Answers.Body: <p>In my answer to <a href="https://math.stackexchange.com/questions/897092/who-has-a-winning-strategy-in-knight-and-why">the same question</a> over on math.stackexchange, I discussed general m by n boards. Here I will just give a winning strategy for Player 2 on the ordinary 8 by 8 chessboard. It's based on the following pairing of squares:</p>
  
  <p>a1c2, b1d2, c1a2, d1b2, e1g2, f1h2, g1e2, h1f2,<br>
  a3c4, b3d4, c3a4, d3b4, e3g4, f3h4, g3e4, h3f4,<br>
  a5c6, b5d6, c5a6, d5b6, e5g6, f5h6, g5e6, h5f6,<br>
  a7c8, b7d8, c7a8, d7b8, e7g8, f7h8, g7e8, h7f8.</p>
  
  <p>That is, square a1 is paired with square c2, b1 is paired with d2, and so on. Note that the 64 squares are partitioned into nonoverlapping pairs, and each set of paired squares are a knight's move apart. The winning strategy for Player 2 is: WHEREVER PLAYER 1 PUTS THE KNIGHT, MOVE IT TO THE OTHER SQUARE IN THE SAME PAIR. For instance, if Player 1 starts the game by dropping the knight on e8, Player 2 replies by playing Ng7. If Player 1 now plays Ne6, Player 2 plays Ng5, and so on. Here is a sample game, with Player 1 making random moves, and Player 2 following the winning strategy described above:</p>
  
  <ol>
  <li>Ne8 Ng7 2. Ne6 Ng5 3. Nf3 Nh4 4. Nf5 Nh6 5. Nf7 Nh8 6. Ng6 Ne5 7. Nc6 Na5 8. Nb3 Nd4 9. Ne2 Ng1 10. Nh3 Nf4 11. Nh5 Nf6 12. Ng4 Ne3 13. Nc5 Na3 14. Nb1 Nd2 15. Nf1 Nh2 and Player 2 wins.</li>
  </ol>
  
  <p><strong>P.S.</strong> Of course the strategy still works if White is allowed to move the knight to <strong>any</strong> square not already visited, and only Black is limited to making knight moves.</p>
  
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-03-21T04:09:20.247Z
Answers.LastActivityDate: 2016-03-12T06:05:20.027Z
Answers.LastEditDate: 2017-04-13T12:19:24.717Z
Answers.UserDisplayName: bof
Answers.Body: <p>I would assume that the optimal game would follow the knight's tour. The comment be CognisMantis is correct and the game would last 64 half-moves or 32 complete moves. </p>
  
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2016-03-12T09:06:14.637Z
Answers.Text: Player 2 has a winning strategy even if Player 1 sabotages a complete knight's tour, see bof's answer.
Answers.UserDisplayName: Dag Oskar Madsen
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2016-03-12T07:59:21.737Z
Answers.LastActivityDate: 2016-03-12T07:59:21.737Z
Answers.UserDisplayName: Mike Jones
Body: <p>The rules of "knight" : Player 1 puts a knight on some square. Then the players,
   beginning with Player 2, alternately move the knight. The knight cannot be
   moved to a square already visited, including the starting square. The player,
   who cannot move the knight anymore, loses.</p>
  
  <p>Which player has a winning strategy ?</p>
  
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-03-20T18:30:57.477Z
Comments.Text: I would guess that the second player wins. Assume the knight starts out on a white square. Therefore the second player will always move the knight on the dark squares and the first player will always move the knight on the white squares.
Comments.UserDisplayName: CognisMantis
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-03-20T18:54:09.360Z
Comments.Text: @CognisMantis I tried this and the game went Ne1 Nc2 Nd4 Nb3? Na1! with the first player winning, so I guess some strategy is needed.
Comments.UserDisplayName: bof
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-03-20T19:10:15.397Z
Comments.Text: yeah, my assumption was that with ideal play, the game will go over all the squares. This assumption is likely wrong.
Comments.UserDisplayName: CognisMantis
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-03-20T19:15:08.513Z
Comments.Text: I suggest visiting the mathematics forum for this problem
Comments.UserDisplayName: CognisMantis
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-03-20T23:26:27.357Z
Comments.Text: I have done this, without any result so far. On a 4x4-board and on a 4x6-board, Player 2 has a winning strategy.
Comments.UserDisplayName: Peter
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-03-21T01:22:03.473Z
Comments.Text: If you have to post the same question in two different sites, shouldn't you at least include a [reference](http://math.stackexchange.com/questions/897092/who-has-a-winning-strategy-in-knight-and-why) from one to another? I posted an answer in math.se, am I supposed to post it here too?
Comments.UserDisplayName: bof
Tags: <chess-variants>
Title: Who wins in the game "knight" if both players play optimal?
ThreadText: # Who wins in the game "knight" if both players play optimal?
  
  (Asked by: username_0 on 2015-03-20)
  
  The rules of "knight" : Player 1 puts a knight on some square. Then the players,
  beginning with Player 2, alternately move the knight. The knight cannot be
  moved to a square already visited, including the starting square. The player,
  who cannot move the knight anymore, loses.
  
  Which player has a winning strategy ?
  
  **Tags:** chess-variants
  
  > **Comments:**
  > **username_1**: I would guess that the second player wins. Assume the knight starts out on a white square. Therefore the second player will always move the knight on the dark squares and the first player will always move the knight on the white squares.  
  > **username_2**: @username_1 I tried this and the game went Ne1 Nc2 Nd4 Nb3? Na1! with the first player winning, so I guess some strategy is needed.  
  > **username_1**: yeah, my assumption was that with ideal play, the game will go over all the squares. This assumption is likely wrong.  
  > **username_1**: I suggest visiting the mathematics forum for this problem  
  > **username_0**: I have done this, without any result so far. On a 4x4-board and on a 4x6-board, Player 2 has a winning strategy.  
  > **username_2**: If you have to post the same question in two different sites, shouldn't you at least include a [reference](http://math.stackexchange.com/questions/897092/who-has-a-winning-strategy-in-knight-and-why) from one to another? I posted an answer in math.se, am I supposed to post it here too?  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_2 on 2015-03-21. Score: 13
  
  In my answer to [the same question](https://math.stackexchange.com/questions/897092/who-has-a-winning-strategy-in-knight-and-why) over on math.stackexchange, I discussed general m by n boards. Here I will just give a winning strategy for Player 2 on the ordinary 8 by 8 chessboard. It's based on the following pairing of squares:
  
  a1c2, b1d2, c1a2, d1b2, e1g2, f1h2, g1e2, h1f2,  
  a3c4, b3d4, c3a4, d3b4, e3g4, f3h4, g3e4, h3f4,  
  a5c6, b5d6, c5a6, d5b6, e5g6, f5h6, g5e6, h5f6,  
  a7c8, b7d8, c7a8, d7b8, e7g8, f7h8, g7e8, h7f8.
  
  That is, square a1 is paired with square c2, b1 is paired with d2, and so on. Note that the 64 squares are partitioned into nonoverlapping pairs, and each set of paired squares are a knight's move apart. The winning strategy for Player 2 is: WHEREVER PLAYER 1 PUTS THE KNIGHT, MOVE IT TO THE OTHER SQUARE IN THE SAME PAIR. For instance, if Player 1 starts the game by dropping the knight on e8, Player 2 replies by playing Ng7. If Player 1 now plays Ne6, Player 2 plays Ng5, and so on. Here is a sample game, with Player 1 making random moves, and Player 2 following the winning strategy described above:
  
  1. Ne8 Ng7 2. Ne6 Ng5 3. Nf3 Nh4 4. Nf5 Nh6 5. Nf7 Nh8 6. Ng6 Ne5 7. Nc6 Na5 8. Nb3 Nd4 9. Ne2 Ng1 10. Nh3 Nf4 11. Nh5 Nf6 12. Ng4 Ne3 13. Nc5 Na3 14. Nb1 Nd2 15. Nf1 Nh2 and Player 2 wins.
  
  **P.S.** Of course the strategy still works if White is allowed to move the knight to **any** square not already visited, and only Black is limited to making knight moves.
  
  
  ---
  
  ### Answer by username_3 on 2016-03-12. Score: -1
  
  I would assume that the optimal game would follow the knight's tour. The comment be username_1 is correct and the game would last 64 half-moves or 32 complete moves.
  
  > **Comments:**
  > **username_4**: Player 2 has a winning strategy even if Player 1 sabotages a complete knight's tour, see bof's answer.  
  
  ---
  
  