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       #Post#: 110--------------------------------------------------
       MAKING MAGIC SQUARES
       By: eba95 Date: July 30, 2010, 6:40 am
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       Making Magic Squares
       A magic square is an NxN matrix in
       which every row, column, and
       diagonal add up to the same
       number. Ever wonder how to
       construct a magic square?
       A silly way to make one is to put the
       same number in every entry of the
       matrix. So, let's make the problem
       more interesting--- let's demand
       that we use the consecutive
       numbers.
       I will show you a method that works
       when N is odd. As an example,
       consider a 3x3 magic square, as in
       Figure 1. Start with the middle entry
       of the top row. Place a 1 there. Now
       we'll move consecutively through
       the other squares and place the
       numbers 2, 3, 4, etc. It's easy: after
       placing a number, just remember to
       always move:
       1. diagonally up and to the right
       when you can,
       2. down if you cannot.
       The only thing you must remember
       is to imagine the matrix has "wrap-
       around", i.e., if you move off one
       edge of the magic square, you re-
       enter on the other side.
       Thus in Figure 1, from the 1 you
       move up/right (with wraparound) to
       the bottom right corner to place a 2.
       Then you move again (with
       wraparound) to the middle left to
       place the 3. Then you cannot move
       up/right from here, so move down
       to the bottom left, and place the 4.
       Continue...
       It's that simple. Doing so will ensure
       that every square gets filled!
       Presentation Suggestions:
       Do 3x3 and 5x5 examples, and then
       let students make their own magic
       squares by using other sets of
       consecutive numbers. How does the
       magic number change with choice of
       starting number? How can you
       modify a magic square and still
       leave it magic?
       The Math Behind the Fact:
       See if you can figure out (prove) why
       this procedure works. Get intuition
       by looking at lots of examples!
       If you are ready for more, you might
       enjoy this variant: take a 9x9 square.
       You already know how to fill this
       with numbers 1 through 81. But let
       me show you another way! View the
       9x9 as a 3x3 set of 3x3 blocks! Now
       fill the middle block of the top row
       with 1 through 9 as if it were its own
       little 3x3 magic square... then move
       to the bottom right block according
       to the rule above and fill it with 10
       through 27 like a little magic square,
       etc. See Figure 2. When finished
       you'll have a very interesting 9x9
       magic square (and it won't be
       apparent that you used any rule)!
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