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)

  • $\begingroup$ does 'infinite chessboard' mean it is infinite in all 4 directions? I.e. there is no corner or no 'back row' (which you can still have in an infinite board) $\endgroup$
    – JimN
    Apr 14, 2021 at 16:18
  • $\begingroup$ @JimN It's infinite in all 4 directions. This is actually a dual problem of the escaping fugitive problem, in discrete version. $\endgroup$
    – Eric
    Apr 14, 2021 at 16:22
  • 2
    $\begingroup$ As you mention the possibility of capturing a king with a king I take it that certain rules of chess are suspended (like a king cannot move into check)? $\endgroup$
    – loopy walt
    Apr 14, 2021 at 16:47
  • $\begingroup$ @loopywalt Yeah, let's ignore those laws. They're just officers and fugitive dressing and acting like chess kings ;-) $\endgroup$
    – Eric
    Apr 14, 2021 at 16:52
  • 2
    $\begingroup$ Who goes first? $\endgroup$
    – gbe
    May 18, 2021 at 19:40


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