Problem 26 · 2013 Math Kangaroo
Stretch
Logic & Word Problems
careful-counting
Several straight lines are drawn in the plane. Line a intersects exactly three other lines, and line b intersects exactly four other lines. Line c intersects exactly n other lines with \(n \ne 3, 4\). How many lines were drawn?
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Answer: C — 6
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Hint 1 of 2
A line misses only the lines parallel to it; group lines by direction.
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Hint 2 of 2
Each intersection count tells you the size of that line's parallel class.
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Approach: parallel classes
- If there are N lines, a line in a class of size s meets N−s others.
- Line a: N−s_a = 3; line b: N−s_b = 4; line c needs a third class with n≠3,4.
- Classes of sizes 3,2,1 give N = 6 and c meets 5 lines, so C.
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