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Assume an \(8\times 8\) chessboard with usual colouring. You may repaint all the squares of a row or column. The goal is to attain one black square. Possible?
 10 months ago
 10 months ago
Assume an \(8\times 8\) chessboard with usual colouring. You may repaint all the squares of a row or column. The goal is to attain one black square. Possible?
 10 months ago
 10 months ago

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FazeelayazBest ResponseYou've already chosen the best response.0
do u need explanation
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.3
What does repaint mean? Invert the colours of a row? (IE turn all light squares dark and vice versa)
 10 months ago

FazeelayazBest ResponseYou've already chosen the best response.0
ok suppose that we want one black square that is how we can do it
 10 months ago

FazeelayazBest ResponseYou've already chosen the best response.0
black square is in the middle lets say at 4rth row and 4rth coloumn
 10 months ago

FazeelayazBest ResponseYou've already chosen the best response.0
first we paint all the rows except 4rth one with white color not 4rth row 4rth row must have one black
 10 months ago

FazeelayazBest ResponseYou've already chosen the best response.0
then we paint all coloumns with white except 4rth coloumn
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.3
So, in simple terms, repainting is a function that maps... dw:1370257845631:dw
 10 months ago

FazeelayazBest ResponseYou've already chosen the best response.0
in this way all the remaining except the the one we said will be white and only one which is at location 4rth row and 4rth coloumn would be white
 10 months ago

FazeelayazBest ResponseYou've already chosen the best response.0
ok repainting means painting what you want to paint if 1 row is to be repainted with white then all the squares in that row would be white
 10 months ago

Ishaan94Best ResponseYou've already chosen the best response.0
I guess so @terenzreignz , because the answer at back is 'no'.
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.3
Well then, \[\large \left[\begin{matrix}0&1&0&1&0&1&0&1 \\1&0&1&0&1&0&1&0\\0&1&0&1&0&1&0&1\\1&0&1&0&1&0&1&0\\0&1&0&1&0&1&0&1\\1&0&1&0&1&0&1&0\\0&1&0&1&0&1&0&1\\1&0&1&0&1&0&1&0\end{matrix}\right]\] Okay, let's start with this... 0 = light square 1 = dark square
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.3
Let's repaint the 1st, 3rd, 5th and 7th rows (starting from the top) 1&0&1&0&1&0&1&0 \[\large \left[\begin{matrix}1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\end{matrix}\right]\]
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.3
Okay, nvm, I lost myself :D
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.3
It might the case that no matter how you repaint, you always add/subtract an even number of dark squares.... so you can't end up with just 1...
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.3
Though you could always repaint the second, fourth, sixth, and eight columns (from the left) and end up with one (really big) dark square XD
 10 months ago
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