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please solve the problem

Consider the squares of an 8 *8 chessboard …filled with the numbers 1 to 64 as in the fi…gure 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 obtained is always 260.

Note by Sayan Chaudhuri
4 years, 5 months ago

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Turn each number into \(8 \cdot x + y\) where \(0 \leq y < 8\). \(x\) turns out to be the row number, starting from \(0\), and \(y\) is the column number, starting from \(1\). Since each row and column is picked exactly once, then ANY configuration's sum would be \(8 \cdot \displaystyle \sum_{i=0}^7 i + \displaystyle \sum_{i=1}^8 i = 8 \cdot 28 + 36 = 260\).

cmiiw. Ronald Kurniawan Tjondrowiguno · 4 years, 5 months ago

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@Ronald Kurniawan Tjondrowiguno i think u r right,hats off Sayan Chaudhuri · 4 years, 5 months ago

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Nice invariant problem. I think of some variants for this problem. a) instead of \(8\times 8\), find a formula for an \(n\times n\) table. b) instead of \(1\) to \(n^2\), try to find a formula for \(n^2\) consecutive integers. c)instead of consecutive integers, try to find a formula for an arithmetic progression with \(n^2\) terms. :) Yong See Foo · 4 years, 5 months ago

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@Yong See Foo the problem turns out 2 B more spiral,for me that would be very difficult 2 solve Sayan Chaudhuri · 4 years, 5 months ago

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@Sayan Chaudhuri its not spiral, its leftmost cell to rightmost cell of each row then going to next row from top to bottom. actually b is the most similar to the original problem. Yong See Foo · 4 years, 5 months ago

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The only way of choosing the 8 numbers is by taking the diagonals. The sum of numbers on the diagonals is 260. Rohan Rao · 4 years, 5 months ago

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@Rohan Rao no, thats wrong Harshit Kapur · 4 years, 5 months ago

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@Harshit Kapur Ah yes, found some more ways, sorry! Rohan Rao · 4 years, 5 months ago

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