Quadramagicology

Perhaps no other area of non-practical mathematics has been so popular for so long as magic squares. Mathematicians, artists and mystics have long been fascinated by the mesmerising patterns that they produce.

A traditional magic square is a square grid of numbers in which the values of each row, column, and diagonal add up to the same sum. A magic square's order is the number of cells in each row. 1) …… For a magic square of consecutive natural numbers starting with 1, let n equal the magic square's order and C equal the magic constant. Then, for an order-3 magic square starting with 1, C = 15.

Mathematicians trace an order-3 magic square back to ancient China, Babylonia, and Mayan culture. In India amulets with magic squares were worn as protective charms, while in western Europe Renaissance astrologers equated them with planets.

According to the legend the Loh Shu was the first magic square, turning up in the dots on the shell of a tortoise that crept out of the Yellow River in China about four millennia ago. 2) …… The Chinese gave it spiritual importance, believing that it encapsulated the harmonies of the universe. Feng Shui, the Chinese system of arranging objects, is in part based on the Lo Shu. 3) …….

The first documented magic square in the West was a 4x4 array of integers one through to 16 which appeared in an engraving Melancholia created by Albrecht Dürer in 1514.

A magic square from the 12th-century temple at Khajuraho in India gives an idea of the extra possibilities offered by a 4-by-4 square. Its magic constant is 34, but there's a new twist. Place the table next to copies of itself, and it creates an infinite "magic carpet": any four adjacent entries along a straight line − horizontal, vertical or diagonal − sum to 34. 4) …….

Squares with the magic-carpet property were called panmagic. Soon they were joined by antimagic squares − in which all the row, column and diagonal sums were different − and by nested and knight's-tour magic squares, in which each number from 1 to 64 is a chess knight's move apart from the next one.

Some great mathematicians studied magic squares – such as Leonhard Euler in the 18th century, and Édouard Lucas and Arthur Cayley in the 19th. The American statesman and scientist Benjamin Franklin liked to spend his spare time constructing innovative variations of magic squares. 5) ……..

Today, magic squares are studied in relation to factor analysis, combinatorial mathematics, matrices, modular arithmetic, and geometry.