Problem 30 · 2024 Math Kangaroo
Stretch
Counting & Probability
caseworkcareful-counting
Consider n different straight lines in a plane, labelled \(\ell_1, \ldots, \ell_n\). The line \(\ell_1\) intersects 5 of the other lines, the line \(\ell_2\) intersects 9 of the other lines, and the line \(\ell_3\) intersects 11 of the other lines. Which of the following is a possible value of n?
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Answer: B — 12
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Hint 1 of 2
A line misses only the lines parallel to it, so 'intersects m others' means it has \(n-1-m\) lines parallel to it.
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Hint 2 of 2
Since one line already intersects 11 others, n cannot be smaller than 12, so test n = 12 first.
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Approach: turn intersection counts into parallel-class sizes and check the smallest n
- A line parallel to none of the others would meet all \(n-1\); since \(\ell_3\) meets 11, we need \(n-1\ge 11\), so \(n\ge 12\).
- Try \(n=12\): then \(\ell_3\) is in a direction by itself (meets all 11), \(\ell_2\) has \(12-1-9=2\) lines parallel to it, and \(\ell_1\) has \(12-1-5=6\) lines parallel to it, so the directions group as \(7+3+1+1=12\) lines.
- That grouping is consistent, so \(n=12\) works, answer B.
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