Phoenix Rising

Magic Squares

Definition:
A magic square is an arrangement of the numbers from 1 to n^2 (n-squared) in an ‘n x n‘ matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same (known as the ‘Magic Constant’).

The ‘Magic Constant’ (or sum of the entries) can be arrived at by applying the following equation:

Where ‘M’ is the ‘Magic Constant’ and ‘n‘ is the number of rows and columns.

Calculating the ‘Magic Constant’
Take a 3×3 matrix. From equation (1) above, we get:

The simplest magic square is of 1 single row and column, containing just the single entry ‘1′.

figure (i)

The following is the next simplest of a 3×3 matrix, where the ‘Magic Constant’ = 15.

figure (ii)

The next by progression is the 4×4 matrix (also know as the Dürer Magic Square albeit his one is the ‘upside down’ version of the one below).

figure (iii)

The original Dürer magic square was created in an engraving titled ‘Melancholia’ by Albrecht Dürer, and showed the ‘15′ next to the ‘14′ in the bottom row. This being the date of the engraving.

figure (iv)

This magic square (figure (iii)) has a number of very interesting properties, not only do the horizontal, vertical and diagonal all add up the the ‘Magic Constant’ of 34, but:

The four corners add to 34.
The four numbers in the center add to 34.
The 15 and 14 in the top row and the 3 and 2 facing them in the bottom row add to 34.
The 12 and 8 in the first column and the 9 and 5 facing them in the last column add to 34.
The four squares in the corners add to 34.
If you go clockwise around the square and choose the first squares away from the corners (15,9,2,8), they add to 34. The same holds if you go counterclockwise.
Pretty neat huh?!?