AMC 8 · 2019 · #12

Grade 0 geometry-3dlogic

face-adjacencyspatial-visualizationlogical-deduction caseworkphysical-representation ↑ Prerequisites: spatial-visualization

📏 Medium solution 💡 3 insights 📊 Diagram

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Problem

The faces of a cube are painted in six different colors: red $(R)$ , white $(W)$ , green $(G)$ , brown $(B)$ , aqua $(A)$ , and purple $(P)$ . Three views of the cube are shown below. What is the color of the face opposite the aqua face?

Pick an answer.

(A)

$text{ red}$

(B)

$text{ white}$

(C)

$text{ green}$

(D)

$text{ brown}$

(E)

$text{ purple}$

View mode:

Toolkit + CCSS Solution

Understand

Restated: A cube has six faces, each painted a different color: Red (R), White (W), Green (G), Brown (B), Aqua (A), and Purple (P). We are given three different views of the same cube; in each view, three mutually adjacent faces are visible (the three faces that meet at a single corner). Using these three snapshots, figure out which color is on the face directly opposite the Aqua face.

Givens: The cube has $6$ different-colored faces: R, W, G, B, A, P; View 1 shows the corner where R, B, G meet; View 2 shows the corner where W, B, R meet; View 3 shows the corner where P, R, G meet; In any single view, the three visible faces share a corner and are therefore mutually adjacent; Aqua (A) is not visible in any of the three views

Unknowns: The color of the face opposite the Aqua face

Understand

Plan

Primary tool: #17 Visualize Spatial Relationships

Secondary: #10 Create a Physical Representation, #16 Change Focus / Count the Complement

This is a 3D cube problem, so Tool #17 (Visualize Spatial Relationships) is the natural fit — we need to picture how the faces sit around the cube. A real die or a paper cube (Tool #10) makes the adjacency facts undeniable for a young learner. Notice that Red appears in all three views, so Red is the perfect "pivot" face: by collecting the neighbors of Red across the three views, we can build the full list of faces adjacent to Red. The clever move is then Tool #16 (Change Focus / Complement): instead of trying to locate Aqua directly, we find the face opposite Red (the one missing from Red's neighbor list), and then use the symmetry of "opposite" to flip the question around.

Execute — Answer: A

#17 Visualize Spatial Relationships K.G.B.4 Step 1

Recall the cube's structure.
A cube has $6$ faces.
Any single face touches $4$ other faces along its edges (its neighbors) and faces $1$ remaining face across the cube (its opposite).

$$6 = 1 \text{ (itself)} + 4 \text{ (neighbors)} + 1 \text{ (opposite)}$$

💡 Knowing that every face of a cube has exactly $4$ neighbors and $1$ opposite is just basic 3D-shape knowledge from Kindergarten geometry.

#10 Create a Physical Representation K.G.B.4 Step 2

Read the three views and note which faces sit at each visible corner.
Three faces meeting at one corner of a cube are pairwise adjacent, so each view gives us a small batch of "X is next to Y" facts.

View 1 corner: $\{R, B, G\}$. View 2 corner: $\{W, B, R\}$. View 3 corner: $\{P, R, G\}$.

💡 Holding a real block and matching each picture to a corner makes the adjacency facts obvious by touch.

#17 Visualize Spatial Relationships K.G.B.4 Step 3

Use Red as the pivot.
Red appears in every view, so list everything that touches Red across the three views.

Neighbors of R from the views: $\{B, G\} \cup \{W, B\} \cup \{P, G\} = \{B, G, W, P\}$

💡 Combining the neighbor sets from the three corners builds the full "around Red" picture from small pieces.

#16 Change Focus / Count the Complement K.OA.A.2 Step 4

Find the face opposite Red by elimination.
Red has $4$ neighbors (B, G, W, P) and $1$ opposite face.
The six colors are $\{R, W, G, B, A, P\}$; remove Red itself and its four neighbors and only one color is left.

$\{R, W, G, B, A, P\} \setminus \{R, B, G, W, P\} = \{A\}$, so the face opposite Red is Aqua.

💡 Out of $6$ colors, $5$ are accounted for, so the $1$ left over (Aqua) must be the opposite face — a Kindergarten "missing addend" idea.

#17 Visualize Spatial Relationships K.G.B.4 Step 5

Flip the relation.
"Opposite" is symmetric: if Aqua is opposite Red, then Red is opposite Aqua.
So the face opposite the Aqua face is Red — answer choice (A).

$$\text{Aqua opposite Red} \;\Longleftrightarrow\; \text{Red opposite Aqua} \;\Rightarrow\; \textbf{(A)}$$

💡 On a cube, opposite faces come in mutual pairs, so naming one direction names the other.

[1] #17 K.G.B.4 Recall the cube's structure. A cube has $6$ faces. Any single face touches $4$ o

[2] #10 K.G.B.4 Read the three views and note which faces sit at each visible corner. Three face

[3] #17 K.G.B.4 Use Red as the pivot. Red appears in every view, so list everything that touches

[4] #16 K.OA.A.2 Find the face opposite Red by elimination. Red has $4$ neighbors (B, G, W, P) an

[5] #17 K.G.B.4 Flip the relation. "Opposite" is symmetric: if Aqua is opposite Red, then Red is

Review

Reasonableness: Cross-check with a second pivot. From View 2, Brown touches Red and White; from View 1, Brown also touches Green. That makes Brown adjacent to $\{R, W, G\}$ in our three views — no contradiction with Aqua being elsewhere on the cube. Aqua never shows up in any view, which fits perfectly with Aqua sitting on the back of the cube (opposite Red, hidden by the Red face in every snapshot). Answer Red (A) is consistent with every view.

Alternative: Tool #10 (Physical Representation): grab a small cube or sugar-cube and stick six color stickers on it. Place R on top, then use View 1 to put B in front and G on the right. Check View 2 — rotate so the R-B edge stays visible and W lands on the left; that works. Check View 3 — rotate so R-G edge stays visible and P lands on the back; that works too. The only face never used is the bottom, opposite Red — which must be Aqua. So opposite Aqua is Red.

CCSS standards used (min grade 0)

K.G.B.4 Analyze and compare two- and three-dimensional shapes (Using basic cube structure — $6$ faces, each face with $4$ neighbors and $1$ opposite — and recognizing that three faces meeting at a corner are mutually adjacent.)
K.OA.A.2 Solve addition and subtraction word problems within 10 (Eliminating the $5$ accounted-for colors (Red plus its $4$ neighbors) from the set of $6$ to find the single remaining color (Aqua) opposite Red.)

⭐ This AMC 8 problem only needs Kindergarten cube sense and "$6$ minus $5$ leaves $1$" counting you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 0)

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