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Sum Square Grid Row & Column ≠ 0 (Posted on 2010-04-29) Difficulty: 3 of 5
Each cell of a 1997x1997 square grid contains either +1 or -1, with no cell being vacant.

The product of all the numbers in the ith row, and the product of all the numbers in the ith column are respectively denoted by Ri and Ci.

               1997
Prove that  Σi=1(Ri + Ci) is always nonzero.

No Solution Yet Submitted by K Sengupta    
Rating: 2.0000 (2 votes)

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Solution Solution | Comment 1 of 6
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I think this should do it. Consider any grid with X rows and Y columns where all cells contain the value 1.  Σi=1(Ri + Ci) = X + Y.

<o:p> </o:p>

From here we have two possible moves:

<o:p> </o:p>

(i)              If we change the sign of cell (Xi , Yi ) where Ri and Ci have opposite signs, the total will remain unchanged, as we are only swapping the positions of the +1 and -1.

<o:p> </o:p>

(ii)            If we change the sign of cell (Xi , Yi ) where Ri and Ci have the same sign, the total will either increase by 4 (if we go from two -1s to two +1s) or decrease by 4.

<o:p> </o:p>

Thus, we can only ever change the total in even amounts of 4.

<o:p> </o:p>

In the case of a 1997 x 1997 grid.

Σi=1(Ri + Ci)     = 1997 + 1997

                        = 1997 * 2

= A number which clearly does not have 4 as a factor, so we cannot lower the total to 0.


  Posted by farcear on 2010-04-29 12:18:11
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