According to Chinese legend a turtle like the one above crept out of the Yellow River about 4000 years ago. It looks like it is riddled with spots, or bullet holes. But if you look carefully, the dots on its back represent the digits from 1 to 9 arranged in the following way:

492 357 816

If you add the numbers in each row together, they are all equal to 15. For example 4 + 9 + 2 = 15, and so on.

If you add the columns, they sum to 15 also. For example, 4 + 3 + 8 = 15. And yes, you guessed it, the diagonals do too.

A grid containing consecutive numbers starting from 1 such that rows, columns and diagonals all add up to the same number is known as a magic square. The 3x3 square on turtle is known in China as the lo shu.

Magic squares have long fascinated soothsayers, herpetologists, mystics, architects, soldiers, artists, mathematicians…and now, stamp collectors. Macau, the former Portuguese colony now a part of China, has just issued a set of magic square stamps that, it claims, not only promotes Chinese culture but also creates a “unique product in the history of philately.”

Each stamp in the collection features a magic square. More curiously, the stamps have a monetary value of between 1 to 9 patacas (the Macau currency), and you will be able to buy them in a sheet where the first row has the 4, 9 and 2 pataca stamps, the second has the 3, 5 , 7 pataca stamps and the third has the 8, 1 and 6. In other words, the stamps are themselves arranged in a magic square, which is none other than the lo shu!

Meta!!! Amazed no postal service ever thought of this before…

In fact there will be ten stamps in the set, including the 12 pataca stamp above with the turtle on it. Scroll down to see how they form the lo shu. Meanwhile, here’s a quick guide to the images on the stamps, which provides a mini-history of this mathematical curiosity.

4 pataca: Dürer

Magic squares can be bigger than three rows and columns. The best known of the 4 x 4 squares was immortalized by German artist Albrecht Dürer in his wood carving Melencolia 1. The magic square appears written in the background, behind a sulky angel. Each row, column and diagonal adds up to 34. In fact, many more combinations of four numbers add up to 34, such as the outer corners, some of the 2x2 subsquares, and many more. But the geekiest aspect of the Dürer square is that it includes the date of when he thought it up – 1514, which we see on the bottom row. Happy 500th birthday Melencolia 1! Cheer up, little angel!

The 4 pataca and the 9 pataca stamps: both the German renaissance and the French enlightenment loved magic squares.

9 pataca: de la Loubère



There are many ways to create magic squares. One of the most famous methods is named after Simon de la Loubère, a seventeenth century French diplomat who spent time in what used to be called Siam, now Thailand. The method only works for squares that have an odd number of rows/columns. You start with a 1 halfway along a side, as the stamp shows, and then progress diagonally (NE) with the rule that if you leave the square on the top, you reappear on the bottom, and if you leave the square on the right you reappear on the left. Each free square you reach you must write down the next number up, and if a square is not free, you place the new number on the square below it.

2 pataca – Sator

Okay, this isn’t a magic square. But is a square of ancient mystical interest whose power comes from its playful arrangement of letters. It contains the Latin words

SATOR (sower)



AREPO (Arepo, probably a proper name)



TENET (holds)



OPERA (the works)



ROTAS (rolling)



Which can be read forwards, backwards, downwards and upwards. The meaning is unclear, but suggestions have been made like “The sower Arepo keeps the world rolling.” Several Sator squares have been found in excavations, including one in the ruins of Pompeii.

The 2 pataca and the 3 pataca stamps: Ancient Rome and the USA.

3 pataca – Franklin



Has there ever been an overachiever like Benjamin Franklin? Thinker, politician, scientist, Founding Father, musician, inventor, statesman, author…and magic square legend! One of his inventions was the ‘broken diagonal’. The 8x8 square in the stamp is not strictly a magic square since the full diagonals do not add up to 260. But the rows, columns and broken diagonals – colour-coded in the stamp – do.

5 pataca – Su Hui

Again, not a magic square, but pretty amazing all the same. It is a ‘palindromic poem’ composed by the Chinese poet Su Hui around the fourth century AD. In the full version, the poem is a 29 x 29 square where each position has a single Chinese character. The poem can be read forwards, backwards, upwards and downwards. In fact, there are 2848 different ways to read it. The stamp contains the 15x15 central section of the full poem. Su Hui is said to have written the poem to her husband who had moved to live far from her, and then married another woman. When the husband read it, he returned to Su Hui.

The 5 and 7 pataca stamps: poetry and geometry.

7 pataca – Sallows



Lee Sallows, a British recreational mathematician living in the Netherlands, has invented a whole new type of magic square. In a ‘geomagic square’ the shapes in each row, column and diagonal can be reassembled into the same master shape. The stamp shows a 3x3 geomagic square in the middle, and around the outside are how the constituent parts fit into the master shape, which is a 4x4 square with one unit taken out.

The stamps for the 8, 1 and 6 pataca stamps are due to be released next year, and when they are you will be able to buy a set of them that form their own lo shu magic square:

The currently most satisfying way to make up 30 patacas.

Further reading: I have a more detailed section about magic squares in Alex’s Adventures in Numberland, and I wrote an article about geomagic squares in the Observer.

The Zen of Magic Squares by Cliff Pickover

Legacy of the Luoshu, Frank Swetz.

To buy the stamps: Macau Post