## Ishaan94 2 years 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?

1. Fazeelayaz

yes

2. Fazeelayaz

i think

3. Fazeelayaz

do u need explanation

4. terenzreignz

What does repaint mean? Invert the colours of a row? (IE turn all light squares dark and vice versa)

5. Fazeelayaz

ok suppose that we want one black square that is how we can do it

6. Fazeelayaz

black square is in the middle lets say at 4rth row and 4rth coloumn

7. Jack1

yes

8. Fazeelayaz

first we paint all the rows except 4rth one with white color not 4rth row 4rth row must have one black

9. Fazeelayaz

then we paint all coloumns with white except 4rth coloumn

10. terenzreignz

So, in simple terms, repainting is a function that maps... |dw:1370257845631:dw|

11. Fazeelayaz

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

12. Fazeelayaz

so ans is yes

13. Fazeelayaz

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

14. Ishaan94

I guess so @terenzreignz , because the answer at back is 'no'.

15. terenzreignz

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

16. terenzreignz

Let's repaint the 1st, 3rd, 5th and 7th rows (starting from the top) 1&0&1&0&1&0&1&0 $\large \left[\begin{matrix}1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\\1&0&1&0&1&0&1&0\end{matrix}\right]$

17. terenzreignz

Okay, nvm, I lost myself :D

18. terenzreignz

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...

19. terenzreignz

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

20. Ishaan94

That's it.