Exact Opponent Attacks

[!IMPORTANT] Colour the pieces black and white, and draw an edge whenever opposite-colour pieces attack each other. Then the puzzle becomes a bipartite graph problem with an exact degree condition.

Important properties

• If every piece attacks exactly $k$ opponent pieces, then every vertex has degree $k$.
• Because each edge joins one white piece to one black piece, the total edge count is both $kW$ and $kB$, where $W$ and $B$ are the numbers of white and black pieces. So $W=B$ whenever $k>0$.
• Knights give local sparse patterns, while queens create long line-of-sight interactions that can be controlled by blockers.
• The same graph idea explains the old degree-2 knight problems and the new knight-and-queen problems.

The challenge is not just counting. It is realizing the required regular bipartite graph inside the actual geometry of the chessboard.