Results 1 to 2 of 2

Math Help - Chessboard

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    19

    Chessboard

    I have a question. A castle and bishop are placed on different squares of a chessboard. What is the probability that one piece threatens the other? If I remember correctly one of them moves diagonally and the other one moves only horizontally and vertically and I think that there are 64 squares on a chessboard. I don't have the slightest clue how to go about solving the problem. Could anyone help please?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by sweeetcaroline View Post
    I have a question. A castle and bishop are placed on different squares of a chessboard. What is the probability that one piece threatens the other? If I remember correctly one of them moves diagonally and the other one moves only horizontally and vertically and I think that there are 64 squares on a chessboard. I don't have the slightest clue how to go about solving the problem. Could anyone help please?
    A castle (usually known as a rook) moves laterally and a bishop moves diagonally. That means that they can never both threaten each other. So the probability that one piece threatens the other is equal to the probability that the rook threatens the bishop, plus the probability that the bishop threatens the rook.

    Wherever the rook is placed on the board, it threatens 14 squares (7 on its rank and 7 on its file). So if the rook and the bishop are placed on different squares, the probability that the rook threatens the bishop is 14/63, or 2/9.

    For the bishop, the situation is more complicated. If it is placed on one of the 28 squares on the edge of the board, it threatens 7 squares. If it is on one of the 20 squares one rank or file away from the edge, it threatens 9 squares. On one of the 12 squares that are two ranks or files from the edge, it threatens 11 squares. Finally, if it is on one of the 4 central squares, it threatens 13 squares. So the average number of squares threatened by a bishop is \frac{7\times28 + 9\times20 + 11\times12 + 13\times4}{64} = \frac{35}4. Therefore the probability that the bishop threatens the rook is \frac{35}{4\times63} = \frac5{36}.

    Adding those results together, you see that the probability that one piece threatens the other is \frac{13}{36}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. chessboard problem
    Posted in the Statistics Forum
    Replies: 2
    Last Post: January 5th 2011, 02:00 AM
  2. ChessBoard
    Posted in the Statistics Forum
    Replies: 7
    Last Post: September 1st 2010, 08:55 AM
  3. Tiling a chessboard
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: June 3rd 2010, 04:46 PM
  4. Chessboard
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: December 3rd 2009, 11:16 AM
  5. the chessboard
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: May 2nd 2009, 09:01 AM

Search Tags


/mathhelpforum @mathhelpforum