Competition · AMC preparation · step 4 of 4
AMC 8 · 2015 · #12
Grade 4 geometry-3dcounting
Pick an answer.
AMC 8 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Sort the 12 edges by direction
Grab any box: a cube's 12 edges point in just 3 directions, 4 per direction, so they split into 3 groups of 4.
Touching the cube makes the three directions obvious — and "same direction" is exactly what "parallel" means.
4.G.A.1Create A Physical RepresentationHandle one direction at a time
Treat one direction at a time: the whole count becomes three copies of the same small question, how many pairs from 4 edges?
Splitting one hard count into three identical easy counts is the Tool #7 subproblems move.
4.OA.A.3Identify SubproblemsList the pairs in one group
List one group's pairs in order: {1,2},{1,3},{1,4},{2,3},{2,4},{3,4} — that's 6 pairs, with none missed or repeated.
A systematic list with a clear ordering rule is the safest way to count pairs without double-counting.
Among the 4 parallel edges that point in one direction, there are exactly 6 different pairs of edges.
▸ Why?
Label the 4 edges 1,2,3,4 and list each pair with a fixed rule — always write the smaller label first — giving {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, which is 6 pairs with none missed and none repeated.
▸ Why?
Forcing the smaller label before the larger one makes every two-edge pair appear in the list exactly once, so the listed entries match the real pairs one for one.
▸ Why?
The pairs split by their smaller edge into separate groups of 3, 2, and 1 with no overlap, so the whole count is 3 + 2 + 1 = 6.
Multiply by the 3 directions
The 3 directions each give the same 6 non-overlapping pairs, so the total is 3 × 6 = 18 pairs.
"3 groups of 6" is exactly what multiplication means in Grade 3.
3.OA.A.1Identify SubproblemsPick 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!
- Sort the 12 edges by direction
- Handle one direction at a time
- List the pairs in one group
- Multiply by the 3 directions
A parent dashboard for the family lives at sensimlab.com.
More like this
Same problem type. Tags show what each one shares with this problem.
- AMC 8 2003 #13 Gr 4Similar level
Same sub-type: Grid / Solid Cell PositionSame techniqueSystematic Enumeration CaseworkFourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the…
- AMC 10 2023B #10 Gr 4Similar levelSame sub-type: Grid / Solid Cell PositionSame techniqueCasework
You are playing a game. A 2 × 1 rectangle covers two adjacent squares (oriented either horizontally or vertically) of a…
- AMC 8 2000 #11 Gr 4Similar levelSame techniqueSystematic Enumeration Casework
The number 64 has the property that it is divisible by its unit digit. How many whole numbers between 10 and 50 have thi…