# Proof of irreversibility of 50 move rule

I recently edited the Wikipedia article 50 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.

• 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.
– ETD
Feb 22 '15 at 17:19
• 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 saunalahti.fi/~stniekat/pccc/codex.htm
– phs
Feb 22 '15 at 19:42
• 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). Feb 23 '15 at 7:15
• 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?
– JiK
Feb 23 '15 at 18:20
• Yeah, you're right... I think I need both Feb 23 '15 at 18:38

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.

• I hope this gets accepted. +1! Nov 2 '18 at 5:07
• @user45266 The question was asked 3 years ago, it's probably not going to be accepted haha. Nov 2 '18 at 5:07
• Hey, you never know. Nice answer though! Nov 2 '18 at 5:11

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.

• I'm using the word "proof" in a mathematical sense Feb 23 '15 at 0:20
• My point is that there is no 'proof'. There is an arbitrary rule. This rule is well defined. Feb 23 '15 at 2:50
• 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. Feb 23 '15 at 10:44
• I am looking for a formal proof that captures and pawn moves are irreversible. Feb 23 '15 at 13:47
• That part is arbitrary, agreed. But not the "pawn moved or piece captured" part. Feb 23 '15 at 17:50

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.

• "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)
– ETD
Feb 22 '15 at 18:02
• (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.
– ETD
Feb 22 '15 at 18:03
• 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. Feb 13 '16 at 23:16