Tiling for $mtimes n$ chessboard problem for both $m, n$ are odd
$begingroup$
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?
combinatorics chessboard
edited Nov 30 '18 at 18:15
Lorenzo B.
1,8402520
asked Nov 30 '18 at 17:23
DummKorfDummKorf
335
$endgroup$
add a comment |
$begingroup$
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?
combinatorics chessboard
edited Nov 30 '18 at 18:15
Lorenzo B.
1,8402520
asked Nov 30 '18 at 17:23
DummKorfDummKorf
335
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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?
combinatorics chessboard
edited Nov 30 '18 at 18:15
Lorenzo B.
1,8402520
asked Nov 30 '18 at 17:23
DummKorfDummKorf
335
$endgroup$
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
combinatorics chessboard
edited Nov 30 '18 at 18:15
Lorenzo B.
1,8402520
asked Nov 30 '18 at 17:23
DummKorfDummKorf
335
edited Nov 30 '18 at 18:15
Lorenzo B.
1,8402520
asked Nov 30 '18 at 17:23
DummKorfDummKorf
335
edited Nov 30 '18 at 18:15
Lorenzo B.
1,8402520
edited Nov 30 '18 at 18:15
Lorenzo B.
1,8402520
edited Nov 30 '18 at 18:15
Lorenzo B.
1,8402520
1,8402520
asked Nov 30 '18 at 17:23
DummKorfDummKorf
335
asked Nov 30 '18 at 17:23
DummKorfDummKorf
335
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
1
active
oldest
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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 . . . . . .
. . .
. . .
answered Nov 30 '18 at 18:04
Mike EarnestMike Earnest
20.9k11951
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
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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 . . . . . .
. . .
. . .
answered Nov 30 '18 at 18:04
Mike EarnestMike Earnest
20.9k11951
$endgroup$
add a comment |
$begingroup$
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 . . . . . .
. . .
. . .
answered Nov 30 '18 at 18:04
Mike EarnestMike Earnest
20.9k11951
$endgroup$
add a comment |
$begingroup$
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 . . . . . .
. . .
. . .
answered Nov 30 '18 at 18:04
Mike EarnestMike Earnest
20.9k11951
$endgroup$
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 . . . . . .
. . .
. . .
answered Nov 30 '18 at 18:04
Mike EarnestMike Earnest
20.9k11951
edited Dec 1 '18 at 3:38
answered Nov 30 '18 at 18:04
Mike EarnestMike Earnest
20.9k11951
answered Nov 30 '18 at 18:04
Mike EarnestMike Earnest
20.9k11951
answered Nov 30 '18 at 18:04
Mike EarnestMike Earnest
20.9k11951
20.9k11951
add a comment |
add a comment |
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$begingroup$
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