AMC 8 · 2015 · #12
Grade 4 geometry-3dcountingProblem
How many pairs of parallel edges, such as and or and , does a cube have?
Pick an answer.
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
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) $36$
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
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 $4$.
💡 Touching the cube makes the three directions obvious — and "same direction" is exactly what "parallel" means.
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.
💡 Splitting one hard count into three identical easy counts is the Tool #7 subproblems move.
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.
💡 A systematic list with a clear ordering rule is the safest way to count pairs without double-counting.
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.
💡 "$3$ groups of $6$" is exactly what multiplication means in Grade 3.
4.G.A.1 Pick up a tissue box or any cube-shaped object and look at its $12$ edges. They 4.OA.A.3 Now treat one direction at a time as its own little problem. The question "how m 4.OA.A.3 List the pairs inside one group. Label the $4$ edges in a single direction as $1 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.1Interpret 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.1Draw 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.3Solve 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!
⭐ 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!
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