document.write( "Question 1208189: How many possible are there to place two queens on an empty chessboard such that neither threatens the other? \n" ); document.write( "
Algebra.Com's Answer #846455 by mccravyedwin(407)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Here is my approach. In the diagram below, I have placed the number of\r\n" );
document.write( "forbidden places for the second queen in the event that we choose to put the\r\n" );
document.write( "first queen in that space. I will explain below, where I got them. \r\n" );
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document.write( "21 21 21 21 21 21 21 21\r\n" );
document.write( "21 23 23 23 23 23 23 21\r\n" );
document.write( "21 23 25 25 25 25 23 21\r\n" );
document.write( "21 23 25 27 27 25 23 21\r\n" );
document.write( "21 23 25 27 27 25 23 21\r\n" );
document.write( "21 23 25 25 25 25 23 21\r\n" );
document.write( "21 23 23 23 23 23 23 21\r\n" );
document.write( "21 21 21 21 21 21 21 21\r\n" );
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document.write( "Case 1. We put a queen on the edge of the outer 8x8 square.\r\n" );
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document.write( "If we put a queen on the edge of the outer 8x8 square, there are 21 \r\n" );
document.write( "forbidden places for the other queen, 7 on the same row, 7 on the same\r\n" );
document.write( "column, and 7 on the same diagonal.  That's 21 forbidden places.  That's\r\n" );
document.write( "why I put 21's around the outer edge of the 8x8 square and colored them\r\n" );
document.write( "red. So for each of the 28 places we choose for one queen, there are \r\n" );
document.write( "63-21=42 safe places to put the other queen. That's (28)(42)=1176.\r\n" );
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document.write( "Case 2. We put a queen on the edge of the inner 6x6 square.\r\n" );
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document.write( "If we put a queen on the edge of the inner 6x6 square, there are 23 \r\n" );
document.write( "forbidden places for the other queen, 7 on the same row, 7 on the same\r\n" );
document.write( "column, and 9 on the same diagonal, since we pick up 2 more forbidden\r\n" );
document.write( "places on the diagonals. That's 23 forbidden places.  That's\r\n" );
document.write( "why I put 23's around the edge of the 6x6 square and colored them green.\r\n" );
document.write( "So for each of the 20 places we choose for one queen, there are 63-23=40 \r\n" );
document.write( "safe places to put the other queen. That's (20)(40)=800. \r\n" );
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document.write( "Case 3. We put a queen on the edge of the inner 4x4 square.\r\n" );
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document.write( "If we put a queen on the edge of the inner 4x4 square, there are 25 \r\n" );
document.write( "forbidden places for the other queen, 7 on the same row, 7 on the same\r\n" );
document.write( "column, and 11 on the same diagonal, since we pick up 2 more forbidden\r\n" );
document.write( "places on the diagonals. That's 25 forbidden places.  That's\r\n" );
document.write( "why I put 25's around the edge of the 4x4 square and colored them red.\r\n" );
document.write( "So for each of the 12 places we choose for one queen, there are 63-25=38 \r\n" );
document.write( "safe places to put the other queen. That's (12)(38)=456. \r\n" );
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document.write( "Case 4. We put a queen on the center 2x2 square.\r\n" );
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document.write( "If we put a queen on the edge of the center 2x2 square, there are 27 \r\n" );
document.write( "forbidden places for the other queen, 7 on the same row, 7 on the same\r\n" );
document.write( "column, and 13 on the same diagonal, since we pick up 2 more forbidden\r\n" );
document.write( "places on the diagonals. That's 27 forbidden places.  That's\r\n" );
document.write( "why I put 27's for the center 2x2 square and colored them green.\r\n" );
document.write( "So for each of the 4 places we choose for one queen, there are 63-27=36 \r\n" );
document.write( "safe places to put the other queen. That's (4)(36)=144.\r\n" );
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document.write( "Total for the 4 cases = 1176 + 800 + 456 + 144 = 2576.\r\n" );
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document.write( "Since we can swap the queens, we take half, and get 1288.\r\n" );
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document.write( "Edwin
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