# One knight and one queen on an infinite chess-board (a simple game)

On an infinite chessboard player A places the queen wherever he wants. Then player B places the knight wherever he wants. Then the game starts.

One rule is that the square where player A places his queen in the game is removed from the chessboard.

Can A always check-mate B (the both are playing their best strategies)?

A check-mate here is the position in which A attacks knight of B but knight of B has nowhere to jump because the squares on which he can jump are all removed.

The second rule is that queen cannot "go over" (cannot "hover over") the removed squares and cannot attack the knight "over the removed squares".

The third rule is that knight cannot capture the queen.

So, queen should be careful in she´s strategy because if she removes "too much" squares that would restrict possible movements of her over the board but she can travel long distances.

On the other hand, knight jumps only short distances.

• Well, $A$ can start in the upper left hand corner and systematically delete squares in a spiral towards the center, trapping the knight. – lulu Jun 30 at 17:04
• @lulu What is the meaning of "upper left hand corner" if we are on the infinite chessboard? – Grešnik Jun 30 at 17:05
• Ah, I missed that it was infinite. So, then, this is a variation of Conway's Angel Game. Hence, possibly difficult. – lulu Jun 30 at 17:08
• @lulu But decidable? – Grešnik Jun 30 at 17:15
• Not sure what you mean. There must be a winning strategy for one or the other player. But, Conway's problem is fairly hard. I have not reviewed any of the accepted proofs that the $2-$dim angel can always win (in your variant, the knight is similar to the angel), so that bodes well for your knight. Indeed, your devil (the queen) is quite a bit weaker than Conway's but then your knight is (slightly) weaker then his angel too. My guess is that your knight always wins, and I'd try to adapt one of the known resolutions of the Conway game. – lulu Jun 30 at 17:20