Exercise 6. Let $n$ be a non-zero positive integer. Suppose there exists a set of $n$ distinct lines in the plane such that each one intersects exactly ten others. What are the possible values for $n$?
Solution to Exercise 6 Let $S$ be the set of considered lines.

Consider any line $(d)$ in $S$. We then distinguish in $S \backslash\{(d)\}$ the $p$ lines parallel to $(d)$ in $S$ and the 10 lines in $S$ that intersect $(d)$.

We obtain the equality $n-1=p+10$ hence $p=n-11$.

The number $p$ of lines parallel to $(d)$ is therefore constant regardless of the choice of the line $(d)$! Any line $(d)$ in the set $S$ is thus parallel to exactly $n-11$ other lines (and intersects 10 other lines).

Let's group the lines in $S$ into bundles of parallel lines: by denoting $k$ as the number of different directions taken by the lines in $S$, we obtain the identity $n=k \cdot(1+(n-11))$ (each bundle contains exactly 1 line with its $n-11$ parallels).

Thus, $n$ is a multiple of $n-10$. Therefore, $n-10$ is a divisor of $n-(n-10)=10$.

The only divisors (positive) of 10 are $1,2,5,10$ so $n \in\{11 ; 12 ; 15 ; 20\}$.

Conversely, we can indeed construct sets of lines that satisfy the property that any line intersects exactly 10 other lines in the set, thanks to the properties demonstrated on these sets.

- For $n=11$, it suffices to take 11 lines in general position (no pair of lines is formed by two parallel lines).
- For $n=12$, it suffices to take 6 pairs of parallel lines such that no pair is parallel to another.
- For $n=15$, we use the same type of construction: we take 5 lines parallel to each other, 5 other lines parallel to each other but not parallel to the first ones, and 5 other lines parallel to each other that are not parallel to any of the previously constructed lines.
- Finally, for $n=20$, we take two bundles of 10 parallel lines such that two lines from two different bundles are never parallel.

The set of possible values for $n$ is therefore indeed $\{11 ; 12 ; 15 ; 20\}$ (we have not counted any extra values).