How n Lines Can Meet¶
Pencil types of line arrangements and the integer sequences they generate.
Draw \(n\) straight lines on a sheet of paper, no two of them parallel. Every pair of lines meets somewhere, but the meeting points need not be distinct: three, four or more lines may pass through the same point. At each intersection point write down how many lines go through it. The resulting multiset of numbers is the pencil type of the drawing. The ten lines in the picture meet in one point on five lines, ten points on three lines and five ordinary crossings, so their type is \(\{5, 3^{10}, 2^5\}\). Point at an intersection to see which lines pass through it.
The question is as simple as it sounds: which pencil types occur, and how many are there for each \(n\)?
Three tests separate the possible from the impossible. Arithmetic first: a point on \(k\) lines uses up \(\binom k2\) of the \(\binom n2\) pairs of lines, so the pencil sizes must fit into the available pairs. Combinatorics second: two lines meet only once, so two pencils cannot share two lines. Geometry last: the lines have to exist in the real plane. Counting what survives each test gives three integer sequences, and a fourth one counts how many different numbers of intersection points \(n\) lines can have.
| sequence | \(n = 1, 2, 3, \dots\) | status |
|---|---|---|
| \(a(n)\), types allowed by arithmetic | 1, 1, 2, 4, 7, 15, 32, 66, 141, 295, 619, 1286, 2654, … | known: A072964, shifted by one |
| \(b(n)\), ways \(n\) lines can meet, consistent with "two lines meet only once" | 1, 1, 2, 3, 5, 9, 17, 29, 56, 106, 202, 384 | computed through \(n = 12\); not in the OEIS |
| \(c(n)\), ways \(n\) lines in the plane can meet | 1, 1, 2, 3, 5, 9, 16, 28, 53, 98, 180, 336 | computed through \(n = 12\) (see the sequence page); not in the OEIS |
| \(d(n)\), different possible numbers of intersection points | 1, 1, 2, 3, 5, 9, 13, 18, 24, 30, 38, 46, 56 | computed through \(n = 13\); not in the OEIS |
That three sequences arising from a question a schoolchild can ask are missing from the On-Line Encyclopedia of Integer Sequences is the reason for this site. The missing entries are not an oversight that a formula would fix: the sequence \(c(n)\) contains the orchard problem of Sylvester (how many triple points can \(n\) lines have?), which was settled only for large \(n\) by Green and Tao in 2013.1 The list of realizable types determines the answer to it, so a complete description of that list, not just its size, would solve the orchard problem for every \(n\).
What is on this site¶
- The question, stated carefully, with the notation of Grünbaum's Arrangements and Spreads and an explanation of the three sieves.
- One page per sequence, with the status of every term: proved, computer-verified, bounded, or conjectured.
- Theorems and proofs: what is known classically, what we proved, and what the computer checked. Among the new results: for \(n \ge 13\) lines the number of intersection points is \(1\), \(n\), \(2n-4\), \(2n-2\), or at least \(2n-1\); in particular, for \(n \ge 6\), \(2n-3\) intersection points are impossible exactly when \(n \ge 12\).
- Algorithms and data: how the tests were computed, how every example is certified by exact arithmetic, and how to check everything independently.
- A gallery of every type known to be realizable for 1 to 14 lines (complete up to 12 lines), in symmetric drawings where we found one and from exact coordinates, and a page on what cannot be drawn and why.
- Open problems.
Status of this write-up¶
This write-up is still being extended. The questions were posed by Petra Radko; the mathematics, programs, computations and text are the work of Claude Code (see About). Everything stated as a theorem is either proved on the theorems page or cited with a full reference; everything computed can be reproduced from the repository, whose continuous integration re-verifies every certificate on every change.
How to cite¶
Petra Radko, How n Lines Can Meet: pencil types of line arrangements and the integer sequences they generate, 2026, https://gwynerva.github.io/line-pencils/ (code: https://github.com/Gwynerva/line-pencils).
Text, figures and data are licensed CC BY 4.0; code is MIT.
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B. Green, T. Tao, On sets defining few ordinary lines, Discrete & Computational Geometry 50 (2013), 409–468, doi:10.1007/s00454-013-9518-9. ↩