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Magic Squares When I was a college freshman (1956), I went through some books being tossed by the school library, and became intrigued by a small volume about magic squares. From that book I learned how to construct magic squares having any ODD number of squares on a side. In all the years since, I have yet to personally meet anyone else who knew how to do that. With the advent of the Internet, however, I find that many people have studied magic squares to a degree far beyond my level of interest.This will, then, be merely an introduction to the subject of magic squares, the history of which can be traced back centuries. Here are magic squares... a 3 X 3, a 5 X 5, and a 7 X 7 Each was constructed using the system you'll learn here. A magic square, of ANY size with an ODD number of squares on a side, can be constructed using this method. Note that each row, each column, and both diagonals add up to the same value.  The Method: Notice that, for these squares, 1 is always placed in the center of the upper row. Then... the rules are applied. Rule 1: Move to the next square up diagonally to the right (if you can). Rule 2: When a number falls outside a cell above a column, place it in the bottom cell of that column. Rule 3: When a number falls outside a cell to the right of a row, place it in the cell to the extreme left of that row. Rule 4: If the cell we wish to use is already occupied, the number is placed immediately below the previous number. Benjamin Franklin was a serious dabbler in magic squares, and created a number of unique ones. He constructed this "panmagic" square having magic constant 260. Any half-row or half-column in this square totals 130, and the four corners plus the middle total 260. In addition, bent diagonals (such as 52-3-5-54-10-57-63-16) also total 260 (Madachy 1979, p. 87).
Albrecht Dürer's famous engraving of" Melancholia" made in 1514 (which, by the way, is owned by the Minneapolis Institute of Arts) includes a picture of a 4 X 4 magic square.Interestingly, this square can be created using a simple "mod 4" method, but Dürer switched the 2 center columns so that it would read 1514, the date of his engraving. The Mod-4 method - used to create magic squares that are 4-square, 8-square, 12-square, etc. The method involves marking diagonals across each 4-square section, then simply entering numbers, starting with 1 in a corner, entering only those digits falling in squares the diagonal(s) cross. Then, starting at the opposite corner from 1, number again, entering only those digits that fall in squares the diagonal(s) do NOT cross.
I derived a small formula that will predict the "sum value" for any 4-square or odd-sized squares: Where n = the number of squares on a side... (n^3 + n)/2
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