# Magic Squares

Magic squares have intrigued people for thousands of years and in ancient times they were thought to be connected with the supernatural and hence, magical. Today, magic squares are considered magical because there are so many relationships between the sums of the numbers in the squares. So, what is a magic square? A magic square is an arrangement of the numbers from 1 to n2 in an n x n matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same (Alejandre), called the magic constant or sum. The magic sum can be found using the formula (n(n2 + 1))/2 for any n x n matrix. There are many different variations of magic squares, of which, some have more ways of finding the magic sum in the square and others use geometric shapes or number words.

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The earliest known magic square was found in a Chinese book, Yih King, in which the legend of “Lo Shu” is told. This magic square is a 3 x 3 matrix, with numbers 1-9, and the magic sum is 15. “The legend of “Lo Shu” or “scroll of the river Lo” tells the story of a huge flood that destroyed crops and land. The people offered a sacrifice to the river god to calm his anger. Every time the river flooded, there emerged a turtle that would walk around the sacrifice. It wasn’t until a child noticed a unique pattern on the turtle’s shell that told the people how many sacrifices (15) to make for the river god to accept their sacrifice.” (http://plaza.ufl.edu/ufkelley/magic/index.htm) This kind of magic square is known as the “traditional” magic square since it has no other special properties besides the ones noted above. The magic square is still common in China today. It is found on buildings and in artistic designs, and fortune tellers uses them in their trade.

The magic squares then found their way to India. Here, the magic squares were not only used to spread mathematical knowledge but also had spiritual purposes. For example, a magic square was found in a medical book as a way to ease childbirth. The oldest magic square of order four (4 x 4) was found inscribed in Khajuraho, India dating back to the 11th or 12th century.

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This kind of magic square has many more properties than a traditional magic square. In addition to the rows, columns, and diagonals, but also the broken diagonals (4+6+13+11=34), of any 2 x 2 block of number (9+6+4+15=34), the four corners (9+16+7+2), the corners of any 3 x 3 block (9+3+14+8=34) , and the sum of the middle two entries of the two outer columns and rows (6+3+ 12+13 = 34) all have the same sum.

The first recorded magic square in Europe and most famous 4 x 4 magic square is found in the famous painting, Melancholia, by German artist Albrecht Duerer (1471 – 1528). The painting is said to “depict the indecision of the intellectual” (Britton). A unique “magic” property of this magic square is that the center entries of the bottom row are 15 and 14, which is the year the painting was made (1514).

This magic square shares the same properties as the other mentioned 4 x 4 magic square only the rows and columns are interchanged in such a way to remain the constant sum of 34. We can use some properties of magic squares to construct more squares from other manufactured squares; 1. Magic square will remain magic if any number is added to every number of a magic square. 2. A magic square will remain magic if any number multiplies every number of a magic square. 3. A magic square will remain magic if two rows, or columns, equidistant from the center are interchanged. 4. An even order magic square ( n x n where n is even) will remain magic if the quadrants are interchanged. 5. An odd order magic square will remain magic if the partial quadrants and the row is interchanged.” (Hawley)

In the seventeenth century, a Frenchman named Antoine de la Loubere created a method for constructing any odd n x n magic squares using consecutive numbers starting with 1. The general rule is that you move diagonal upwards and to the right. However, there are two exceptions. If your move is outside of the magic square, you put your entry on the opposite side of the row or column and if when you move, you land on an occupied space, you put the entry under your last entry. Many ingenious methods for constructing magic squares have been devised but all methods are only for specific cases or

different types of magic squares.

There are many famous names associated with magic squares including Martin Gardner, Leonard Euler, and Benjamin Franklin. Benjamin Franklin is known for his invention of large ordered magic squares. In his youth, he created an 8 x 8 magic square with magic sum of 260 and a 16 x 16 magic square with sum 2,056. These magic squares have the same properties as the traditional magic square except the diagonals do not add up to the magic sum but, other special properties exist. For example in the 8 x 8 magic square, the sum of half of a row or column is equal to half of 206 and each of the “bent” rows (as Franklin called them) of 8 numbers total to 206.

Many more variations of magic squares and their constructions exist as well as different properties. Attractive patterns are seen by connecting consecutive numbers in a magic square. Alphamagic squares are constructed by using the number of letters in the word for each number, which generates another magic square.