Let $m,n,u,v \in \mathbb{N}$ be parameters with $m,n \geq 3$. Suppose two players play a game on a $m \times n$ chess board and we denote the squares of the board by the set of points $ (i,j) $ such that $1 \leq i \leq m, 1 \leq j \leq n $. Player 1 has $u$ queens, and player 2 has $v$ knights. The initial configuration is some given subset of the board where no piece is placed at a position that can be immediately attacked by another piece of the opposing player, and the player with at least one piece when the game terminates is the winner. Both players must make a move when it is their turn. The game terminates when either player captures all of the pieces of the opposing player. If Player 1 goes first, under what circumstances does either player have a winning strategy? What can be said about the probability as a function of the parameters given with both players playing optimally, that the game will terminate in finitely many steps?

I have obtained the result for $m = 3$ and arbitrary $n$ with $u= v = 1$. Ed Dean proved that Player 1 always has a winning strategy for arbitrary $m,n$ and $u = v = 1$ (see below), and he sketched a winning strategy for Player 1 in the $u = 1, v = 2$ case. He also gave an argument for the lack of a winning strategy for Player 1 for the $u = 1, v = 3$ case. All other cases are still unknown as of yet.

Edit: Previous suggestion on a way to show in the $u = v = 1$ case that the knight can avoid capture for sufficiently large m,n is removed.

Edit 2: Thanks to Ed Dean for resolving some cases, as indicated above.

Edit 3: Included a new question regarding a related probability distribution.

Edit 4: Moved edit 3 to a new question: A random variable in a game of knights and queens