Can you place 21 knights on a 11x11 chess board, such that every empty cell is under attack? Good luck!
Here is a similar question for 10x10: Knights covering a 10x10 chess board
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Sign up to join this communityCan you place 21 knights on a 11x11 chess board, such that every empty cell is under attack? Good luck!
Here is a similar question for 10x10: Knights covering a 10x10 chess board
You can solve the problem via integer linear programming as follows. Define a graph with one node per cell and an edge for each pair of cells that are a knight's move away from each other. For node $i\in N$, let $N_i \subset N$ be the neighbors of $i$, and let binary decision variable $x_i$ indicate whether node $i$ is selected. The problem is to minimize $\sum_{i\in N} x_i$ subject to $$x_i + \sum_{j\in N_i} x_j \ge 1 \quad \text{for $i\in N$}.$$ The constraint enforces that either node $i$ or one of its neighbors is selected.
Here's one optimal solution:
. . . . . . . . . . . . . X . . . . . X . . . X X X . . X X X X . . . . . . . . . . . . . . . . X . . . . . . . . . . . . . . . . . . . X . . . . . . . . . . X . X . . . X X . . . X . . . . . X X . . . X . X . . X . . . . . . . . . . . . . .
The minimum such domination number for an $n \times n$ board is in OEIS A006075.
@RobPratt's answer is correct. Here are the alternative arrangements I've found, all of which are only slightly different from each other.
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Picking exactly one white knight from each color group will lead to a valid arrangement, giving a total of $5 \cdot 5 \cdot 2 \cdot 2 \cdot 8 = 100$ arrangements unique up to reflection and rotation.