Answers.Body: <p>The two irreversible moves can be proved irreversible (it is impossible to replicate that board state later in the game) like so:</p>
  
  <p>Assumptions:
   - Pawns can't move backward.
   - Every pawn move moves it forward.
   - Pieces can't be added to the board.
   - There is no way to create a pawn (unlike queens, bishops, knights, and rooks which can be created via pawn promotion).</p>
  
  <p>Firstly, captures can be proved irreversible because they always remove a piece from the board. If a board state has N pieces and one is removed you now have N-1 pieces. There is no way to increase the number of pieces so you can never return to N pieces.</p>
  
  <p>Secondly, since every pawn move moves it forward, a pawn can never return to its original position. The only way to have a pawn in that position later on is to have it be occupied by another pawn. But for another pawn to occupy the position <em>that</em> pawn has to move. There is only a finite number of pawns so after every pawn has moved to fill the position left by the previous pawn there are no pawns left to fill the position of the last pawn, preventing the board state from being reached again.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-02T05:07:00.283Z
Answers.Text: *I hope this gets accepted. +1!*
Answers.UserDisplayName: user45266
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-02T05:07:38.347Z
Answers.Text: @user45266 The question was asked 3 years ago, it's probably not going to be accepted haha.
Answers.UserDisplayName: Omegastick
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-02T05:11:16.493Z
Answers.Text: Hey, you never know. Nice answer though!
Answers.UserDisplayName: user45266
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Answers.CreationDate: 2018-11-01T09:55:39.643Z
Answers.LastActivityDate: 2018-11-01T09:55:39.643Z
Answers.UserDisplayName: Omegastick
Answers.Body: <p>The only 'formal proof' you need is a reference to the rule book of the appropriate governing body. In the US, this could be the USCF.  For international events, it would be FIDE.</p>
  
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-02-23T00:20:36.610Z
Answers.Text: I'm using the word "proof" in a mathematical sense
Answers.UserDisplayName: StaleMate
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-02-23T02:50:00.473Z
Answers.Text: My point is that there is no 'proof'. There is an arbitrary rule. This rule is well defined.
Answers.UserDisplayName: Tony Ennis
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-02-23T10:44:45.887Z
Answers.Text: Tony is absolutely right. Chess is a human construct and its rules a social convention. There are many different variants and there is no compulsion to play according to FIDE or USCF rules unless you are playing in one of their tournaments. There can be no proof in the sense of pure mathematics because the game and the rules are contingent.
Answers.UserDisplayName: Brian Towers
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-02-23T13:47:00.057Z
Answers.Text: I am looking for a formal proof that captures and pawn moves are irreversible.
Answers.UserDisplayName: StaleMate
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-02-23T14:10:13.717Z
Answers.Text: I strongly disagree that it is an "arbitrary rule". It is very clear that the rule is based on irreversibility as an indication of progress in the game.  I could write a proof myself but that wouldn't be acceptable to Wikipedia, being that it would be "original research"
Answers.UserDisplayName: StaleMate
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-02-23T17:48:39.703Z
Answers.Text: Why 50 moves?  Why not 51? 49? Variable depending on game length?  It's arbitrary.
Answers.UserDisplayName: Tony Ennis
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-02-23T17:50:55.967Z
Answers.Text: That part is arbitrary, agreed.  But not the "pawn moved or piece captured" part.
Answers.UserDisplayName: StaleMate
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-01T09:30:19.170Z
Answers.Text: OP has been asked to provide a reference for the *rationale* for that rule, not for the rule itself. The only way to do it would be obtaining an explicit statement by the person who introduced that rule (which, I agree, seems very hard to get): "yes, I confirm that it is what I was thinking about when I wrote the rule".
Answers.UserDisplayName: Federico Poloni
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Answers.CreationDate: 2015-02-22T17:04:32.077Z
Answers.LastActivityDate: 2015-02-22T17:04:32.077Z
Answers.UserDisplayName: Tony Ennis
Answers.Body: <p>Whoever gave the reason they did for removing your edit was being kind. Your supposed  rationale for the rule is not correct. The 50 move rule and the 3 fold repetition rule are there to cut short games which are going nowhere. </p>
  
  <p>If there are captures or pawn moves then the nature of the position is changing and there is the possibility of one side or the other winning or losing via these moves. In blocked positions where such moves are either impossible for either side or suicidal it makes no sense to just go on pushing wood. Irreversibility has nothing whatsoever to do with it.</p>
  
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-02-22T18:02:52.170Z
Answers.Text: "Your supposed rationale for the rule is nonsense." This seems too harsh to me. I agree completely with your statement that, "If there are captures or pawn moves then the nature of the position is changing and there is the possibility of one side or the other winning," but I would argue that it is exactly the *irreversibility* of such moves, as noted by the OP, that makes them signs of progress.  ... (cont'd)
Answers.UserDisplayName: ETD
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Answers.CreationDate: 2015-02-22T18:03:00.120Z
Answers.Text: (cont'd) ... Players can shuffle pieces around forever (ignoring repetition for the moment), but there is a clear finite limit on the number of pawn moves and captures that can be made (*because* these are irreversible), and that is why such moves makes sense as a measure of progress occurring in the game.
Answers.UserDisplayName: ETD
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2016-02-13T23:16:11.173Z
Answers.Text: The comments can be understood in relation to the 75 move rule (9.6), if not to the 50 move rule.  Two players who conspire to have an infinite game against an arbiter who wants it to be finite, will end up drawn after less than something like 130 * 75 moves.  This does relate to reversibility, but also to an intentionally low number of irreversible moves that the rule recognizes as a proof of progress.
Answers.UserDisplayName: Jirka Hanika
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2015-02-22T17:39:10.870Z
Answers.LastActivityDate: 2015-02-23T06:54:27.030Z
Answers.LastEditDate: 2015-02-23T06:54:27.030Z
Answers.UserDisplayName: Brian Towers
Body: <p>I recently edited the Wikipedia article <a href="https://en.wikipedia.org/wiki/Fifty-move_rule" rel="nofollow">50 Move Rule</a> as follows:    </p>
  
  <blockquote>
    <p>The rationale for this rule is that pawn moves and captures are the
    only '''irreversible''' types of moves.  That is, after each pawn move
    or capture, it is impossible for any previous position in the game to
    be repeated.</p>
  </blockquote>
  
  <p>Unfortunately, my edit was quickly removed with the comment: <code>This is unreferenced. Come up with a reference if you can.</code></p>
  
  <p>I think this is pretty clear just by thinking about it, but is there any "formal proof" published anywhere that what I tried to add to the article is true?</p>
  
  <p><strong>EDIT:</strong> The words "the only" should be deleted.  In addition to captures and pawn moves, losing the right to castle on either or both sides (by moving a rook or king), or failing to capture a certain pawn "en passant" (thus giving up the right to do so later) are also irreversible.  These logically should also reset the 50-move count but are omitted from the rules for reasons of simplicity.</p>
  
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-02-22T17:19:30.783Z
Comments.Text: For which part of what you tried to add are you seeking proof of its truth? That those are the only sorts of irreversible moves, or that that is really the rationale for the rule? Either way, your title is a little misleading and might be best reworded to indicate which sort of proof you are looking for.
Comments.UserDisplayName: ETD
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-02-22T19:42:14.623Z
Comments.Text: Castling and declining to capture en-passant are also irreversible moves. In chess problems, the 50-move rule counts castling as a way of resetting the counter, see Codex at http://www.saunalahti.fi/~stniekat/pccc/codex.htm
Comments.UserDisplayName: phs
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-02-23T07:15:13.830Z
Comments.Text: If you want a moment of glory editing the Wikipedia entry for the 50 move rule then you should know that this phrase in the current article is wrong - "Theoretically, a game could continue indefinitely this way". The introduction of the 75 move rule (a 75 move equivalent to the 50 move rule which allows the arbiter to intervene and declare the game a draw) in the July 1 2014 amendments means that in practice the game would not continue indefinitely even if both players were incalcitrant (neither player prepared to accept a draw) or ignorant (players don't know the rules).
Comments.UserDisplayName: Brian Towers
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-02-23T18:20:32.007Z
Comments.Text: The question sounds like you want references to the fact that what you wrote about irreversibility _is the rationale for this rule_, but your comments to Tony Ennis's anwer below seem to indicate that you want a proof that what you wrote about irreversibility _is true_. Which one do you want?
Comments.UserDisplayName: JiK
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2015-02-23T18:38:20.093Z
Comments.Text: Yeah, you're right... I think I need both
Comments.UserDisplayName: StaleMate
Tags: <rules><50-move-rule>
Title: Proof of irreversibility of 50 move rule
ThreadText: # Proof of irreversibility of 50 move rule
  
  (Asked by: username_0 on 2015-02-22)
  
  I recently edited the Wikipedia article [50 Move Rule](https://en.wikipedia.org/wiki/Fifty-move_rule) as follows:
  
  > The rationale for this rule is that pawn moves and captures are the
  > only '''irreversible''' types of moves. That is, after each pawn move
  > or capture, it is impossible for any previous position in the game to
  > be repeated.
  
  Unfortunately, my edit was quickly removed with the comment: `This is unreferenced. Come up with a reference if you can.`
  
  I think this is pretty clear just by thinking about it, but is there any "formal proof" published anywhere that what I tried to add to the article is true?
  
  **EDIT:** The words "the only" should be deleted. In addition to captures and pawn moves, losing the right to castle on either or both sides (by moving a rook or king), or failing to capture a certain pawn "en passant" (thus giving up the right to do so later) are also irreversible. These logically should also reset the 50-move count but are omitted from the rules for reasons of simplicity.
  
  **Tags:** rules, 50-move-rule
  
  > **Comments:**
  > **username_1**: For which part of what you tried to add are you seeking proof of its truth? That those are the only sorts of irreversible moves, or that that is really the rationale for the rule? Either way, your title is a little misleading and might be best reworded to indicate which sort of proof you are looking for.  
  > **username_2**: Castling and declining to capture en-passant are also irreversible moves. In chess problems, the 50-move rule counts castling as a way of resetting the counter, see Codex at http://www.saunalahti.fi/~stniekat/pccc/codex.htm  
  > **username_3**: If you want a moment of glory editing the Wikipedia entry for the 50 move rule then you should know that this phrase in the current article is wrong - "Theoretically, a game could continue indefinitely this way". The introduction of the 75 move rule (a 75 move equivalent to the 50 move rule which allows the arbiter to intervene and declare the game a draw) in the July 1 2014 amendments means that in practice the game would not continue indefinitely even if both players were incalcitrant (neither player prepared to accept a draw) or ignorant (players don't know the rules).  
  > **username_4**: The question sounds like you want references to the fact that what you wrote about irreversibility _is the rationale for this rule_, but your comments to username_7's anwer below seem to indicate that you want a proof that what you wrote about irreversibility _is true_. Which one do you want?  
  > **username_0**: Yeah, you're right... I think I need both  
  
  ---
  
  ## 3 answers
  
  ### Answer by username_5 on 2018-11-01. Score: 8
  
  The two irreversible moves can be proved irreversible (it is impossible to replicate that board state later in the game) like so:
  
  Assumptions:
  - Pawns can't move backward.
  - Every pawn move moves it forward.
  - Pieces can't be added to the board.
  - There is no way to create a pawn (unlike queens, bishops, knights, and rooks which can be created via pawn promotion).
  
  Firstly, captures can be proved irreversible because they always remove a piece from the board. If a board state has N pieces and one is removed you now have N-1 pieces. There is no way to increase the number of pieces so you can never return to N pieces.
  
  Secondly, since every pawn move moves it forward, a pawn can never return to its original position. The only way to have a pawn in that position later on is to have it be occupied by another pawn. But for another pawn to occupy the position *that* pawn has to move. There is only a finite number of pawns so after every pawn has moved to fill the position left by the previous pawn there are no pawns left to fill the position of the last pawn, preventing the board state from being reached again.
  
  > **Comments:**
  > **username_6**: *I hope this gets accepted. +1!*  
  > **username_5**: @username_6 The question was asked 3 years ago, it's probably not going to be accepted haha.  
  > **username_6**: Hey, you never know. Nice answer though!  
  
  ---
  
  ### Answer by username_7 on 2015-02-22. Score: 3
  
  The only 'formal proof' you need is a reference to the rule book of the appropriate governing body. In the US, this could be the USCF. For international events, it would be FIDE.
  
  > **Comments:**
  > **username_0**: I'm using the word "proof" in a mathematical sense  
  > **username_7**: My point is that there is no 'proof'. There is an arbitrary rule. This rule is well defined.  
  > **username_3**: Tony is absolutely right. Chess is a human construct and its rules a social convention. There are many different variants and there is no compulsion to play according to FIDE or USCF rules unless you are playing in one of their tournaments. There can be no proof in the sense of pure mathematics because the game and the rules are contingent.  
  > **username_0**: I am looking for a formal proof that captures and pawn moves are irreversible.  
  > **username_0**: I strongly disagree that it is an "arbitrary rule". It is very clear that the rule is based on irreversibility as an indication of progress in the game.  I could write a proof myself but that wouldn't be acceptable to Wikipedia, being that it would be "original research"  
  > **username_7**: Why 50 moves?  Why not 51? 49? Variable depending on game length?  It's arbitrary.  
  > **username_0**: That part is arbitrary, agreed.  But not the "pawn moved or piece captured" part.  
  > **username_8**: OP has been asked to provide a reference for the *rationale* for that rule, not for the rule itself. The only way to do it would be obtaining an explicit statement by the person who introduced that rule (which, I agree, seems very hard to get): "yes, I confirm that it is what I was thinking about when I wrote the rule".  
  
  ---
  
  ### Answer by username_3 on 2015-02-22. Score: 3
  
  Whoever gave the reason they did for removing your edit was being kind. Your supposed rationale for the rule is not correct. The 50 move rule and the 3 fold repetition rule are there to cut short games which are going nowhere.
  
  If there are captures or pawn moves then the nature of the position is changing and there is the possibility of one side or the other winning or losing via these moves. In blocked positions where such moves are either impossible for either side or suicidal it makes no sense to just go on pushing wood. Irreversibility has nothing whatsoever to do with it.
  
  > **Comments:**
  > **username_1**: "Your supposed rationale for the rule is nonsense." This seems too harsh to me. I agree completely with your statement that, "If there are captures or pawn moves then the nature of the position is changing and there is the possibility of one side or the other winning," but I would argue that it is exactly the *irreversibility* of such moves, as noted by the OP, that makes them signs of progress.  ... (cont'd)  
  > **username_1**: (cont'd) ... Players can shuffle pieces around forever (ignoring repetition for the moment), but there is a clear finite limit on the number of pawn moves and captures that can be made (*because* these are irreversible), and that is why such moves makes sense as a measure of progress occurring in the game.  
  > **username_9**: The comments can be understood in relation to the 75 move rule (9.6), if not to the 50 move rule.  Two players who conspire to have an infinite game against an arbiter who wants it to be finite, will end up drawn after less than something like 130 * 75 moves.  This does relate to reversibility, but also to an intentionally low number of irreversible moves that the rule recognizes as a proof of progress.  
  
  ---
  
  