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# Chess

Chess is no joke: it has more possible sequences of moves than the number of atoms in the observable universe, but working through chess puzzles is a great way to gain insightful strategies.

Above is a \(3\times3\) board with 4 knights two white knights and two black knights. As in a standard game of chess, the knight can move only two steps in the horizontal or vertical direction and then one step in the other direction for one move. Define an **action** as moving a knight of any color.

The objective of the game is to interchange the position of both the black and white knights while alternately moving a knight of different color. The final state of the board is:

Using only actions, what is the minimum number of **actions** required to complete the game?

Note: Only one king on the board with the opponent only having queens.

White and Black start at the beginning of the game. Black is to mirror the moves with the same type of chess piece that White makes. For instance, if White plays a4, then Black plays a5.

It's White's turn. What is the minimum number of moves to win the game?

\[\] **Bonus:** How many different mirror mates with the minimum moves are there?

What is the biggest number of White Rooks that can be placed on a chessboard in such a way that a Black King, also placed on the board, is not under attack? You can place the King wherever you think is more profitable.

**Bonus:** Generalize it for a \(m\times n\) chessboard.

Whose move is it now?

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