I have two questions related to defining distances in chess. On this wiki page, this subject is briefly touched, by saying that the taxicab (also called Manhattan) distance is used for rooks, which as shown here, is given by: |a_2-_a_1|+|b_2-b_1| where (a1,b1) is the coordinate of first square, and (a2,b2) that of the second. Now this is easy to see because the chessboard can be mapped to xy coordinate system, with square a1 being (1,1), etc. But for the bishops, it says, that the taxicab distance is again used with the difference of only counting squares of same color and by rotating the chessboard 45 degrees, while using the diagonals as axes now.

  1. Is there a simple formula giving the taxicab distance for bishops similar to that of rooks, using coordinates?

  2. Lastly, the same article mentions the Chebyshev distance (minimum number of moves for the king to go from one square to another), being used for both the kings and queens on board. But knowing the queen moves both like the bishops and rooks, wouldn't a variant of the taxicab distance be more meaningful here?

  3. Any links, or literature suggestions on such ideas (defining distances etc) in chess, would be much appreciated.

  1. It would seem to be max( |a_2-a_1|, |b_2-b_1| ). Whichever of the horizontal or vertical distances is greater, the bishop will always travel over at least that many squares and does not need to travel over any more than that many squares.

  2. This article seems to use distance to mean "number of squares passed over while moving the piece from the source to the destination". In that sense the distance from a1 to c3 for a queen is 2 (Chebyshev distance), not 4 (Manhattan distance); it goes to b2 and then c3, not through b1, c1, and c2.

  • @TowersOfHanoi The Chebyshev distance on the regular board is the same as the taxicab distance on the rotated-by-45-degrees-and-only-using-one-color board.
    – dfan
    Aug 21 '15 at 16:28

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