AMC 8 · 2015 · #12

Easy mode Grade 4

Problem

Picture a cube. A cube has $12$ edges in total.

Two edges are a "parallel pair" if they point in the same direction and never touch each other. For example, in the figure, $\overline{AB}$ and $\overline{GH}$ are a parallel pair, and so are $\overline{EH}$ and $\overline{FG}$ .

How many different parallel pairs of edges does the cube have?

$\textbf{(A) }6\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad \textbf{(E) }36$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: A cube has $12$ edges. Count how many unordered pairs of those edges are parallel to each other — pairs like $\overline{AB}$ and $\overline{GH}$, or $\overline{EH}$ and $\overline{FG}$.

Givens: The shape is a cube, with $12$ edges total; Two edges count as a "parallel pair" if they point in the same direction (or exactly opposite); Pairs are unordered: $\{e_1, e_2\}$ is the same pair as $\{e_2, e_1\}$; Answer choices: (A) $6$, (B) $12$, (C) $18$, (D) $24$, (E) $36$

Unknowns: The total number of unordered parallel-edge pairs on the cube

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #10 Create a Physical Representation, #2 Make a Systematic List

Trying to scan all $12$ edges and check every pair would mean checking $66$ pairs — too many to do safely by eye. Tool #10 (Physical Representation) helps first: pick up any box or cube and look at it. The $12$ edges fall into exactly $3$ directions (length, width, height), with $4$ parallel edges in each direction. Tool #7 (Identify Subproblems) turns the big count into three identical mini-counts: "how many pairs from $4$ edges?" Tool #2 (Systematic List) handles each mini-count by listing the pairs in order, so we never double-count or miss one. Add the three mini-counts and we're done.

Execute — Answer: C

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

Pick up a tissue box or any cube-shaped object and look at its $12$ edges.
They sort into $3$ directions: $4$ edges run left–right, $4$ run front–back, and $4$ run up–down.
Edges in the same direction are parallel; edges in different directions are not.
So the $12$ edges split cleanly into $3$ groups of $4$.

$$12 \text{ edges} = 3 \text{ directions} \times 4 \text{ edges per direction}$$

💡 Touching the cube makes the three directions obvious — and "same direction" is exactly what "parallel" means.

#7 Identify Subproblems 4.OA.A.3 Step 2

Now treat one direction at a time as its own little problem.
The question "how many parallel pairs total?" becomes three copies of "how many pairs from $4$ edges?" plus a sum at the end.

$$\text{total pairs} = 3 \times (\text{pairs from one group of } 4)$$

💡 Splitting one hard count into three identical easy counts is the Tool #7 subproblems move.

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

List the pairs inside one group.
Label the $4$ edges in a single direction as $1, 2, 3, 4$ and list every unordered pair in order — pair the smallest with each bigger one, then the next smallest, and so on: $\{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\}$.
That's $6$ pairs, and the ordering rule guarantees no repeats and no misses.

$$\{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\} \;\Rightarrow\; 6 \text{ pairs}$$

💡 A systematic list with a clear ordering rule is the safest way to count pairs without double-counting.

#7 Identify Subproblems 3.OA.A.1 Step 4

Each of the $3$ directions contributes the same $6$ pairs, and a pair from one direction cannot equal a pair from another (the edges point different ways).
So the total is $3$ groups of $6$ pairs.

$$3 \times 6 = 18 \;\Rightarrow\; \textbf{(C)}$$

💡 "$3$ groups of $6$" is exactly what multiplication means in Grade 3.

[1] #10 4.G.A.1 Pick up a tissue box or any cube-shaped object and look at its $12$ edges. They

[2] #7 4.OA.A.3 Now treat one direction at a time as its own little problem. The question "how m

[3] #2 4.OA.A.3 List the pairs inside one group. Label the $4$ edges in a single direction as $1

[4] #7 3.OA.A.1 Each of the $3$ directions contributes the same $6$ pairs, and a pair from one d

Review

Reasonableness: Sanity check by counting differently. The $12$ edges form $66$ unordered pairs in all ($12 \times 11 / 2 = 66$). Of those, the parallel ones are $18$, the perpendicular ones (edges in two different directions) are $3 \times (4 \times 4) = 48$, and $18 + 48 = 66$. The bookkeeping closes, so $18$ is consistent. Also: $18$ sits squarely between the small answers ($6, 12$) and the inflated traps ($24, 36$). Choice (D) $24$ is what you get if you accidentally treat $(e_1, e_2)$ and $(e_2, e_1)$ as different pairs in $2$ groups; choice (E) $36$ does it for all three groups. Choice (A) $6$ counts only one group, and (B) $12$ confuses pairs with edges.

Alternative: Tool #1 (Draw a Diagram) gives a face-by-face version. A cube has $3$ pairs of opposite (parallel) faces. Each pair of opposite faces shares $4$ parallel edges going around them, contributing $\binom{4}{2} = 6$ parallel pairs in that direction. With $3$ direction-pairs of faces, total $= 3 \times 6 = 18$. Same answer, drawn instead of touched.

CCSS standards used (min grade 4)

3.OA.A.1 Interpret products of whole numbers (e.g., $5 \times 7$ as $5$ groups of $7$) (Reading $3 \times 6 = 18$ as "$3$ groups of $6$ pairs" when combining the three directional groups into the final total.)
4.G.A.1 Draw and identify points, lines, line segments, rays, angles, perpendicular and parallel lines in two-dimensional figures (Recognizing that two edges are parallel exactly when they point in the same direction — and sorting the cube's $12$ edges into the three directional groups based on that property.)
4.OA.A.3 Solve multistep word problems using the four operations (Splitting the count into three identical subproblems (one per direction), listing the $6$ pairs inside one group, and combining the partial results into one total.)

⭐ Pick up a box and look at its edges — they only point in $3$ directions, so the cube problem is really just "$6$ pairs, three times" — pure Grade 3 multiplication!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 4)

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