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Somewhere I came across a description of street chess played in India (I think) with a number of non-standard rules. One of them was that when a pawn reaches the 8th rank, it must promote according to its file: a/h: rook, b/g: knight, c/f: bishop, d/e: queen.

Is there a name for the variant comprising just this rule?

It would seem to be a pretty interesting, mild variant that would still be very "chess-like", but would potentially throw out the opening theory, and completely upset some endgame theory. (For instance, K vs KPP where those pawns are in the b and g files would be a draw, and passed pawns in the b, c, f or g files would be valuable, but not necessarily game-winning). And there could be interesting strategy around trying to get c or f pawns into the d or e files, even if sacrificing pieces to make it happen.

Have any serious tournaments been played in this variant?

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I have played such games with elderly retired soldiers. The variation is known as Indian chess, and apart from the promotion rule, the pawns always move one step. I am not sure if any tournaments have been played in this format. But what I remember is that fianchetto was often used. The game was slow, and pawn promotion was pretty infrequent. Having played international format, one felt that one is stuck in a dated enterprise.

You are right about the changes in the end game but one plays with what one has. The practice makes the game slower, thus, it can not have the usual international time limits as in international chess.

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Is there a name for the variant comprising just this rule?

While not "just" this rule to distinguish it from modern chess, Murray's A History of Chess describes this version, naming it Hindustani Chess. (Chapter 4, p. 81).

In that version, promotion is only possible if the player has already lost a piece "that the Pawn must adopt on reaching the eighth row". And in the case of bishops, the lost piece must be the one of the appropriate color (light square or dark square).

The game he describes also varies from modern chess in the lack of a double-step option for pawns and the lack of castling (instead the King is allowed once per game to move as a knight if he has not ever been checked so far, with a question as to whether or not it is legal to capture with such a move).

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