Answers.Body: <p>It's in theory possible if the king starts right next to the knights. I guess that means the &quot;number&quot; is 1.</p>
  <pre><code>[fen &quot;8/8/8/3k4/2NNN3/3N1N2/3N4/4K3 w - - 0 1&quot;]
  
  1. Nb4#
  </code></pre>
  
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Answers.Text: You can remove the left-most knight.
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Answers.Body: <p>If the king is at least three squares away from the infinite east-west line of knights then imagine the king starts running away to the north. At some point, after making n moves the king must be prevented from moving north by having all three squares to the north blocked. This will require at least two knights which would have to have made about 2*n/2 = n moves.</p>
  <p>But in this case, the king can capture one of the knights instead. So there must be three knights moving north instead, which would take about 3*n/2 &gt; n moves, so the king can outrun them.</p>
  <p>The sacrifice of the knight is never advantageous for the pursuers because it’s cost about n/2 moves for the sacrificial knight to move up but it only cost 2 moves for the king to eat it.</p>
  <p>So the king cannot be captured even by an infinite line of knights.</p>
  
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Answers.Text: Very good. Then, of course, one's mind wanders to "suitably curved" lines of knights, to approximately beat that advantage. :)  But, yes, very good.
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Answers.Text: Excellent answer.  (+1)
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Answers.Text: What about a scenario where there are two infinite lines of knights a finite distance apart, and a king starts somewhere between them?  How densely packed would the knights have to be?
Answers.UserDisplayName: supercat
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Answers.Text: @supercat if the king is trapped/enclosed, you're done (you have infinite time to finish the mate). So whichever way the king goes towards, you need to be able to assemble a shield to make a barrier - prevent the king from going through the line. I'm guessing the further the king is to the knights, the less densely packed they need to be, since you can predict the impact point more in advance and defend accordingly.
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Answers.Text: @NikanaReklawyks: If horizontal east-west lines that are n units aparty to the north and south of the king had a knight on every space, one could build a line far enough to the west that it would be complete before the king could get there, and then build a line to the east, and then inch the lines toward each other.  If, however, there was only a knight on every tenth space, then if the king started N spaces from the line, there would be 2N+1 squares at which the king could reach the line, and it's not obvious that line could be made solid at any position the king might reach.
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Answers.Text: I think your argument for why the king can escape the single infinite line of knights can be done simpler (at least I think it's simpler). The amount of movement north that the king has done is always at least as much as the second most advanced horse (more precisely the mean of the two most advanced horses). And each move you do with a third knight just increases this lead by one square. So there can only ever be at most a single knight that has come far enough to cover spaces _north_ of the king.
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Body: <p>On an infinite (perhaps only in one direction?) chessboard, what kind of configuration of (possibly a very large number of) knights could checkmate a lone king? I'm really thinking of an infinite-in-two-directions chessboard... And the salient point in my mind is that, although an individual knight can move faster than a king, a larger group of knights cannot. So, for example, it seems to me that an infinite line of knights could not checkmate a king that was (what number?) a few squares away from that line.</p>
  <p>I'm quite curious as to whether there's a conceptual way to quantify this.</p>
  
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Comments.Text: Did you look at the related question of how many knights aranged in a loose circle around the king it takes to mate the king?
Comments.UserDisplayName: quarague
Tags: <analysis><checkmate><chess-variants><mathematics><knights>
Title: When can legions of knights on an infinite chessboard checkmate a lone king?
ThreadText: # When can legions of knights on an infinite chessboard checkmate a lone king?
  
  (Asked by: username_0 on 2024-03-15)
  
  On an infinite (perhaps only in one direction?) chessboard, what kind of configuration of (possibly a very large number of) knights could checkmate a lone king? I'm really thinking of an infinite-in-two-directions chessboard... And the salient point in my mind is that, although an individual knight can move faster than a king, a larger group of knights cannot. So, for example, it seems to me that an infinite line of knights could not checkmate a king that was (what number?) a few squares away from that line.
  
  I'm quite curious as to whether there's a conceptual way to quantify this.
  
  **Tags:** analysis, checkmate, chess-variants, mathematics, knights
  
  > **Comments:**
  > **username_1**: Did you look at the related question of how many knights aranged in a loose circle around the king it takes to mate the king?  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_4 on 2024-03-15. Score: 16 | Accepted answer
  
  If the king is at least three squares away from the infinite east-west line of knights then imagine the king starts running away to the north. At some point, after making n moves the king must be prevented from moving north by having all three squares to the north blocked. This will require at least two knights which would have to have made about 2\*n/2 = n moves.
  
  But in this case, the king can capture one of the knights instead. So there must be three knights moving north instead, which would take about 3\*n/2 > n moves, so the king can outrun them.
  
  The sacrifice of the knight is never advantageous for the pursuers because it’s cost about n/2 moves for the sacrificial knight to move up but it only cost 2 moves for the king to eat it.
  
  So the king cannot be captured even by an infinite line of knights.
  
  > **Comments:**
  > **username_0**: Very good. Then, of course, one's mind wanders to "suitably curved" lines of knights, to approximately beat that advantage. :)  But, yes, very good.  
  > **username_5**: Excellent answer.  (+1)  
  > **username_6**: What about a scenario where there are two infinite lines of knights a finite distance apart, and a king starts somewhere between them?  How densely packed would the knights have to be?  
  > **username_7**: @username_6 if the king is trapped/enclosed, you're done (you have infinite time to finish the mate). So whichever way the king goes towards, you need to be able to assemble a shield to make a barrier - prevent the king from going through the line. I'm guessing the further the king is to the knights, the less densely packed they need to be, since you can predict the impact point more in advance and defend accordingly.  
  > **username_6**: @NikanaReklawyks: If horizontal east-west lines that are n units aparty to the north and south of the king had a knight on every space, one could build a line far enough to the west that it would be complete before the king could get there, and then build a line to the east, and then inch the lines toward each other.  If, however, there was only a knight on every tenth space, then if the king started N spaces from the line, there would be 2N+1 squares at which the king could reach the line, and it's not obvious that line could be made solid at any position the king might reach.  
  > **username_8**: I think your argument for why the king can escape the single infinite line of knights can be done simpler (at least I think it's simpler). The amount of movement north that the king has done is always at least as much as the second most advanced horse (more precisely the mean of the two most advanced horses). And each move you do with a third knight just increases this lead by one square. So there can only ever be at most a single knight that has come far enough to cover spaces _north_ of the king.  
  
  ---
  
  ### Answer by username_2 on 2024-03-18. Score: 3
  
  It's in theory possible if the king starts right next to the knights. I guess that means the "number" is 1.
  [fen "8/8/8/3k4/2NNN3/3N1N2/3N4/4K3 w - - 0 1"]
  
  1. Nb4#
  
  > **Comments:**
  > **username_3**: You can remove the left-most knight.  
  
  ---
  
  