Problem 12 · 2015 AMC 8
Medium
Geometry & Measurement
counting-pairscube-structure
How many pairs of parallel edges, such as AB and GH, or EH and FG, does a cube have?
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Answer: C — 18 pairs.
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Hint 1 of 2
Every edge of a cube points along one of just three directions (left-right, front-back, up-down). Two edges are parallel exactly when they share a direction — so sort the 12 edges into three groups of 4 and only compare within a group.
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Hint 2 of 2
A 'pair' means choosing 2 of the 4 same-direction edges, with order not mattering — that's C(4,2) = 6 per group. Three groups gives the answer. (Choosing an unordered pair is the C(n,2) move; it's also why the count-each-twice method must divide by 2.)
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Approach: group edges by direction
- Sort the 12 edges by direction: 3 directions, 4 parallel edges each. Parallel pairs can only form within a direction-group, so the three groups don't interact.
- Within one group, a pair is any 2 of the 4 edges (order doesn't matter): C(4, 2) = 6.
- Total: 3 × 6 = 18.
- Why this transfers: 'how many unordered pairs share property X' → bucket the objects by X, then sum C(size, 2) over the buckets — no double counting to clean up.
Another way — double-count:
- Each of the 12 edges has 3 edges parallel to it.
- Total ordered pairs: 12 × 3 = 36; each unordered pair counted twice, so 36/2 = 18.
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