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. 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

References: 1) Alejandre, S. "Suzanne Alejandre 's Magic Squares." http://mathforum.org/alejandre/magic.square.html. 2) Anderson, Dawn. http://illuminations.nctm.org/LessonDetail.aspx?id=L263 3) Britton, Jill. 2005. http://britton.disted.camosun.bc.ca/jbmelancholia.htm. 4) Grogono, A. W. "Magic Squares by Grog." http://www.grogono.com/magic/. 5) Hawley, D. "Magic Squares." http://www.nrich.maths.org.uk/mathsf/journalf/aug98/art1/. 6) Magic Squares: A Simple Webquest. http://plaza.ufl.edu/ufkelley/magic/index.htm.