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Math Help - 2 questions 6 digit number and neighbour squares

  1. #1
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    2 questions 6 digit number and neighbour squares

    Neighbour Squares
    (4)On a standard chessboard how many different ways can you select a block of five neighbour squares?

    Two squares are neighbours if they have a common side.

    (If the problem was asked for a 3x3 board the answer would be 49.)


    enjoy it
    Last edited by Innsmouth; April 2nd 2009 at 09:20 PM.
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  2. #2
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    Hello, Innsmouth!

    Neighbour Squares
    On a standard chessboard how many different ways
    . . can you select a block of five neighbour squares?
    Two squares are neighbours if they have a common side.

    (If the problem was asked for a 3x3 board the answer would be 49.)

    I finally found the 49 ways on a 3x3 board!
    Code:
          *-----------*
          |:::::::::::|
          |:::::::*---*
          |:::::::|   |    16 ways
          *---*---*   |
          |           |
          *-----------*
    Code:
          *-----------*
          |:::|:::|:::|
          *:::*---*:::*
          |:::|   |:::|    8 ways
          *---*   *---*
          |           |
          *-----------*
    Code:
          *-------*---*
          |:::::::|   |
          *---*:::*---*
          |   |:::::::|    8 ways
          |   |:::*---*
          |   |:::|   |
          *---*---*---*
    Code:
          *-----------*
          |:::::::::::|
          |:::*-------*
          |:::|       |    4 ways
          |:::|       |
          |:::|       |
          *---*-------*
    Code:
          *-----------*
          |:::::::::::|
          *---*:::*---*
          |   |:::|   |    4 ways
          |   |:::|   |
          |   |:::|   |
          *---*---*---*

    Code:
          *---*-------*
          |   |:::::::|
          *   |:::*---*
          |   |:::|   |    4 ways
          *---*:::|   *
          |:::::::|   |
          *---*---*---*
    Code:
          *---*-------*
          |:::|       |
          |:::*---*   |
          |:::::::|   |    4 ways
          *---*:::*---*
          |   |:::|:::|
          *---*---*---*
    Code:
          *---*---*---*
          |   |:::|   |
          *---*:::*---*
          |:::::::::::|    1 way
          *---*:::*---*
          |   |:::|   |
          *---*---*---*


    On an 8x8 board, these would have many more placements.

    Plus, we must place the other four "Pentominoes".
    Code:
          *-------*
          |:::::::|
          *---*:::*-------*
              |:::::::::::|
              *-----------*
    
    
          *---*
          |:::|
          |:::*-----------*
          |:::::::::::::::|
          *---------------*
    
    
              *---*
              |:::|
          *---*:::*-------*
          |:::::::::::::::|
          *---------------*
    
    
          *-------------------*
          |:::::::::::::::::::|
          *-------------------*


    That's a
    LOT of counting!

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  3. #3
    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by Soroban View Post
    Hello, Innsmouth!


    I finally found the 49 ways on a 3x3 board!
    Code:
          *-----------*
          |:::::::::::|
          |:::::::*---*
          |:::::::|   |    16 ways
          *---*---*   |
          |           |
          *-----------*
    Code:
          *-----------*
          |:::|:::|:::|
          *:::*---*:::*
          |:::|   |:::|    8 ways
          *---*   *---*
          |           |
          *-----------*
    Code:
          *-------*---*
          |:::::::|   |
          *---*:::*---*
          |   |:::::::|    8 ways
          |   |:::*---*
          |   |:::|   |
          *---*---*---*
    Code:
          *-----------*
          |:::::::::::|
          |:::*-------*
          |:::|       |    4 ways
          |:::|       |
          |:::|       |
          *---*-------*
    Code:
          *-----------*
          |:::::::::::|
          *---*:::*---*
          |   |:::|   |    4 ways
          |   |:::|   |
          |   |:::|   |
          *---*---*---*
    Code:
          *---*-------*
          |   |:::::::|
          *   |:::*---*
          |   |:::|   |    4 ways
          *---*:::|   *
          |:::::::|   |
          *---*---*---*
    Code:
          *---*-------*
          |:::|       |
          |:::*---*   |
          |:::::::|   |    4 ways
          *---*:::*---*
          |   |:::|:::|
          *---*---*---*
    Code:
          *---*---*---*
          |   |:::|   |
          *---*:::*---*
          |:::::::::::|    1 way
          *---*:::*---*
          |   |:::|   |
          *---*---*---*


    On an 8x8 board, these would have many more placements.

    Plus, we must place the other four "Pentominoes".
    Code:
          *-------*
          |:::::::|
          *---*:::*-------*
              |:::::::::::|
              *-----------*
    
    
          *---*
          |:::|
          |:::*-----------*
          |:::::::::::::::|
          *---------------*
    
    
              *---*
              |:::|
          *---*:::*-------*
          |:::::::::::::::|
          *---------------*
    
    
          *-------------------*
          |:::::::::::::::::::|
          *-------------------*


    That's a
    LOT of counting!

    If you won't mind me asking , how much time did it take to

    post that

    after you solved it ...

    Absolutely fabulous!!
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  4. #4
    Super Member

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    May 2006
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    Lexington, MA (USA)
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    Hello again, Innsmouth!

    I was wrong . . . It didn't take that much work to count them.


    The 8 pentominos can be placed in a 3x3 grid in 49 ways.
    The 3x3 grid can be located in: 6^2 = 36 positions.

    Hence, the 8 forms can be placed in: . 49\cdot36 \:=\:{\color{blue}1764} ways.



    Consider 3 of the remaining 4 pentominoes.
    Code:
          *-------*
          |:::::::|
          *---*:::*-------*
              |:::::::::::|
              *-----------*
    
          *---*
          |:::|
          |:::*-----------*
          |:::::::::::::::|
          *---------------*
    
              *---*
              |:::|
          *---*:::*-------*
          |:::::::::::::::|
          *---------------*

    Horizontal: these occupy a 2x4 rectangle; each has 4 orientations.
    The 2x4 rectangles have 5 horizontal locations and 7 vertical locations.
    . . The rectangles have 35 possible positions.
    So, each form has: . 4\cdot35 \:=\:140 ways.

    Hence, the three horizontal forms have: . 3 \times 140 \:=\:420 ways.


    Vertical: the forms occupy a 2x4 rectangle, with 4 orientations.
    The rectangles have 7 horizontal locations and 5 vertical locations.

    Hence, the three vertical forms have: .  3 \times 140\:=\:420 ways.


    Therefore, these three forms have: . 420 + 420 \:=\:{\color{blue}840} ways.



    Code:
          *-------------------*
          |:::::::::::::::::::|
          *-------------------*

    The horizontal 1x5 bar has 4 horizontal locations and 8 vertical locations.
    . . Hence, it has 32 possible locations.
    The vertical 5x1 bar also has 32 possible locations.

    Hence, it has 32 + 32 \:=\:{\color{blue}64} ways.


    Therefore, there are: . 1764 + 840 + 64 \:=\:\boxed{2668} ways
    . . to place the twelve pentominos on a chessboard.



    I'm grateful they didn't ask for "six neighbour squares".
    There would be 35 hexominos to consider.


    Thanks for the praise, ARDASH. I didn't pay much attention,
    but it probably took an hour (most work on diagrams, using
    COPY and PASTE over and over).
    It's certainly obvious by now, isn't it? . . . I truly enjoy doing this.
    .
    Last edited by Soroban; March 28th 2009 at 04:31 PM.
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