Mixing
Coloring of cells in a grid
$begingroup$
In a grid a corner cell is colored red and all the other cells are colored blue. Consider the following recoloring operation: In any row or column colors of the cells are interchanged – red cells are colored blue and blue cells are colored red. Which of the following can be done through the recoloring operation?
I. All the corner cells red.
II. All the corner cells blue.
III. One of the corner cells blue and the other corner cells red.
combinatorics
asked Jan 31 at 17:57
Chand16Chand16
256
$endgroup$
add a comment |
$begingroup$
In a grid a corner cell is colored red and all the other cells are colored blue. Consider the following recoloring operation: In any row or column colors of the cells are interchanged – red cells are colored blue and blue cells are colored red. Which of the following can be done through the recoloring operation?
I. All the corner cells red.
II. All the corner cells blue.
III. One of the corner cells blue and the other corner cells red.
combinatorics
asked Jan 31 at 17:57
Chand16Chand16
256
$endgroup$
4
$begingroup$
It seems to me that all other cells besides the corner cells are irrelevant, and so we can consider a 2x2 square instead of a grid. Am I missing something?
$endgroup$
– Todor Markov
Jan 31 at 18:04
3
$begingroup$
What are your ideas? Considering that only corner cells are important, can you reduce the general problem to a smaller one?
$endgroup$
– Ingix
Jan 31 at 18:04
add a comment |
$begingroup$
In a grid a corner cell is colored red and all the other cells are colored blue. Consider the following recoloring operation: In any row or column colors of the cells are interchanged – red cells are colored blue and blue cells are colored red. Which of the following can be done through the recoloring operation?
I. All the corner cells red.
II. All the corner cells blue.
III. One of the corner cells blue and the other corner cells red.
combinatorics
asked Jan 31 at 17:57
Chand16Chand16
256
$endgroup$
In a grid a corner cell is colored red and all the other cells are colored blue. Consider the following recoloring operation: In any row or column colors of the cells are interchanged – red cells are colored blue and blue cells are colored red. Which of the following can be done through the recoloring operation?
I. All the corner cells red.
II. All the corner cells blue.
III. One of the corner cells blue and the other corner cells red.
combinatorics
combinatorics
asked Jan 31 at 17:57
Chand16Chand16
256
asked Jan 31 at 17:57
Chand16Chand16
256
asked Jan 31 at 17:57
Chand16Chand16
256
asked Jan 31 at 17:57
Chand16Chand16
256
asked Jan 31 at 17:57
Chand16Chand16
256
256
4
$begingroup$
It seems to me that all other cells besides the corner cells are irrelevant, and so we can consider a 2x2 square instead of a grid. Am I missing something?
$endgroup$
– Todor Markov
Jan 31 at 18:04
3
$begingroup$
What are your ideas? Considering that only corner cells are important, can you reduce the general problem to a smaller one?
$endgroup$
– Ingix
Jan 31 at 18:04
add a comment |
4
$begingroup$
It seems to me that all other cells besides the corner cells are irrelevant, and so we can consider a 2x2 square instead of a grid. Am I missing something?
$endgroup$
– Todor Markov
Jan 31 at 18:04
3
$begingroup$
What are your ideas? Considering that only corner cells are important, can you reduce the general problem to a smaller one?
$endgroup$
– Ingix
Jan 31 at 18:04
4
4
$begingroup$
It seems to me that all other cells besides the corner cells are irrelevant, and so we can consider a 2x2 square instead of a grid. Am I missing something?
$endgroup$
– Todor Markov
Jan 31 at 18:04
$begingroup$
It seems to me that all other cells besides the corner cells are irrelevant, and so we can consider a 2x2 square instead of a grid. Am I missing something?
$endgroup$
– Todor Markov
Jan 31 at 18:04
3
3
$begingroup$
What are your ideas? Considering that only corner cells are important, can you reduce the general problem to a smaller one?
$endgroup$
– Ingix
Jan 31 at 18:04
$begingroup$
What are your ideas? Considering that only corner cells are important, can you reduce the general problem to a smaller one?
$endgroup$
– Ingix
Jan 31 at 18:04
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I assume that the grid is a rectangle. Let $C$ be a set consisting of four corner cells of the grid. Initially $C$ has odd numbers both of red and blue cells. It is easy to check that the recoloring doesn’t change the parity of cells of each color, so situations I and II are impossible. Situation III is possible, after the first step. If we recolor a column or a row containing only blue cells.
answered Feb 2 at 13:38
Alex RavskyAlex Ravsky
42.9k32483
$endgroup$
$begingroup$
Alex Ravsky thank you
$endgroup$
– Chand16
Feb 6 at 16:34
add a comment |
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$begingroup$
I assume that the grid is a rectangle. Let $C$ be a set consisting of four corner cells of the grid. Initially $C$ has odd numbers both of red and blue cells. It is easy to check that the recoloring doesn’t change the parity of cells of each color, so situations I and II are impossible. Situation III is possible, after the first step. If we recolor a column or a row containing only blue cells.
answered Feb 2 at 13:38
Alex RavskyAlex Ravsky
42.9k32483
$endgroup$
$begingroup$
Alex Ravsky thank you
$endgroup$
– Chand16
Feb 6 at 16:34
add a comment |
$begingroup$
I assume that the grid is a rectangle. Let $C$ be a set consisting of four corner cells of the grid. Initially $C$ has odd numbers both of red and blue cells. It is easy to check that the recoloring doesn’t change the parity of cells of each color, so situations I and II are impossible. Situation III is possible, after the first step. If we recolor a column or a row containing only blue cells.
answered Feb 2 at 13:38
Alex RavskyAlex Ravsky
42.9k32483
$endgroup$
$begingroup$
Alex Ravsky thank you
$endgroup$
– Chand16
Feb 6 at 16:34
add a comment |
$begingroup$
I assume that the grid is a rectangle. Let $C$ be a set consisting of four corner cells of the grid. Initially $C$ has odd numbers both of red and blue cells. It is easy to check that the recoloring doesn’t change the parity of cells of each color, so situations I and II are impossible. Situation III is possible, after the first step. If we recolor a column or a row containing only blue cells.
answered Feb 2 at 13:38
Alex RavskyAlex Ravsky
42.9k32483
$endgroup$
I assume that the grid is a rectangle. Let $C$ be a set consisting of four corner cells of the grid. Initially $C$ has odd numbers both of red and blue cells. It is easy to check that the recoloring doesn’t change the parity of cells of each color, so situations I and II are impossible. Situation III is possible, after the first step. If we recolor a column or a row containing only blue cells.
answered Feb 2 at 13:38
Alex RavskyAlex Ravsky
42.9k32483
answered Feb 2 at 13:38
Alex RavskyAlex Ravsky
42.9k32483
answered Feb 2 at 13:38
Alex RavskyAlex Ravsky
42.9k32483
answered Feb 2 at 13:38
Alex RavskyAlex Ravsky
42.9k32483
42.9k32483
$begingroup$
Alex Ravsky thank you
$endgroup$
– Chand16
Feb 6 at 16:34
add a comment |
$begingroup$
Alex Ravsky thank you
$endgroup$
– Chand16
Feb 6 at 16:34
$begingroup$
Alex Ravsky thank you
$endgroup$
– Chand16
Feb 6 at 16:34
$begingroup$
Alex Ravsky thank you
$endgroup$
– Chand16
Feb 6 at 16:34
add a comment |
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4
$begingroup$
It seems to me that all other cells besides the corner cells are irrelevant, and so we can consider a 2x2 square instead of a grid. Am I missing something?
$endgroup$
– Todor Markov
Jan 31 at 18:04
3
$begingroup$
What are your ideas? Considering that only corner cells are important, can you reduce the general problem to a smaller one?
$endgroup$
– Ingix
Jan 31 at 18:04