $4 - 2 = 2$ "middle" unit cubes between its two corner unit cubes), $6$ face-centers, and $1$ deep interior block.
A unit cube's iced-face count equals the number of iced big-cube faces it touches.

$$\text{corners}=8,\;\;\text{edge middles}=12\times 2 = 24,\;\;\text{face centers}=6\times 4=24,\;\;\text{interior}=2\times 2\times 2=8$$

💡 Sorting the small cubes into corner / edge / face / interior families is the kindergarten skill of analyzing a 3D shape's parts.

#7 Identify Subproblems K.G.B.4 Step 2

Split into three sub-questions, one per location type that could give exactly $2$ iced faces: (a) cubes on the $4$ top edges where the iced top meets an iced side, (b) cubes on the $4$ vertical edges where two iced sides meet, (c) cubes at the $4$ bottom corners where two iced sides meet the un-iced bottom.

$$\text{cases} = \{\text{top edges},\;\text{vertical edges},\;\text{bottom corners}\}$$

💡 Breaking the cube's surface into pieces I can count separately is the Tool #7 sub-problems move on a 3D shape.

#2 Make a Systematic List 3.OA.A.1 Step 3

Count case (a) — top edges.
There are $4$ edges around the top face.
Each edge has $2$ middle unit cubes (the $2$ end unit cubes are top corners, which touch $3$ iced faces).
Each middle cube touches the iced top and one iced side: exactly $2$ iced faces.

$$4 \text{ top edges} \times 2 \text{ middle cubes} = 8 \text{ cubes}$$

💡 Multiplying "$4$ groups of $2$" is the third-grade meaning of multiplication.

#2 Make a Systematic List 3.OA.A.1 Step 4

Count case (b) — vertical edges.
There are $4$ vertical edges joining top corners to bottom corners.
The top end of each is a top corner (3 iced faces) and the bottom end is a bottom corner (handled below).
The $2$ middle cubes on each vertical edge touch two iced side faces: exactly $2$ iced faces.

$$4 \text{ vertical edges} \times 2 \text{ middle cubes} = 8 \text{ cubes}$$

💡 Same "$4$ groups of $2$" pattern as case (a) — multiplication keeps the count tidy.

#2 Make a Systematic List 3.OA.A.1 Step 5

Count case (c) — bottom corners.
There are $4$ unit cubes at the bottom corners of the big cube.
Each touches two iced side faces and the un-iced bottom, so each has exactly $2$ iced faces.
(The $4$ top corners are excluded because they touch the iced top plus two iced sides, giving $3$ iced faces.) Also, the $2$ middle cubes on each of the $4$ bottom edges touch only one iced side, so they have just $1$ iced face — not counted.

$$4 \text{ bottom corners} \times 1 = 4 \text{ cubes}$$

💡 Listing each of the $4$ bottom corners once and checking its iced faces is straight systematic counting.

#7 Identify Subproblems 3.OA.D.8 Step 6

Add the three cases to get the total number of unit cubes with exactly two iced faces.

$$8 + 8 + 4 = 20 \;\Rightarrow\; \textbf{(D)}$$

💡 Combining sub-problem answers with a single addition is the wrap-up step of any multi-step word problem in Grade 3.

[1] #10 K.G.B.4 Picture the cube and label its $6$ kinds of positions: $8$ corners, $12$ edges (

[2] #7 K.G.B.4 Split into three sub-questions, one per location type that could give exactly $2

[3] #2 3.OA.A.1 Count case (a) — top edges. There are $4$ edges around the top face. Each edge h

[4] #2 3.OA.A.1 Count case (b) — vertical edges. There are $4$ vertical edges joining top corner

[5] #2 3.OA.A.1 Count case (c) — bottom corners. There are $4$ unit cubes at the bottom corners

[6] #7 3.OA.D.8 Add the three cases to get the total number of unit cubes with exactly two iced

Review

Reasonableness: Sanity check the parts. The full big cube has $4 \cdot 4 \cdot 4 = 64$ unit cubes. Of these, $8$ are deep inside (no icing), $4 \cdot 4 = 16$ 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 ($4 \cdot 2 = 8$ 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.

Alternative: Tool #16 (Change Focus / Complement) is a clean alternative. If the cake had icing on all $6$ faces, the standard "2-faces-iced" count would be the $12$ edges $\times$ $2$ middle cubes $= 24$. Removing icing from the bottom changes two categories: the $8$ bottom-edge middles drop from "$2$-iced" to "$1$-iced" ($-8$), and the $4$ bottom corners drop from "$3$-iced" to "$2$-iced" ($+4$). Net change: $-8 + 4 = -4$, giving $24 - 4 = 20$ — the same answer (D).

CCSS standards used (min grade 3)

K.G.B.4 Analyze and compare two- and three-dimensional shapes (Identifying the corner, edge, face-center, and interior positions of the $4 \times 4 \times 4$ cube so each unit cube can be classified by how many big-cube faces it touches.)
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups (Counting each case as "$4$ groups of $2$" (top edges and vertical edges) using the grade-3 meaning of multiplication.)
3.OA.D.8 Solve two-step word problems using four operations within 100 (Combining the three case counts $8 + 8 + 4 = 20$ to finish the multi-step word problem.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 3)

More like this