Figure 1
Figure 2

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 "wraparound", i.e., if you move off one edge of
the magic square, you reenter 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)!
How to Cite this Page:
Su, Francis E., et al. "Making Magic Squares."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
