AMC 8 · 2019 · #12

Grade 0 geometry-3dlogic

Archetype Net Folding / Face Adjacency └ Sub-archetype Deduce Face Adjacency from Multiple 3D Views

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

📏 Medium solution 💡 3 insights 📊 Diagram

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Problem

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.

Pick an answer.

(A)

$text{ red}$

(B)

$text{ white}$

(C)

$text{ green}$

(D)

$text{ brown}$

(E)

$text{ purple}$

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Visualize Spatial Relationships

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.

1STEP 1

Every face of a cube has exactly 4 neighbors (sharing an edge) and 1 opposite face across the cube.

6 = 1 (itself) + 4 (neighbors) + 1 (opposite)

2STEP 2

At each corner the three visible faces are pairwise adjacent, so every view hands you a batch of "X touches Y" facts.

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

3STEP 3

Use Red as the pivot: across the three views, Red touches Brown, Green, White, and Purple.

Neighbors of R from the views: {B, G} ∪ {W, B} ∪ {P, G} = {B, G, W, P}

4STEP 4

Red already claims itself plus those 4 neighbors, so the only leftover color, Aqua, must be the face opposite Red.

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

5STEP 5

"Opposite" is mutual: since Aqua sits opposite Red, the face opposite Aqua is Red — choice (A).

Aqua opposite Red ⟺ Red opposite Aqua → (A)

Answer

text{ red}

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.

💡Key takeaway

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

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