Sure, here's a possible ruleset for the new variant of chess played on a board with alternating black, white, and grey squares:

Board: The board consists of 8x8 alternating black, white, and grey squares, arranged in a repeating pattern.

Pieces: The pieces are the same as in standard chess, with the exception of the following rules for bishops:

Bishops: Bishops can move diagonally any number of spaces on black and white squares, but can only move onto and off of grey squares as if they were the king piece. Bishops cannot capture pieces on grey squares.
Objectives: The objective of the game is the same as in standard chess: to checkmate the opponent's king.

Movement: The movement of the pieces is the same as in standard chess, with the following modifications for bishops:

Bishops: Bishops can move diagonally any number of spaces on black and white squares, but can only move onto and off of grey squares as if they were the king piece. Bishops cannot capture pieces on grey squares.
Special Moves: The following special moves from standard chess are allowed:
En passant: If a pawn moves two squares on its first move and lands next to an opponent's pawn, the opponent's pawn can capture it as if it had moved only one square.

Castling: Kings can castle with their rooks once per game, provided that they meet the following conditions:

The king and the rook have not moved previously in the game.
There are no pieces between the king and the rook.
The king is not in check, and does not pass through or land on a square that is attacked by an opponent's piece.
Check and Checkmate: The rules for check and checkmate are the same as in standard chess.
Overall, this ruleset would create a new and interesting variant of chess that would require players to think strategically about the use of grey squares and the movements of their bishops. It would also add an additional defensive layer to the game, as players would need to protect their pieces from attacking bishops on the grey squares.
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