Non-attacking knights and rooks on a chessboard

Question

This question from math contest olympiad phystech I tried to find the solution but I can't find it. Please help me to find solution of this problem. Given a board with a size of $11 × 11$ cells. Jack wants to put $N$ of rooks and $2N$ of horses on the board so that none of the pieces beats any other. At what greatest $N$ can he do this?

Answer 1

Since the $N$ rooks take away $N$ ranks and $N$ files, only $(11-N)^2\ge2N$ squares are remaining for the knights. $N=8$ leaves only $9$ squares, but $N=7$ leaves $16$, which is enough. Put rooks on c5, d4, e3, f6, g9, h8, and i7 and you could put knights on a1, a2, b1, b2, a10, a11, b10, b11, j1, j2, k1, k2, j10, j11, k10, and k11 and satisfy all constraints.

EDIT: Actually on a $k\times k$ board with $N$ rooks where $k\le2N+1\le2k+1$ you can always find squares for the $(k-N)^2$ knights: first fill the black long diagonal with rooks, then remove every other rook until only $N$ are left. All squares occupied by rooks or available to knights are now black squares so the knights, attacking only white squares, can't attack anything. E.g. on a $17\times17$ board put rooks on a1, b2, c3, d4, e5, f6, g7, h8, i9, j10, k11, l12, m13, n14, o15, p16, and q17. Then remove the a1, c3, e5, g7, and i9 rooks and you have $25$ black squares you can put knights on.