AMC 10 · 2019 · #11
Grade 10 geometry-3dPick an answer.
Tool #17 (Visualize Spatial Relationships) comes first: this is a "how many" question, but nothing can be counted until we know which edge pairs are coplanar, and that is a picture-in-your-head fact about lines in space. Tool #7 (Identify Subproblems) then splits the good pairs into two clean cases — edges sharing a face, and parallel edges not sharing a face. Tool #2 (Make a Systematic List) does the actual counting inside each case and guarantees nothing is double counted. Tool #16 (Count the Complement) is held in reserve as an independent check: count the bad (skew) pairs instead and subtract.
When do two lines share a plane
Meeting or being parallel means one plane.
Think of laying a flat sheet of paper on the cube: it can rest on two edges that cross or on two edges that run side by side, but not on two edges that pass by each other at different heights.
Two lines share a plane exactly when they cross or when they run side by side.
▸ Why?
Two lines that never meet but keep a constant gap still lie on one flat sheet.
▸ Why?
Every pair falls into exactly one of the two cases, so the counts add with nothing missed.
Count all pairs first
Count all pairs first.
Count everything once, then decide what to remove — an upper bound catches an over-count immediately.
7.SP.C.8Make A Systematic ListCase 1: both edges on one face
Same-face pairs are most of them.
A face is already a plane, so every pair of edges drawn on it is free — no checking needed.
4.G.A.1Identify SubproblemsCase 2: far-apart parallel twins
The far parallel twins count too.
Every edge has one far-away twin pointing the same way, and parallel lines always share a plane even when they are nowhere near each other.
4.G.A.2Make A Systematic ListAdd the two cases
Adding the two cases gives 42.
Two buckets that cannot overlap and miss nothing means the totals simply add.
7.SP.C.8Make A Systematic ListTwo edges share a plane only if they cross or run parallel, so count the 36 pairs sitting on a single face, add the 6 parallel twins across the cube, and you get (D) 42 — the only idea beyond careful listing is knowing that skew edges have no plane.
- When do two lines share a plane
- Count all pairs first
- Case 1: both edges on one face
- Case 2: far-apart parallel twins
- Add the two cases
More like this
Same archetype — closest grade level first.
