On an infinite chessboard there's a single white king and N black kings. The nearest black king must be K moves away from the white king. Given N, white dictates the value of (finite) K, then black places their kings.
Question 1: Can white always force a draw without capturing any black kings? (by forcing a draw I mean white king is never captured if the game goes on forever)
Question 2: If instead of kings we have power-one rooks, which can only move one square, can white rook avoid being captured without capturing black pieces?
Related: Can the fugitive escape? (continuous version dual problem)