26. On a plane, $N$ lines are drawn, no two of which are parallel and no three intersect at the same point. For which $N$ is it always possible to place one of the numbers $1,2, \ldots, N-1$ at each intersection point of the lines so that on any line all these numbers appear exactly once?

## 9th grade
89.26. Answer. This is only possible for even $N$. Consider all points marked with a one. Two lines pass through each of them, and therefore $N$ must be divisible by two.

Now let $N$ be even. Number the lines from 1 to $N$ and for any $i$ and $j$ less than $N$, at the intersection of the $i$-th and $j$-th lines, write the positive remainder of the number $i+j$ when divided by $N$ (i.e., instead of 0, we will write $N$). Near the intersection of the $N$-th line and the $i$-th line, write the remainder of the number $2i$ when divided by $N$.