Tiling for $mtimes n$ chessboard problem for both $m, n$ are odd












2












$begingroup$


Consider an $mtimes n$ chessboard in which both $m$ and $n$ are odd. The board has
one more square of one color, say, black, than of white. Show that, if exactly
one black square is forbidden on the board, the resulting board has a tiling with
dominoes.



So if at least one of the $m$ or $n$ are even, you can fully tile the chessboard. But what about odd?










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$endgroup$












  • $begingroup$
    jade-cheng.com/uh/coursework/math-475/homework-01.pdf This should help you understand the problem. Exercise $2$ is what you're looking for
    $endgroup$
    – Sauhard Sharma
    Nov 30 '18 at 18:30
















2












$begingroup$


Consider an $mtimes n$ chessboard in which both $m$ and $n$ are odd. The board has
one more square of one color, say, black, than of white. Show that, if exactly
one black square is forbidden on the board, the resulting board has a tiling with
dominoes.



So if at least one of the $m$ or $n$ are even, you can fully tile the chessboard. But what about odd?










share|cite|improve this question











$endgroup$












  • $begingroup$
    jade-cheng.com/uh/coursework/math-475/homework-01.pdf This should help you understand the problem. Exercise $2$ is what you're looking for
    $endgroup$
    – Sauhard Sharma
    Nov 30 '18 at 18:30














2












2








2


1



$begingroup$


Consider an $mtimes n$ chessboard in which both $m$ and $n$ are odd. The board has
one more square of one color, say, black, than of white. Show that, if exactly
one black square is forbidden on the board, the resulting board has a tiling with
dominoes.



So if at least one of the $m$ or $n$ are even, you can fully tile the chessboard. But what about odd?










share|cite|improve this question











$endgroup$




Consider an $mtimes n$ chessboard in which both $m$ and $n$ are odd. The board has
one more square of one color, say, black, than of white. Show that, if exactly
one black square is forbidden on the board, the resulting board has a tiling with
dominoes.



So if at least one of the $m$ or $n$ are even, you can fully tile the chessboard. But what about odd?







combinatorics chessboard






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edited Nov 30 '18 at 18:15









Lorenzo B.

1,8402520




1,8402520










asked Nov 30 '18 at 17:23









DummKorfDummKorf

335




335












  • $begingroup$
    jade-cheng.com/uh/coursework/math-475/homework-01.pdf This should help you understand the problem. Exercise $2$ is what you're looking for
    $endgroup$
    – Sauhard Sharma
    Nov 30 '18 at 18:30


















  • $begingroup$
    jade-cheng.com/uh/coursework/math-475/homework-01.pdf This should help you understand the problem. Exercise $2$ is what you're looking for
    $endgroup$
    – Sauhard Sharma
    Nov 30 '18 at 18:30
















$begingroup$
jade-cheng.com/uh/coursework/math-475/homework-01.pdf This should help you understand the problem. Exercise $2$ is what you're looking for
$endgroup$
– Sauhard Sharma
Nov 30 '18 at 18:30




$begingroup$
jade-cheng.com/uh/coursework/math-475/homework-01.pdf This should help you understand the problem. Exercise $2$ is what you're looking for
$endgroup$
– Sauhard Sharma
Nov 30 '18 at 18:30










1 Answer
1






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$begingroup$

Here is a crude drawing of a 7 by 11 chessboard which I will use to illustrate the construciton. The black squares are labeled A and B, because these two categories of black square use different constructions.



A   A   A   A   A   A   A 
B B B B B B
A A A A A A A
B B B B B B
A A A A A A A
B B B B B B
A A A A A A A


If the selected black square, "*," is an "A," then tile the remainder of its row and column with dominoes, then the remaining even by even rectangles with dominoes:



          1  
1
3 3 2 2 * 4 4 5 5 6 6
7
7
8
8


If the selected black square is a "B," then first surround that square with four dominoes as shown, then break the remaining grid 8 rectangles, each with at least one even (possibly zero) dimension:



       . . . 
. . .
. . 3 4 4 . . . . . .
. . 3 * 1 . . . . . .
. . 2 2 1 . . . . . .
. . .
. . .





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    2












    $begingroup$

    Here is a crude drawing of a 7 by 11 chessboard which I will use to illustrate the construciton. The black squares are labeled A and B, because these two categories of black square use different constructions.



    A   A   A   A   A   A   A 
    B B B B B B
    A A A A A A A
    B B B B B B
    A A A A A A A
    B B B B B B
    A A A A A A A


    If the selected black square, "*," is an "A," then tile the remainder of its row and column with dominoes, then the remaining even by even rectangles with dominoes:



              1  
    1
    3 3 2 2 * 4 4 5 5 6 6
    7
    7
    8
    8


    If the selected black square is a "B," then first surround that square with four dominoes as shown, then break the remaining grid 8 rectangles, each with at least one even (possibly zero) dimension:



           . . . 
    . . .
    . . 3 4 4 . . . . . .
    . . 3 * 1 . . . . . .
    . . 2 2 1 . . . . . .
    . . .
    . . .





    share|cite|improve this answer











    $endgroup$


















      2












      $begingroup$

      Here is a crude drawing of a 7 by 11 chessboard which I will use to illustrate the construciton. The black squares are labeled A and B, because these two categories of black square use different constructions.



      A   A   A   A   A   A   A 
      B B B B B B
      A A A A A A A
      B B B B B B
      A A A A A A A
      B B B B B B
      A A A A A A A


      If the selected black square, "*," is an "A," then tile the remainder of its row and column with dominoes, then the remaining even by even rectangles with dominoes:



                1  
      1
      3 3 2 2 * 4 4 5 5 6 6
      7
      7
      8
      8


      If the selected black square is a "B," then first surround that square with four dominoes as shown, then break the remaining grid 8 rectangles, each with at least one even (possibly zero) dimension:



             . . . 
      . . .
      . . 3 4 4 . . . . . .
      . . 3 * 1 . . . . . .
      . . 2 2 1 . . . . . .
      . . .
      . . .





      share|cite|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        Here is a crude drawing of a 7 by 11 chessboard which I will use to illustrate the construciton. The black squares are labeled A and B, because these two categories of black square use different constructions.



        A   A   A   A   A   A   A 
        B B B B B B
        A A A A A A A
        B B B B B B
        A A A A A A A
        B B B B B B
        A A A A A A A


        If the selected black square, "*," is an "A," then tile the remainder of its row and column with dominoes, then the remaining even by even rectangles with dominoes:



                  1  
        1
        3 3 2 2 * 4 4 5 5 6 6
        7
        7
        8
        8


        If the selected black square is a "B," then first surround that square with four dominoes as shown, then break the remaining grid 8 rectangles, each with at least one even (possibly zero) dimension:



               . . . 
        . . .
        . . 3 4 4 . . . . . .
        . . 3 * 1 . . . . . .
        . . 2 2 1 . . . . . .
        . . .
        . . .





        share|cite|improve this answer











        $endgroup$



        Here is a crude drawing of a 7 by 11 chessboard which I will use to illustrate the construciton. The black squares are labeled A and B, because these two categories of black square use different constructions.



        A   A   A   A   A   A   A 
        B B B B B B
        A A A A A A A
        B B B B B B
        A A A A A A A
        B B B B B B
        A A A A A A A


        If the selected black square, "*," is an "A," then tile the remainder of its row and column with dominoes, then the remaining even by even rectangles with dominoes:



                  1  
        1
        3 3 2 2 * 4 4 5 5 6 6
        7
        7
        8
        8


        If the selected black square is a "B," then first surround that square with four dominoes as shown, then break the remaining grid 8 rectangles, each with at least one even (possibly zero) dimension:



               . . . 
        . . .
        . . 3 4 4 . . . . . .
        . . 3 * 1 . . . . . .
        . . 2 2 1 . . . . . .
        . . .
        . . .






        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 1 '18 at 3:38

























        answered Nov 30 '18 at 18:04









        Mike EarnestMike Earnest

        21.3k11951




        21.3k11951






























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