Answers.Body: <p>Maximum:</p>
  <pre><code>[FEN &quot;PppppppP/pppppppp/8/8/8/8/PPPPPPPP/pPPPPPPp w KQkq - 0 1&quot;]
  </code></pre>
  <p>The board must be right side up because the armies cannot switch sides
  without some capturing.</p>
  <p>Given this, all the pawns must be on their initial squares.
  Since nothing has been captured, the only pieces that could have moved are
  the Rooks (each shuffling between what used to be called its R1 and N1)
  and the Knights.  So each of the pieces in the corner is a Knight,
  and each piece on a Knight's square is a Rook.</p>
  <p>Moreover we can tell that it is White's move (by the usual parity trick)
  and that there are no castling rights.</p>
  <pre><code>[FEN &quot;NrbqkbrN/pppppppp/8/8/8/8/PPPPPPPP/nRBQKBRn w KQkq - 0 1&quot;]
  </code></pre>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-06-26T23:54:21.083Z
Answers.Text: Perhaps it's nice to move the rooks to demonstrate that that possibility was taken into account, but a slightly more straightforward solution might be to leave the rooks alone and just switch the knight colors.
Answers.UserDisplayName: A. Rex
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-06-27T03:21:31.673Z
Answers.Text: @A.Rex I think the point is that if the rooks are on the starting squares you can't distinguish between the positions with and without castling rights.
Answers.UserDisplayName: Chessanator
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-06-28T12:03:45.290Z
Answers.Text: Just to point other solutions are possible, I believe that the same position with wPa2, wRa1, bNb1 instead of wPa2, wRb1, bNa1 (and the other bN could be on a1 instead on h1) also works...
Answers.UserDisplayName: Evargalo
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-06-26T01:20:57.177Z
Answers.LastActivityDate: 2024-06-26T01:20:57.177Z
Answers.UserDisplayName: Noam D. Elkies
Answers.Body: <p>I claim that Noam’s maximum is also a minimum. I.e. there is no unambiguous position with 3-31 checkers. If a unit is missing then it might be a pawn captured by a pawn and we can have up to 3 promotions resulting. Even if the other 6 columns are pristine, I see no way to constrain these promotions.</p>
  <p>Nor is there a trick available with just 2+1 checkers on the board.</p>
  <p>So 32 it is, although I think there is more variety possible in the positions than in Noam’s example. Rook pawns can be inched forward a square and knights can be scattered over the board.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-06-27T14:53:20.063Z
Answers.Text: Why not 16+1?  We'd have to assume the board orientation.  But we can still move the Rooks to remove castling rights; and we can make it the bare King's move by putting it in check.
Answers.UserDisplayName: Noam D. Elkies
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-06-27T16:46:42.783Z
Answers.Text: @NoamD.Elkies Because we have to explain 8 exact promotions
Answers.UserDisplayName: Laska
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-06-27T17:13:09.523Z
Answers.Text: Oh I see, once we get rid of most of Black's army there's nothing stopping one or more wP's from promoting and coming back to rank 1 or 2.
Answers.UserDisplayName: Noam D. Elkies
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-06-28T12:10:22.133Z
Answers.Text: "and knights can be scattered over the board" doesn't sound correct to me. If a token is not hidden behind the enemy pawn wall, it could a pawn whose spot is occupied by a knight (we can add more shuffling) : you cannot distinguish bPb4 & bNb7 from bPb7 & bNb4
Answers.UserDisplayName: Evargalo
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-06-27T14:17:47.193Z
Answers.LastActivityDate: 2024-06-27T14:17:47.193Z
Answers.UserDisplayName: Laska
Body: <p><a href="https://youtu.be/_Ntn4jEv7rE" rel="nofollow noreferrer">Inspired by a video</a> where top players guess which famous game a position is from, except pieces in the position are replaced by checker pieces (so they can see the piece color but not type). Of course they remember the games, but if it's a position you don't remember, in which cases can you deduce a position only from the piece colors and locations but not piece types? Assuming the position is reached legally, but you don't know who made the last move or whether the game ended or how it ended.</p>
  <p>If you don't need to deduce (don't know or care) who's to move or who has castling rights etc, and only want to know which pieces are represented by which checker pieces on the board. A minimum number of pieces solution might be with only two kings on the board. An obvious case might be the starting position if all pieces are replaced by checkers, but it's not possible to tell if a rook and knight have switched places. I think if the knights are placed in the middle in front of pawns and other pieces are in their starting positions, then it's unique, but I might be wrong. Are there other cases? What's the minimum and maximum number of pieces that can give a unique solution, except for the case with only two kings?</p>
  <p>Note: in the video they were often given hints, like which side of the board they are looking at (white/black), sometimes who's to move or what's the last piece that moved, and the type of one or two pieces, or even that white/black resigned afterwards. But considering that would be too many different questions to consider the different cases, I decided to only ask about the basic case - no hints and ignoring who's to move and other information like castling/en passant rights.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-06-25T21:55:56.330Z
Comments.Text: When you say "If you don't need to deduce who's to move...", do you also mean that you know who's to move?
Comments.UserDisplayName: Benjamin Wang
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-06-25T22:22:58.103Z
Comments.Text: Please limit yourself to one question per post.
Comments.UserDisplayName: Brian Towers
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-06-26T13:13:27.633Z
Comments.Text: Maybe I've misunderstood the "if the knights are placed in the middle in front of pawns and other pieces are in their starting positions, then it's unique" situation, but in this case, can you always distinguish the knights from the pawns? That is, without some extra clues (such as you get in Noam D. Elkies' answer) I think it can sometimes be the case that two pawns move out of the way, and then the knights move to where they were, and you can't tell that situation from the one where the pawns remain unmoved and just the knights are out?
Comments.UserDisplayName: psmears
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-06-27T01:42:31.670Z
Comments.Text: I think there are 2 different questions here, the title one (which is currently shown on HNQ) "*Can a unique position be deduced*" and the question body "*What's the minimum and maximum number of pieces that **can give a unique solution***". As far as I understand, Noam's answer only referred to the question in the body, but not the title, e.g. find a counterexample.
Comments.UserDisplayName: Andrew T.
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-06-27T22:52:20.383Z
Comments.Text: [Can Magnus Carlsen Win A Chess Game WITHOUT Knowing The Pieces? - 2023Jul26](https://www.youtube.com/watch?v=uFee4xB_P1I)
Comments.UserDisplayName: BCLC
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-06-27T22:55:08.320Z
Comments.Text: @BCLC I meant this one: https://youtu.be/_Ntn4jEv7rE
Comments.UserDisplayName: alices_and_bobs
Tags: <problems><mathematics><retrograde-analysis>
Title: Can a unique position be deduced if pieces are replaced by checkers (can see piece color but not type)
ThreadText: # Can a unique position be deduced if pieces are replaced by checkers (can see piece color but not type)
  
  (Asked by: username_0 on 2024-06-25)
  
  [Inspired by a video](https://youtu.be/_Ntn4jEv7rE) where top players guess which famous game a position is from, except pieces in the position are replaced by checker pieces (so they can see the piece color but not type). Of course they remember the games, but if it's a position you don't remember, in which cases can you deduce a position only from the piece colors and locations but not piece types? Assuming the position is reached legally, but you don't know who made the last move or whether the game ended or how it ended.
  
  If you don't need to deduce (don't know or care) who's to move or who has castling rights etc, and only want to know which pieces are represented by which checker pieces on the board. A minimum number of pieces solution might be with only two kings on the board. An obvious case might be the starting position if all pieces are replaced by checkers, but it's not possible to tell if a rook and knight have switched places. I think if the knights are placed in the middle in front of pawns and other pieces are in their starting positions, then it's unique, but I might be wrong. Are there other cases? What's the minimum and maximum number of pieces that can give a unique solution, except for the case with only two kings?
  
  Note: in the video they were often given hints, like which side of the board they are looking at (white/black), sometimes who's to move or what's the last piece that moved, and the type of one or two pieces, or even that white/black resigned afterwards. But considering that would be too many different questions to consider the different cases, I decided to only ask about the basic case - no hints and ignoring who's to move and other information like castling/en passant rights.
  
  **Tags:** problems, mathematics, retrograde-analysis
  
  > **Comments:**
  > **username_1**: When you say "If you don't need to deduce who's to move...", do you also mean that you know who's to move?  
  > **username_2**: Please limit yourself to one question per post.  
  > **username_3**: Maybe I've misunderstood the "if the knights are placed in the middle in front of pawns and other pieces are in their starting positions, then it's unique" situation, but in this case, can you always distinguish the knights from the pawns? That is, without some extra clues (such as you get in username_6' answer) I think it can sometimes be the case that two pawns move out of the way, and then the knights move to where they were, and you can't tell that situation from the one where the pawns remain unmoved and just the knights are out?  
  > **username_4**: I think there are 2 different questions here, the title one (which is currently shown on HNQ) "*Can a unique position be deduced*" and the question body "*What's the minimum and maximum number of pieces that **can give a unique solution***". As far as I understand, Noam's answer only referred to the question in the body, but not the title, e.g. find a counterexample.  
  > **username_5**: [Can Magnus Carlsen Win A Chess Game WITHOUT Knowing The Pieces? - 2023Jul26](https://www.youtube.com/watch?v=uFee4xB_P1I)  
  > **username_0**: @username_5 I meant this one: https://youtu.be/_Ntn4jEv7rE  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_6 on 2024-06-26. Score: 27
  
  Maximum:
  [FEN "PppppppP/pppppppp/8/8/8/8/PPPPPPPP/pPPPPPPp w KQkq - 0 1"]
  
  The board must be right side up because the armies cannot switch sides
  without some capturing.
  Given this, all the pawns must be on their initial squares.
  Since nothing has been captured, the only pieces that could have moved are
  the Rooks (each shuffling between what used to be called its R1 and N1)
  and the Knights.  So each of the pieces in the corner is a Knight,
  and each piece on a Knight's square is a Rook.
  Moreover we can tell that it is White's move (by the usual parity trick)
  and that there are no castling rights.
  [FEN "NrbqkbrN/pppppppp/8/8/8/8/PPPPPPPP/nRBQKBRn w KQkq - 0 1"]
  
  > **Comments:**
  > **username_7**: Perhaps it's nice to move the rooks to demonstrate that that possibility was taken into account, but a slightly more straightforward solution might be to leave the rooks alone and just switch the knight colors.  
  > **username_8**: @A.Rex I think the point is that if the rooks are on the starting squares you can't distinguish between the positions with and without castling rights.  
  > **username_9**: Just to point other solutions are possible, I believe that the same position with wPa2, wRa1, bNb1 instead of wPa2, wRb1, bNa1 (and the other bN could be on a1 instead on h1) also works...  
  
  ---
  
  ### Answer by username_10 on 2024-06-27. Score: 6
  
  I claim that Noam’s maximum is also a minimum. I.e. there is no unambiguous position with 3-31 checkers. If a unit is missing then it might be a pawn captured by a pawn and we can have up to 3 promotions resulting. Even if the other 6 columns are pristine, I see no way to constrain these promotions.
  
  Nor is there a trick available with just 2+1 checkers on the board.
  
  So 32 it is, although I think there is more variety possible in the positions than in Noam’s example. Rook pawns can be inched forward a square and knights can be scattered over the board.
  
  > **Comments:**
  > **username_6**: Why not 16+1?  We'd have to assume the board orientation.  But we can still move the Rooks to remove castling rights; and we can make it the bare King's move by putting it in check.  
  > **username_10**: @NoamD.Elkies Because we have to explain 8 exact promotions  
  > **username_6**: Oh I see, once we get rid of most of Black's army there's nothing stopping one or more wP's from promoting and coming back to rank 1 or 2.  
  > **username_9**: "and knights can be scattered over the board" doesn't sound correct to me. If a token is not hidden behind the enemy pawn wall, it could a pawn whose spot is occupied by a knight (we can add more shuffling) : you cannot distinguish bPb4 & bNb7 from bPb7 & bNb4  
  
  ---
  
  