AMC 8 · 2023 · #17

Grade 6 geometry-3d

spatial-visualizationpolyhedron-netsface-adjacency physical-representationcasework ↑ Prerequisites: spatial-visualizationpolyhedron-nets

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$ ?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: A regular octahedron has eight equilateral-triangle faces, with four faces meeting at each vertex. The flat net shows seven numbered faces ($1$ through $7$) plus one face labeled $Q$. When the net is folded into the octahedron drawn on the right, face $Q$ becomes a slanted upward-pointing triangle on the front of the solid. Which numbered face ends up sharing the right-hand edge of $Q$ (the edge marked "$?$" in the 3D picture)?

Givens: The solid is a regular octahedron: $8$ congruent equilateral-triangle faces, $4$ faces meeting at every vertex; The net (left figure) contains faces $Q, 1, 2, 3, 4, 5, 6, 7$ — eight faces total; In the net, face $Q$ shares an edge with face $6$ and with face $7$, and has one outer (free) edge; In the 3D picture face $Q$ points upward in front; the unknown face $?$ sits on its right; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$

Unknowns: The numbered face that lies to the right of $Q$ on the finished octahedron

Understand

Plan

Primary tool: #10 Create a Physical Representation

Secondary: #17 Visualize Spatial Relationships, #2 Make a Systematic List, #3 Eliminate Possibilities

This is a classic "fold a net into a solid" puzzle, exactly what Tool #10 (Create a Physical Representation) is for: copy the net onto paper, cut it out, and fold — the face that lands on $Q$'s right edge appears in your hands. If no paper is available, walk through the same moves with Tool #17 (Visualize Spatial Relationships). Tool #2 (Make a Systematic List) keeps the search organized — face $Q$ has three edges, two of which are already glued (to $6$ and $7$), so we only need to chase **one** free edge around the net's perimeter. Tool #3 (Eliminate Possibilities) shows that several answer choices ($2, 3, 5$) are already locked into other faces' neighborhoods, leaving only $1$ as a feasible match.

Execute — Answer: A

#10 Create a Physical Representation 6.G.A.4 Step 1

Build the physical model.
Trace the net onto paper or stiff cardstock with the eight triangles labeled $Q, 1, 2, 3, 4, 5, 6, 7$ in their net positions, cut it out as one piece, and fold along every interior edge so each numbered face lifts off the table.
Watch face $Q$ keep its two in-net neighbors $6$ (on its left) and $7$ (on its bottom).
The only edge of $Q$ still loose is its right edge — and that is the edge the question is asking about.

💡 Building a 3D figure from its flat net is exactly the Grade 6 "represent 3D figures using nets" skill.

#2 Make a Systematic List 1.G.A.2 Step 2

Identify the perimeter faces.
Walk around the outside of the net and list every face whose outer (free) edge has not yet been glued to anything: $Q, 6, 4, 1, 2, 3, 5, 7$.
Each of these has one outside edge that must be sealed to another outside edge when the paper closes up into the octahedron.
Face $Q$'s free edge is therefore going to meet the free edge of exactly one face on this list.

$$\text{perimeter faces (with one free edge each):}\ Q,\ 6,\ 4,\ 1,\ 2,\ 3,\ 5,\ 7$$

💡 Listing every outside edge of the flat shape is the same composing-2D-and-3D-shapes idea kids do at Grade 1.

#3 Eliminate Possibilities 6.G.A.4 Step 3

Eliminate faces that are already locked into other neighborhoods.
Walk around the cap that $Q$ sits in: in the net, faces $Q \to 6 \to 4 \to 1$ all meet at one shared vertex (the top of $Q$'s cap), and the four free edges going around this cap must zip together — the free edge of $Q$ has to meet the free edge of $1$.
Faces $2, 3, 5$ live on the **opposite** cap of the octahedron and meet face $Q$ only at distant vertices, never along an edge, so each is eliminated.
Face $4$ is also on $Q$'s cap, but $4$'s free edge zips up with $6$'s, not with $Q$'s — so $4$ is eliminated too.
That leaves only face $1$.

$$\text{eliminated:}\ 2,\ 3,\ 4,\ 5\ \Rightarrow\ \text{remaining candidate:}\ 1$$

💡 Throwing out faces that obviously belong somewhere else on the solid leaves only one possibility — the Grade 6 "net of a 3D figure" idea makes this elimination concrete.

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

Confirm with the physical fold (or its mental version).
As the four perimeter edges of cap $Q$-$6$-$4$-$1$ rise off the table, they swing toward each other and meet at the top vertex of the octahedron.
From the viewing angle of the 3D picture, face $6$ is on $Q$'s left, face $4$ is across the cap (sharing a single vertex, not an edge, with $Q$), and face $1$ swings into place on $Q$'s right.
The four flat corners labeled $Q, 6, 4, 1$ all touch the same point — exactly the "$4$ faces meet at each vertex" property of the regular octahedron stated in the problem.

$$\text{cap vertex:}\ Q \leftrightarrow 6 \leftrightarrow 4 \leftrightarrow 1 \leftrightarrow Q$$

💡 Watching the four flat triangle-corners meet at one point uses the same "compare and analyze 3D shapes" reasoning kids start with in Kindergarten.

#3 Eliminate Possibilities K.OA.A.5 Step 5

Match the surviving candidate to the answer choices.
Face $1$ is the right-hand neighbor of $Q$, so the answer is choice (A).

$$\text{right neighbor of } Q = 1 \;\Rightarrow\; \textbf{(A)}$$

💡 Picking the single matching label from a short list is Kindergarten counting and matching.

[1] #10 6.G.A.4 Build the physical model. Trace the net onto paper or stiff cardstock with the e

[2] #2 1.G.A.2 Identify the perimeter faces. Walk around the outside of the net and list every

[3] #3 6.G.A.4 Eliminate faces that are already locked into other neighborhoods. Walk around th

[4] #17 K.G.B.4 Confirm with the physical fold (or its mental version). As the four perimeter ed

[5] #3 K.OA.A.5 Match the surviving candidate to the answer choices. Face $1$ is the right-hand

Review

Reasonableness: Sanity-check the geometry. A regular octahedron has $6$ vertices and at each vertex exactly $4$ faces meet, so the four faces $Q, 6, 4, 1$ sharing a vertex is consistent with the solid's structure. Each face of $Q$ should have exactly one neighbor across each of its three edges — in the net it already has $6$ and $7$, so it needs exactly one more, found here to be $1$. Faces $2, 3, 5$ together with $7$ form the **opposite** cap of the octahedron (the four faces meeting at the bottom vertex), so they cannot touch $Q$ along an edge — exactly matching the elimination step.

Alternative: Tool #1 (Draw a Diagram) gives a paper-free alternative: re-draw the net and color face $Q$ red and face $5$ yellow. Trace which faces share at least a single vertex with $5$ — you will find that $5$ shares a vertex with $7, Q, 6, 4, 3, 2$ but **not** with face $1$. Since on the octahedron the face directly opposite $5$ (sharing nothing with it, not even a vertex) is the face on the other side of $Q$, face $1$ is the only candidate, again giving (A).

CCSS standards used (min grade 6)

K.OA.A.5 Fluently add and subtract within 5 (Counting down the five answer choices and picking the single matching one once all others have been eliminated.)
K.G.B.4 Analyze and compare two- and three-dimensional shapes (Watching how the four flat triangle-corners $Q, 6, 4, 1$ come together at one vertex when the cap of the octahedron is folded up.)
1.G.A.2 Compose two-dimensional shapes or three-dimensional shapes (Listing all eight perimeter faces of the net whose outer edges must close up when the flat shape is composed into the octahedron.)
6.G.A.4 Represent three-dimensional figures using nets and find surface area (Treating the flat net as a representation of the octahedron, recognizing which net-edges glue together when folded, and identifying the face that ends up adjacent to $Q$ across its free edge.)

⭐ This AMC 8 problem only needs the Grade 6 idea of "a flat net is a 3D figure waiting to be folded" you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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