Magic Squares

Alexander Czik, RHS ‘07

An algorithm for generating inverted magic squares was found in this work. In order to find this algorithm, certain conclusions about the general properties of magic squares were made. These properties were used to create an algorithm for generating normal magic squares. Each value in the magic square could be found as a result of adding or subtracting three distinct variables. When specific numerical values, that were part of the same arithmetic progression, were filled in for these variables, magic squares were generated. An inverted magic square of a similar form as the original magic square was then created. When the two magic squares were multiplied together and the newly created matrix was compared to the numerical values of the identity matrix, three more equations were created. Using these three equations, an inverted magic square could easily be created by using the numerical values of the variables from the original magic square. Other findings should also be noted. In the inverted magic square, the least common denominator for the elements is the same as the determinant, except positive. The magic sum of the inverse is the multiplicative inverse of the magic sum from the original magic square.