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# help

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Dimitri places the maximum possible number of Rooks on a 8 x 8 chessboard in such a way that there's no two Rooks that attack each other. In how many ways can he do that?

Oct 30, 2019

#1
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Dimitri places the maximum possible number of Rooks on a 8 x 8 chessboard in such a way that there's no two Rooks that attack each other. In how many ways can he do that?

A rook can move any distance either horizontally or vertically.

There cannot be two rooks on the same rank or on the same file.

Another way of saying that is there can be only one rook on any rank or file.

Place a rook on each of the eight squares from one corner to the opposite corner.

There are two diagonals in a square chessboard, so the answer would be two ways.

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Oct 30, 2019
#2
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Oh heck, I thought of another way.  Start with a diagonal going from the bottom left to the top right, then move the four, as a unit and keeping the shape, on the top of the board leftward to the top left corner and the four on the bottom of the board rightward to the bottom right corner.  That's one more way, making three.  Then the mirror image of the third way makes a total of four ways

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Guest Nov 3, 2019
#3
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I've thought of yet another way, or, rather, another four ways.  Start with a string of eight rooks in a diagonal direction.  Then take one half of it and rotate that line of four rooks 90o.  Do this at each of the four corners, for an additional four solutions.  Now it's up to EIGHT ways. After all this, I wouldn't be suprised if there are even more ways to position the rooks, but I'm going to stop thinking about it.

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Guest Nov 4, 2019