TITLE: Independence problem: one rook and maximum number of knights on the chessboard $8 \times 8$
QUESTION [10 upvotes]: On the chessboard $8 \times 8$ we can to place one rook and several knights. Find the maximum number of knights, which can be placed on a chessboard along with one rook so that none of the pieces attack each other.
My work. If a rook is placed in a square a8, then we can place the $25$ knights in all the white squares that rook doesn’t attack. I do not know how to prove that $25$ is the maximum number of knights.

REPLY [2 votes]: In each of the 10 cases of rook placement considered by @ArsenBerk, you can get a certificate of optimality via linear programming duality.  After removing the squares attacked by the rook, introduce a decision variable $x_{i,j}$, bounded by 0 and 1, for each remaining square $(i,j)$.  The interpretation is that $x_{i,j}=1$ if and only if square $(i,j)$ contains a knight.  Now the linear programming problem is to maximize $\sum_{i,j} x_{i,j}$ subject to conflict constraints of the form $x_{i,j} + x_{i',j'} \le 1$, where squares $(i,j)$ and $(i',j')$ are a knight's move from each other.  In each case, the optimal objective value is 25, and the optimal dual variables indicate a partition of the board into $s$ single squares and $p$ pairs of conflicting squares with $s+p=25$.  Trivially, each square can contain at most one knight, and each pair can contain at most one knight, so this partition proves that at most 25 knights can be placed.  This idea is very similar to @ArsenBerk's partitions into larger subsets of squares, but here the partition is derived automatically as a by-product of the LP solve, and the certificate does not rely on optimal results for smaller boards.
For example, here is one such partition into 25 regions (21 pairs and 4 singletons) when the rook is placed in square b1:

 1  . 22  2  3  4 23  5 
 6  .  1  4  7  5  3  8 
24  .  6  9  2  8 10 11 
12  . 13  7 14 11 15 16 
13  . 12 17  9 18 19 10 
 .  .  . 14 20 21 16 15 
25  . 20  . 17 19 18 21 
 .  R  .  .  .  .  .  .