Exact Opponent Attacks

These problems use two colours of the same piece. Your goal is to place black and white pieces so that every chosen piece attacks exactly a specified number of opponent pieces.

This turns the board into a graph problem.

Attack Graph

Make a graph whose:

• left side is the white pieces;
• right side is the black pieces;
• edges connect opposite-colour pieces that attack each other.

Then the puzzle condition becomes:

• every vertex has degree exactly $k$.

So you are looking for a regular bipartite attack graph that can actually be realized on the board.

Why The Colours Must Balance

If every white piece has degree $k$, then the total number of edges is

$k \cdot (\text{number of white pieces}).$

If every black piece also has degree $k$, then the same edge count is

$k \cdot (\text{number of black pieces}).$

So for any nonzero $k$, the two colours must contain the same number of pieces.

What Changes From Knights To Queens

• Knights give a sparse local graph, so the main challenge is building repeating degree patterns.
• Queens give a dense line-of-sight graph, so the main challenge is using blockers and geometry to control who sees whom.

In both cases, the method is the same:

1. model the arrangement as a bipartite graph;
2. use degree counting to get structural constraints;
3. build a configuration that realizes the required degree.

Return to the Mathematics in Chess overview for the full collection.