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Hello All,
I reached the chapter in my book on Magic Squares and I've prepared a list of questions that I'm hoping some of you might be able to help me with.
Q1: Let's call a square S. We know the the sum of each row in S is equal to a constant R, and the sum of each column is a constance C. Knowing this information, how can it be proen that S is a magic square (where R=C) ? I just don't see how we can form this conclusion without just saying that R=C, but apparently there must be a way that you can make it always such so that S is magic.
Q2:Using the same square S, lets analyze the first two rows or columns, r1 and r2 or c1 and c2, respectively. Should we swap the contents of r1 with r2, or c1 with c2's, we can form a new Square called X. How is it that X is also a filled magic square?
Any help is appreciated!
Q1: The sum of all the numbers in the square is some integer, call it S. S is equal to the sum of the C's, and is also equal to the sum of the R's. For an nxn square you therefore have S = n x R = n x C, and thus R = C.
Q2: Swapping any two rows has no effect on the sum of any of the columns, nor obviously on the sum of the rows. Hence your R's and C's are all still the same value after you swap two rows. Same thing if you swap columns. It should be made clear however that the sum of the numbers along the diagonal will not be the same after the swap.
I suppose it depends on how your book defines the term "magic square." Perhaps they don't require the diagonals to add to the same number. Here's an easy example to see what I mean: consider a 3x3 magic square:
816
357
492
Note that all rows, columns, and the diagonals add to 15. Now swap the top two rows and you get:
357
816
492
The rows and columns still each add to 15, but the diagonals now add to 6 and 12. To me that's not a magic square, but then again maybe your book says it is.
First question:
First it doesn't follow for just any square that the rows will sum to the same constant (and the same for the columns). Second for a square to be magic,
its rows, columns and diagonals must sum to the same constant (which is the magic sum). Third there is no known simple way of taking a series of numbers to make a magic square out of them.
Second question: "How is it that X is also a filled magic square?"
There's no guarantee of this. Certain row(s) must be switched with certain other row(s) and the same holds for the columns. You won't get another magic square just by swapping the first row with the second row.
You can reference the internet on this and check my article Pieces Of Gold: Magic Squares from the first ezine for more information.