Magic squares have been of interest as a source of recreation for over 4,500 years. A magic square consists of a square array of n[squared] positive and distinct integers arranged so that the sum of any column, row, or main diagonal is the same. In particular, an array of consecutive integers from 1 to n[squared] forming an nxn magic square is called a normalized magic square of order n. Why they possess this property is not, in general, explained; instead, methods of their construction are outlined. Magic squares are divided into three classes: Odd (2m+1), Even-even (4m), and Even-odd (4m+2) with m any positive integer. This paper provides constructions for each class of squares using base-n arithmetic. (Contains 17 figures.)