Time for MATH MINUTE! (provide your favorite theme music here). 

         Get out your paper & pencil, because I have a new puzzle for you!!

PUZZLE #96 Magic Squares?

 

       How could a square be magic?

       In mathematics, a square array of whole numbers is called magic if the sum of the rows and of the columns and of the diagonals all add to the same number S. The numbers used to form the nxn square are usually the numbers 1, 2, . . . , n2. This type of square is called normal.

       For example, here is a 3x3 normal magic square which is built from the numbers 1, 2, . . . 9:

                           

       Notice in this square that all of the sums equal 15.

       The number 15 is not arbitrary if you are using the whole numbers 1, . . . , 9.

       In general this is because there are n rows, so the sum S must be 1/n of the total sum of all of the numbers 1, . . . , n2. So this sum S must satisfy the equation nS = 1+ 2+ . . . + n2 = n2(n2+1)/2, and thus S = n(n2+1)/2. In the 3x3 case above we have S = 3(9+1)/2 = 15.    

       So are there any other 3x3 magic squares out there? How about this one?

                           

       If you study this one and the one above you will see that the relation between the numbers is identical in both, but the presentation is different. In fact, if you flip the 1st one about the diagonal 4-5-6 you will get the 2nd one.

       So you can do some geometric transformations (like rotations, reflections, and half-turns) on magic squares and get another magic square.

       How about this 3x3 square?

                     

       Comparing this square to the one above you see that we have simply added 10 to each entry. This makes a magic square with sum S = 3*10 + 15 = 45. You can add any number to all of the entries in a magic square and get another magic square. There are also other operations which you can do, but these magic squares are not normal since they do not use the numbers 1, . . .  , 9.

       So are there any 3x3 magic squares which are actually different from the ones above? Sadly, the answer is no. They are all obtained from the 1st basic 3x3 normal magic square given above by either transformations or arithmetic operations.

       Sometimes you can find a square which has all of the rows and columns adding up to the same sum, but the diagonals do not add to this sum. Such squares are called semimagic, so you are on the right track if you find one.

 

A.    Try to construct a 3x3 normal magic square on your own, and then show that the result must be a combination of the 3x3 normal magic square above.

 

       All 3x3 normal magic squares will be variations on the first one above.

       Here are a few more:

         

 

       How about larger magic squares? Here is a 4x4 normal magic square (with sum S = 4*(16+1)/2 = 34).

                     

       This is a very famous magic square called the Albrecht Durer magic square. It appeared in his engraving entitled Melancholia I, dated 1514 (notice the 15 and 14 are in juxtaposition on the bottom row - not a coincidence), and it also played an important roll in the recent novel The Lost Symbol by Dan Brown.

       This square is more magic than those above, for notice that the 2x2 corner squares also add to 34, as does the center 2x2 square. In addition, the squares formed by 5-2-12-15, and 3-8-14-9 also sum to 34. That s not all, the rectangles 5-3-12-14, 2-8-15-9, 3-2-14-15, and 8-12-9-5 also sum to 34. In addition, the 4 corners 16-13-1-4 also sum to 34, as does the corners of any 3x3 subsquare. There are other combinations as well. A very magical square indeed.

       You are probably thinking that, as above for 3x3 squares, this is the only example of a 4x4 normal magic square.

       Not so, consider this example:

                     

       So how many normal 4x4 magic squares are there? Would you believe 880?

 

B.    Find a 4x4 normal magic square which is different from those above.

 

       Did you have any luck?

       Here is another 4x4 normal magic square:

             

 

       Who is interested in magic squares? Try the Chinese and Indians as far back as at least 2000BC, as well as Ben Franklin (he found several 8x8 magic squares).

 

       There is a construction technique to the 3x3 normal magic square above which can be used to make bigger normal magic squares of odd order. Here is a step-by-step method for building the 3x3 normal magic square (sometimes called the Siamese method): Starting from the central column of the first row with the number 1, the fundamental movement for filling the squares is diagonally up and right (northeast), one step at a time. If a filled square is encountered, one moves vertically down one square instead, then continuing as before. When a move would leave the entire square, it is wrapped around to the last row or first column, respectively, as if you are on a torus (see J. Weeks: Games on a Torus: http://www.geometrygames.org/TorusGames/ ).

 

             

       Notice that when you get to number 7 you have to move right and up from 6, which takes you to the square occupied by 4, so you have to go down one below 6.

 

C.    This technique can be used to construct normal magic squares of odd order bigger than 3. Try it on this 5x5!

             

       The sum for this square is S = 5(25+1)/2 = 65, right?

 

      

       Above is a step-by-step of the 5x5 using the method for the 3x3 from above. Notice that when you get to inserting 6 in step 2 the next square is occupied by 1, so you have to go down one square. The same thing happens when you get to 11. When you get to 16 in step 3 the next square would be that occupied by 11, so you have to go down one square.

 

       We have only scratched the surface here. Check out the references below for much more info on magic squares.

 

       The material for the above puzzle can be found in Wikipedia:

http://en.wikipedia.org/wiki/Magic_squares#Method_for_constructing_a_magic_square_of_odd_order

       Mathforum:  

http://mathforum.org/alejandre/magic.square.html

       and Mathematical Vignettes:

http://www.jcu.edu/math/vignettes/magicsquares.htm

       as well as other sources.

 

       Have fun!

 

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