Numbers – an austere beauty

There is something arcane and magical about mathematics: the study of quantity, structure, space, and change. If you get it, all well and good. But, for some, maths is a riddle, wrapped in a mystery, inside an enigma.

Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. In Mysticism and Logic (1918), English philosopher, logician and mathematician Bertrand Russell wrote, “Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture.” He probably knew all about magic squares.

Magic squares were familiar to Chinese mathematicians as early as 650 BCE and to Arab mathematicians possibly as early as the 7th century. The first magic squares of the order 5 and 6 appear in an encyclopaedia from Baghdad circa 983 CE, the Encyclopedia of the Brethren of Purity.

In 1514, the German artist and mathematician Albrecht Dürer produced his most famous work, Melencolia I, a woodcut portraying a troubled-looking angel surrounded by scientific objects, which included a 4×4 square:

16	3	2	13
5	10	11	8
9	6	7	12
4	15	14	1

This square is particularly amazing: not only do the rows, lines, and diagonals add up to 34, but so do the four corners, the four digits in the central square, and the four digits in the top left, top right, bottom left and bottom right quarters. There are other combinations of four numbers in the square that result in 34. Dürer even included the year he made the engraving – 1514 – on the bottom line.

Some great mathematicians studied magic squares – such as Leonhard Euler in the 18th century, and Édouard Lucas and Arthur Cayley in the 19th – but the field has generally been the domain of passionate lay people. The most notable aficionado was Benjamin Franklin, founding father of the USA, who liked to spend his spare time constructing particularly innovative variations. In one evening he composed a 16×16 square that he claimed was “the most magically magical of any magic square ever made by any magician.”

The best-known Latin squares today, however, figure in newspapers and puzzle books. Sudoku consists of a partially completed 9×9 Latin square containing the digits one to nine in each column and row, with the added specification that the 3×3 sub-squares must also contain the numbers from one to nine.

In the mid-19th century in upstate New York, Noyes Palmer Chapman, an amateur puzzle enthusiast, made a physical model of a magic square such that the numbers from 1 to 16 were on small wooden squares that could fit into a 4×4 box. He realised that if he left out one of the squares, it was possible to slide the other 15 around. This became known as the “15 Puzzle”, which was an international fad in 1880 – and is the original sliding block puzzle, versions of which can still be found in toyshops and Christmas crackers. In the 1970s, Hungarian designer Ernö Rubik was trying to reinvent the 15 Puzzle in three dimensions and came up with the idea of his famous Cube. Wonderful! But if you’re like me, it’s still a riddle, wrapped up in a mystery…