Mystical Patterns?

Almost everyone has heard of magic squares. They occur when the totals of all columns of a square - horizontal, vertical and diagonal - are the same. Examples of magic squares have been found from over 3000 years ago, while in the middle Ages they were considered to be prophylactics against the plague. If we could go back to Ancient China or India, we would probably see people wearing stone or metal ornaments engraved with an array of numbers similar to those pictured below. Such ornaments were thought to have mystical powers.

The order of a magic square is determined by the number of rows or columns in the square. Thus, a square with 3 rows or columns is said to be a third order magic square. The common sum obtained by adding the elements of a row, column or diagonal is called the constant of the square. Usually a magic square is formed from the consecutive numbers, 1 to n², n representing the order of the square. Consider a third order square as shown on the left. The sum of all the elements of the square will be the sum of all the integers from one to nine, such that a+b+c+d+e+f+g+h+i = 1+2+3+4+5+6+7+8+9 = 45. Since a+b+c = d+e+f = g+h=i, then 3(a+b+c) = 45; a+b+c = 15. Hence the constant for a third order magic square is fifteen. The same method is used for any order square.

There are numerous ways of making a "new" magic square - changing the order of the numbers (Note: there are eight basic arrangements for a third order magic square), by adding a constant throughout, or even by changing units e.g. to Pounds and Pence. Magic squares don't even have to be "addition" squares. The example shown to the right (denoted by (A)) is a multiplying magic square. This type of square can also be modified by changing the order of its numbers, or by multiplying each one by a constant. Example (B) meanwhile is a magic square composed entirely of prime numbers (which are of course numbers divisible only by themselves and the number one).