Magic Square MatrixDate: 03/24/2003 at 10:47:27 From: John Subject: Magic squares Let M be an integer-valued 3x3 matrix whose entries form a magic square. Let s be the sum of all entries in M and d be the determinant of M. Show that d/s is an integer. Date: 03/28/2003 at 07:46:54 From: Doctor Jacques Subject: Re: Magic squares Hi John, We want to show that d = 0 (mod 3x). Without changing d, we can add the left and right columns to the middle one, and do the same with the rows. This yields a matrix: | a x c | | x 0 x | | g x i | (I replaced 3x by 0, since we are interested in the value mod 3x). If you remember that in a magic square, a + e + i = g + e + c, you should be able to complete the proof. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ Date: 03/31/2003 at 05:53:41 From: john Subject: Thank you (magic squares) Doctor Jacques, Thank you, that was a great help. Although I adapted your advice a little (only by leaving the centre value as 3x, and proceeding from there), the method was perfect, and relatively simple compared to what I had been trying. Thanks again. |
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