A magic square is an arrangement of the integers in an MxM square matrix, with each integer occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same.
There are three types of magic squares: 1) M is an odd number (1, 3, 5, 7, etc.); these are referred to as “odd order” magic squares; 2) M is an even number divisible by both 2 and 4 (4, 8, 12, etc.); these are referred to as “doubly even order” magic squares; 3) M is an even number divisible by 2 but not by 4 (6, 10, 14, etc.); these are referred to as “singly even order” magic squares.
Over the past four thousand years numerous methods, some relatively simple and some fairly complex, for generating magic squares of the three different types have been described. I have devised a very simple method of forming any size “odd order” magic square. I named the method “Г + 2” (Gamma plus two) because this name describes the method. I also have generalized an existing procedure, proposed by Ralph Strachey in 1918, of generating “singly even order” magic squares. I called this the “Г + 2+Swap”, or less modestly the “Brumgnach-Strachey” method. I believe that my cell swap portion of the method is innovative and has not been described before. “Doubly even order” magic squares may be easily constructed by a method described by Albrecht Durer in 1514.
The paper describing these innovations is available at:
[ Ссылка ]
