## Ishaan94 Group Title 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? one year ago one year ago

1. Fazeelayaz Group Title

yes

2. Fazeelayaz Group Title

i think

3. Fazeelayaz Group Title

do u need explanation

4. terenzreignz Group Title

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

5. Fazeelayaz Group Title

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

6. Fazeelayaz Group Title

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

7. Jack1 Group Title

yes

8. Fazeelayaz Group Title

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

9. Fazeelayaz Group Title

then we paint all coloumns with white except 4rth coloumn

10. terenzreignz Group Title

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

11. Fazeelayaz Group Title

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 Group Title

so ans is yes

13. Fazeelayaz Group Title

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 Group Title

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

15. terenzreignz Group Title

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 Group Title

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 Group Title

Okay, nvm, I lost myself :D

18. terenzreignz Group Title

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 Group Title

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 Group Title

That's it.