Lines Not Lining Up

For the number of lines needed to be as low as possible, each of them has to intersect as many lines as it can. If there are $n$ lines, each can intersect at most $n-1$ other lines. This generates $n-2$ segments between adjacent intersection points on every line. There will therefore be $n-3$ segments which each skip one of the intersections, $n-4$ of those which skip two intersections, and so on down to finally only one segment which connects the two most distant intersections. So to count all the segments on a given line, we need to add numbers from $1$ to $n-2$ . This is accomplished using the formula: $(n-1)\times(n-2)/2$ . Since there are $n$ lines in all, the number of segments for all of them is given by $n\times(n-1)\times(n-2)/2$ . This needs to be equal to 4,096,769,040. So our equation is:

$n\times(n-1)\times(n-2)=8,193,538,080$

This is a third degree equation, and we could try to solve it as such, but the three numbers ( $n, n-1$ , and $n-2$ ) are very close to each other compared to their size, so they are all close to the third root of the right side, which is equal to $2015.99983\approx 2016$ So our guess would be that the middle of the three numbers is indeed 2016, making the number of lines $n=2017$ . We can easily verify this guess by using 2017 in the equation above.