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The Numbered Chessboard (TOMATO Subjective 188)

Reasoning, Arithmetic.

Problem:  Consider the squares of an \(8 X 8 \) chessboard filled with the numbers 1 to 64 as in the figure below. If we choose 8 squares with the property that there is exactly one from each row and exactly one from each column, and add up the numbers in the chosen squares, show that the sum always adds up to 260.


Solution: The problem asks us to choose numbers selectively such that they are from unique rows and columns. We write the numbers in the table in such a manner that helps us in our calculations. This is how it will be done:

Let the number in the \(i^{th} \) row and \(j^{th}\) column be \(x\). If we carefully observe the table we find an intuitive way of representing \(x\) as follows:

\(x = 8*(i-1) + j \) if \(x\) is the element in the \(i^{th} \) row and \(j^{th}\) column. Now all that is left to do is sum up all such numbers such that no two \(j^{t} \) or \(i^{th}\) value is same. There for the total sum is: \((8*(i_1-1)+ j_1) + (8*(i_2-1) + j_2) + . . . + (8*(i_8-1) + j_8) \)

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