21Mar

A rook is a chess piece that can move any number of spaces either horizontally or vertically on a chess board.?

Author: admin  |  Category: enart.gzfn.com
  • A rook is a chess piece that can move any number of spaces either horizontally or vertically on a chess board. In how many ways can 8 distinct rooks be placed on a chess board so that no rock can capture any other rook?


  • lol. The rook/bishop confusion made me laugh. This reminds me of a puzzle in Professor Layton on the DS. If you've got a mini-Nintendo, check it out.


  • The eight rooks would all have to occupy different ranks and files.

    For the first rook, you have 8x8 = 64 possible locations.
    For the second rook, you have 7x7 = 49 possible locations.
    For the third rook, you have 6x6 = 36 possible locations.
    ... and so on and so forth.

    However, there aren't 8^2 * 7^2 * 6^2 * ... * 1^2 possible placements. Some of these placements will be duplicates. After all, if you place the first rook on a1 and the second on b2, you'll end up with the same position as if you placed the first rook on b2 and the second on a1.

    It turns out that each position is duplicated 8! times (that's 1*2*3*4*5*6*7*8, for the non-mathematically inclined among you).

    When you divide 8*8*7*7*6*6*5*5*4*4*3*3*2*2*1*1 by 8*7*6*5*4*3*2*1, you end up with 8*7*6*5*4*3*2*1, or 8!. That's 40320.

    Thus, there are 40320 distinct positions that satisfy your requirements.


  • The chess board grid is set up as such:

    Files= a, b, c, d, e, f, g, h [ white pieces on rank 1 & 2, a-h]

    Ranks= 1, 2, 3, 4, 5, 6, 7, 8 [ black pieces on rank 7 & 8, a-h ]


    Rb8

    Ra7

    Rc6

    Rd5

    Re4

    Rf3

    Rg2

    Rh1


    As you can see, there are many different ways you can set up the Rooks so that they can not capture eachother! Mathematically the possiblities are greater than you've probably imagined, since each Rook can be exchanged and rearranged and also set on the squares north, south, east, and west...to make similar patterns to mulitply other variables..

    Infact, this illustration in view supports only 1 variable, which can then be multiplied to calculate the availble squares that each Rook can actually occupy as they are allocated to other spots on the board to form other ladder/slant points in the X, Y diagnol variations. LOL!!!

    There are 64 square on a chess/checker board..... knock yourself out!


    PEACE


  • Any good chess program will have a section to explain the moves of the pieces to you as well as there are lots of free sites on the webb to explain all you would want to know.

    here is a good blog about using a chess data base to learn chess

    http://chess-teacher.blogspot.com/

    good luck to you have fun


  • Bramblyspam and mjcdchess have good answers. I thought I would point out a slightly easier way to get the answer. It's clear that we need to place a rook on each row and each column (rank and file if you like). Look at placing the rooks row by row. In the first row, we can place the rook in any of the 8 squares. In the second row, we have 7 places to put a rook. So on to the last row, where we are forced to place the rook in the remaining column. So altogether we have
    8*7*6*5*4*3*2*1
    = 40320
    ways to place the rooks.

    (A similar question for queens is more difficult, you might like to look into that!)


  • I can't get past your decision that a rook can move vertically as a bishop does.


  • That's the "Eight Rooks Problem".

    40320 ways.


  • Good question. I will have to think about this one.


  • along the long diagonals: a1-h8 or a8-h1, the same used when one fianchettos a bishop







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