Reduced Entries Algebraic Magic and Semi-Magic Squares of Orders 3, 5, 7 and 9

In this paper, the author worked with magic and semi-magic squares of orders 3, 5, 7 and 9 for the reduced entries. By reduced or less entries we understand that instead of normal n^2 entries of a magic square of order n, we are using less numbers. Moreover, the entries are no more sequential numbers. These entries are non-sequential positive and negative numbers. In some cases, these may be decimal or fractional values depending on the order of a magic squares.  In this work we bring different ways of writing magic and semi-magic squares of orders 3, 5, 7 and 9. These are based on four types of magic squares, i.e., cornered,  single-digit bordered,    double-digit bordered and pandiagonal. We worked on pandiagonal magic squares of orders 5 and 7.  It is not necessary, but we worked with magic rectangles with equal widths and lengths of the same category within a magic square. If we relax this condition, i.e., by considering only equality in width, still we have good results.  These studies on magic and semi-magic squares are made in algebraic way, i.e., first with letters and then with examples. In all the situations, the conditions for semi-magic squares to become magic squares are also studied. Combining together we have complete work on odd orders magic squares up to order 9. Soon, the work on order 11 shall be presented. For the previous work on even orders magic squares, the reader can follow the reference list.