From Magic Squares to Sudoku

In 1415, the German renaissance painter and mathematician Albrecht Dürer created this busy copperplate engraving of a woman melancholically contemplating amongst a clutter that includes an hour glass, a balance scale, a sphere, a polyhedron (aka philosopher’s stone), a purse, and keys. Like his contemporary’s Mona Lisa, this mysterious painting has been extensively studied, speculated upon and written about. It is believed that it was planned to be the first in a series of paintings describing various temperaments: melancholic, phlegmatic, choleric, and sanguine. It is titled: “Melencolia I”




Among the peculiar geometrical tool in this painting, in the upper right corner, one can find … a magic square!



Wait; weren’t magic squares invented by Ben Franklin three centuries after?

Magic squares are tables of numbers where sums of every row, every column and both diagonals produce the same number. In Drurer’s square the sum is 34:








Magic squares have been around for thousands of years. Magic squares of size 3x3 are mentioned in the Chinese literature dating 650 BC. Magic squares of size 5x5 and 6x6 are found in an encyclopedia from Baghdad written in the ten’s century. They were engraved in stone and metal, worn as talismans, their usage believed to ensure longevity and prevention of diseases (from Wikipedia). Magic squares can be spotted in art all around the globe. The Indian Parshva temple still contains a 4x4 magic square carved in the 10th century. The Passion facade of Gaudi’s Sagrada Família church in Barcelona, designed by sculptor Josep Subirachs, features a 4×4 magic square with a sum of 33 that is the age of Jesus at the time of Passion.



What has fascinated people about Dürer’s square is that in addition to the regular magic properties, it contained more mathematical patterns and messages:

1) Numbers in each of the following five quadrants add up to 34.



2) Any pair of numbers symmetrically placed about the center of the square (such as these) sums to 17:



3) The two numbers in the middle of the bottom row give the date of the engraving: 1514



4) The numbers 1 and 4 at either side of the date correspond to, in English, the letters 'A' and 'D' which are the initials of the artist.

What about Ben Franklin? As a child and an adult he amused himself with creating large size magic squares such as this 8x8:



While his squares are not truly magic squares because sum of each of the diagonals is not equal to the sum of numbers in every row or column, they are still pretty cool. To compensate for a lack of the diagonal property Franklin defined a broken diagonal or “bended rows” properties where the sum of numbers in each highlighted bended rows is the same.



Magic squares could be considered the ancient relatives of the modern Sudoku game. In the late 19th century, French newspapers started publishing matrix puzzles – 9x9 magic squares with some numbers removed. Then the puzzle slowly evolved taking away the sum requirements and instead requiring that each row, column and diagonal have all the numbers from 1 to 9. And finally, the 3x3 sub-division was added producing the modern day Sudoku (you can click to play):



Thousands of Sudoku are published in the world daily. You have likely seen someone doing Sudoku on his way to work, at the playground, laundromat or in the coffee shop. A good story attesting to Sudoku’s popularity was published in the Sydney Morning Herald. In June 2008 an Australian drug-related jury trial costing over Australian $1,000,000 was aborted when it was discovered that five of the twelve jurors had been playing Sudoku instead of listening to the evidence.