← Sensim Lab Sensim Math

AMC 8 · 1999 · #8

Grade 6 geometry-3d
Archetype Net Folding / Face Adjacency
spatial-visualizationpolyhedron-netsface-adjacencynet-folding physical-representationcasework ↑ Prerequisites: spatial-visualization
📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is

Pick an answer.

(A)
B
(B)
G
(C)
O
(D)
R
(E)
Y
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Toolkit + CCSS Solution

Understand

Restated: Six squares — colored Red, Blue, Orange, Yellow, Green, White (each color on both sides) — are hinged together in the net shown and then folded into a cube. Which color ends up on the face opposite White?

Givens: The net has six squares hinged in a fixed layout: bottom row of the net reads $R, B$ (with $B$ to the right of $R$); middle row reads $G, Y, O$ (with $G$ under $B$, $Y$ to the right of $G$, $O$ to the right of $Y$); $W$ sits below $Y$; Each colored square is the same color on the front and the back, so folding direction does not change the color; After folding, the six squares form the six faces of a cube; Answer choices: (A) B, (B) G, (C) O, (D) R, (E) Y

Unknowns: The color of the face directly opposite the White face on the assembled cube

Understand

Restated: Six squares — colored Red, Blue, Orange, Yellow, Green, White (each color on both sides) — are hinged together in the net shown and then folded into a cube. Which color ends up on the face opposite White?

Givens: The net has six squares hinged in a fixed layout: bottom row of the net reads $R, B$ (with $B$ to the right of $R$); middle row reads $G, Y, O$ (with $G$ under $B$, $Y$ to the right of $G$, $O$ to the right of $Y$); $W$ sits below $Y$; Each colored square is the same color on the front and the back, so folding direction does not change the color; After folding, the six squares form the six faces of a cube; Answer choices: (A) B, (B) G, (C) O, (D) R, (E) Y

Plan

Primary tool: #17 Visualize Spatial Relationships

Secondary: #10 Create a Physical Representation, #3 Eliminate Possibilities

The problem hands us a 2D net and asks about a 3D folded cube — the canonical trigger for Tool #17 (Visualize Spatial Relationships). Pick one square as the base, then mentally fold the rest around it and track where each color lands. Tool #10 (Create a Physical Representation) is the safety net: anyone unsure of the mental fold can copy the net onto paper, label the colors, and physically fold it — touching the problem is encouraged for 3D geometry. Tool #3 (Eliminate Possibilities) lets us double-check by noticing that the three faces hinged directly to $W$ in the net ($Y$, plus the two faces reached through $Y$) must be adjacent to $W$, so they cannot be opposite to it — that crosses three answer choices off the list.

Execute — Answer: A

#17 Visualize Spatial Relationships 6.G.A.4 Step 1
  • Pick a base and orient the cube.
  • Choose $Y$ as the front face of the cube.
  • In the net, $G$ sits to the left of $Y$, $O$ sits to the right of $Y$, and $W$ sits directly below $Y$.
  • These three squares are hinged straight onto $Y$, so each folds up to become a side that touches $Y$.
$$\text{base} = Y \;\Rightarrow\; G,\,O,\,W \text{ each share a hinge with } Y$$

💡 Grade 6: a cube net is the unfolded surface of a cube. Whatever square shares a hinge with the base becomes one of the four faces touching the base.

#10 Create a Physical Representation 6.G.A.4 Step 2
  • Fold the three side faces.
  • Rotate $G$ backward along the $Y$-$G$ hinge — $G$ becomes the left face of the cube.
  • Rotate $O$ backward along the $Y$-$O$ hinge — $O$ becomes the right face.
  • Rotate $W$ upward along the $Y$-$W$ hinge — $W$ becomes the bottom face.
$$\text{Front}=Y,\;\text{Left}=G,\;\text{Right}=O,\;\text{Bottom}=W$$

💡 If you cut the net out of paper, this is what your hands do first: lift each neighbor of $Y$ until it stands up perpendicular to $Y$.

#17 Visualize Spatial Relationships 6.G.A.4 Step 3
  • Fold the remaining two faces, which travel together.
  • $B$ is hinged to the top edge of $G$, and $R$ is hinged to the top edge of $B$.
  • Once $G$ becomes the left face of the cube, the top edge of $G$ is exactly the edge where the left face meets the top face.
  • Folding $B$ over that edge places $B$ on top of the cube.
  • Then $R$ is hinged to $B$ on the edge opposite the $B$-$G$ hinge, which on the top face corresponds to the back edge.
  • Folding $R$ down from that back edge places $R$ on the back face of the cube.
$$\text{Top}=B,\;\text{Back}=R$$

💡 Chains of hinged squares fold like a hinge of a folding ruler: each new square swings around the next available outer edge of the partly-built cube.

#3 Eliminate Possibilities 6.G.A.4 Step 4
  • Read off the opposite face of $W$.
  • The three pairs of opposite faces are Top/Bottom, Front/Back, and Left/Right.
  • From the construction: Front $=Y$ and Back $=R$, Top $=B$ and Bottom $=W$, Left $=G$ and Right $=O$.
  • So the face opposite the White face is the Top face, which is $B$.
$$\text{Bottom}=W \;\Rightarrow\; \text{opposite face}=\text{Top}=B \;\Rightarrow\; \textbf{(A)}$$

💡 Cross-check with the answer choices: $Y$, $G$, $O$ all sit next to $W$ in the folded cube, so they cannot be opposite to $W$ — that eliminates (E), (B), (C). And $R$ shares the back-top edge with the top face $B$, so $R$ is adjacent to $B$, not equal to $B$ — that eliminates (D). Only (A) remains.

[1] #17 6.G.A.4 Pick a base and orient the cube. Choose $Y$ as the front face of the cube. In th
[2] #10 6.G.A.4 Fold the three side faces. Rotate $G$ backward along the $Y$-$G$ hinge — $G$ bec
[3] #17 6.G.A.4 Fold the remaining two faces, which travel together. $B$ is hinged to the top ed
[4] #3 6.G.A.4 Read off the opposite face of $W$. The three pairs of opposite faces are Top/Bot

Review

Reasonableness: Adjacency sanity check: $W$ is hinged to $Y$ in the net, so $W$ and $Y$ are adjacent (cannot be opposite). After folding, $W$ is the bottom face and touches every side face except the top, so $W$ is adjacent to $Y$ (front), $G$ (left), $O$ (right), and $R$ (back). The only face $W$ does not touch is the top face, which is $B$ — confirming (A). A second check uses a known cube-net rule: in a straight strip of three squares in a net, the two outer squares fold to opposite faces of the cube. The strip $W$-$Y$-(square above $Y$) in the net would put $W$ opposite that top square. The square above $Y$ in the original net is empty, but the chain $W \to Y \to G \to B$ shows that $B$ rises to the top and $W$ stays at the bottom, again giving $W$ opposite $B$.

Alternative: Tool #10 (Create a Physical Representation): copy the net onto paper, label each square with its color, cut it out, and fold it into a cube along the hinges. With the white square as the bottom, the square sitting on top is $B$ — answer (A). This is the lowest-effort method when mental folding feels uncertain.

CCSS standards used (min grade 6)

  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area (Reading the hinged 2D arrangement as a cube net and folding it (mentally and physically) to identify which colored face lands opposite the white face.)

⭐ Pin one square as the base, fold the rest around it, and read off the opposites — Grade 6 net-folding tells us the face across from $W$ is $B$.

⭐ Pin one square as the base, fold the rest around it, and read off the opposites — Grade 6 net-folding tells us the face across from $W$ is $B$.

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