Magic Square Variations
Date: 08/29/2001 at 21:20:35 From: Anonymous Subject: Magic Squares A magic square consists of numbers arranged in a square so that all rows, columns, and usually the two diagonals will add up to the same sum. Try to create a magic square by arranging the first nine counting numbers in the nine square cells. There is only one possible way. Can you please help me? Thank you, Anonymous
Date: 08/30/2001 at 06:58:00 From: Doctor Jeremiah Subject: Re: Magic Squares Hi there, and thanks for writing. There are actually 8 different ways, but they are all rotations and mirror images of the same one. First consider that if you have 3 cells wide by 3 cells high, you will have to put the numbers 1 through 9 in these nine cells. The sum of the diagonals, rows, and columns will be the same. That means that the sum of all three columns must be the same as the sum of all nine numbers because the nine numbers fit into the three columns. (Let's call the sum of a column, row, or diagonal S): sum of 3 columns = sum of all nine numbers 3S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 3S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 3S = 45 3S / 3 = 45 / 3 S = 15 So the "magic sum" is 15. Let's think a bit about what number must be in the middle. It's an important number because it is used in every sum (all the rows, columns and diagonals). If that middle cell holds 6, then what? In which cell can you put the 9? You can't put it anywhere, because 6+9 = 15 with only two numbers, and we need to make _three_ numbers add to 15. So 6 and above cannot be in the middle cell. If that middle cell holds 4, then what? In which cell can you put the 1? You can't put it anywhere, because 4+1 = 5, and to make 15 you need to put 10 into a cell, but 10 isn't a choice because it's not one of the 9 counting numbers. So 4 and below cannot be in the middle cell. That leaves 5. With 5 in the middle cell the solution is easy (especially if you know the magic sum is 15). Try that and let me know if you get stuck again. And for more information, see Suzanne Alejandre's Web unit: Magic Squares http://mathforum.org/alejandre/magic.square.html How to Construct Magic Squares http://mathforum.org/alejandre/magic.square/adler/adler4.html or search the Dr. Math archives for the keywords magic square (that exact phrase): http://mathforum.org/mathgrepform.html - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/
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