Assume an 8 x 8 chessboard with the usual coloring. "Recoloring" operation changes color of all squares of a row or a column. You can recolor re-peatedly . The goal is to attain just one black square. Show that you cannot achieve the goal . (Hint: If a row or column has b black squares, it changes by (|8 - b) - b|).
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Explanation:
In a chess board no. of squares in any row or column = 8
Total No. of squares = 8 x 8 = 64
No. of black squares = 32
No. of white squares = 32
Let No. of blacks in a selected recoloring row or column = b
No. of black squares after recoloring operation = 8 – b
Initial state b = 32
Desired final state b = 1
Let us tabulate all the possible outcomes after recoloring a row or column.
Difference is the no. of black squares left out in a row or column after recoloring operation. Difference is even. Difference is invariant.
But our goal is one ( 1 is odd) black square, so, it is not possible to attain the goal.
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