Let n be a positive even integer. Consider an n x n grid of squares, with...

 

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Let n be a positive even integer. Consider an n x n grid of squares, with the numbers from 1 to n² written in the squares, starting with 1 in the upper left-hand corner, and where the numbers increase consecutively as one proceeds along each row, with the leftmost entry of each new row equal to the rightmost entry of the previous row: n+1 2n +1 2 n+2 2n+2 TE 3 n+3 2n + 3 ... n²-n+1 n²n+2 n²-n+3 ⠀ ... 72 201 3n n² An allowable coloring of the grid is a way of coloring the squares of the grid where • each square is colored either red or white (not both); in every row, precisely n/2 of the squares are white and precisely n/2 of the squares are red; and • in every column, precisely n/2 of the squares are white and precisely n/2 of the squares are red. For each allowable coloring, the score for that coloring is W-R, where W is the sum of all the numbers on the white squares and R is the sum of all the numbers on the red squares. • If we try all the allowable colorings, what is the largest score that we obtain? • If we try all the allowable colorings, what is the smallest score that we obtain? Let n be a positive even integer. Consider an n x n grid of squares, with the numbers from 1 to n² written in the squares, starting with 1 in the upper left-hand corner, and where the numbers increase consecutively as one proceeds along each row, with the leftmost entry of each new row equal to the rightmost entry of the previous row: n+1 2n +1 2 n+2 2n+2 TE 3 n+3 2n + 3 ... n²-n+1 n²n+2 n²-n+3 ⠀ ... 72 201 3n n² An allowable coloring of the grid is a way of coloring the squares of the grid where • each square is colored either red or white (not both); in every row, precisely n/2 of the squares are white and precisely n/2 of the squares are red; and • in every column, precisely n/2 of the squares are white and precisely n/2 of the squares are red. For each allowable coloring, the score for that coloring is W-R, where W is the sum of all the numbers on the white squares and R is the sum of all the numbers on the red squares. • If we try all the allowable colorings, what is the largest score that we obtain? • If we try all the allowable colorings, what is the smallest score that we obtain?