Long before the Sudoku number puzzle became the crack cocaine of brain teasers, a bored Benjamin Franklin jotted down a couple of much more complex number puzzles of his own.
But, exactly how the U.S. inventor, printer and statesman devised his so-called magic squares 270 years ago, without the help of a computer, and how many permutations are possible under his distinct mathematical design have stumped mathematicians ever since.
Now, a trio of Canadian number crunchers have used modern-day technology to come up with one of the answers.
Using Franklin's rules, there are 1,105,920 variations of his magic square.
That is according to Peter Loly, a professor in the University of Manitoba's department of physics and astronomy, who co-authored a study published online this week by the Proceedings of the Royal Society.
The answer came as a surprise to Prof. Loly and two graduate students who started working on the problem in 2004. After all, a previous estimate pegged the answer at something more than a handful and less than 228 trillion.
It would also likely come as a surprise to Franklin, who later wondered if puzzle- making was the best use of his time.
Franklin was an innovator, but he wasn't the first to design magic squares. Math historians place the first recorded magic squares as Chinese, dating back to 2,800 BC. They were the simple three-column by three-row variety. Over the years, rows and columns were added to magic squares, rules developed around them as well as competition to figure out how many variations of each magic square are possible.
In the 1600s, for example, mathematician Bernard Frénicle de Bessy outlined all 880 variations of the 4x4 magic square, which centuries earlier was deemed to have "mystical properties."
A computer algorithm designed to generate the 9x9 Sudoku puzzles, which appear in newspapers (including this one) and are sold at bookstands, could pump out 6 sextillion unique puzzles, according to its creator who spent six years working on the program.
In 1736-37, Franklin was working as a clerk in the Pennsylvania Assembly when, to amuse himself, he doodled an 8x8 magic square -- the same size as a chess board.
Franklin's puzzle placed the digits one through 64 in each box on the grid. According to his rules, each row and each column must add up to 260.
But within his puzzle, there are also other puzzles.
Any 2x4 block of rows and columns within the grid must also add up to 260. At the same time, "bent rows" or V-shapes, which can go sideways or upside down and form other patterns on the grid, must also add up to 260.
Franklin also created a 16x16 puzzle, in which the sum is 2,056, and which has similarly complicated rules. That large square was a feat Franklin himself described as "the most magically magical of any square ever made by any magician."
Paul Pasles, a mathematician at Villanova University in Pennsylvania, the state where Franklin spent most of his life, is considered one of the leading scholars on Franklin's magic square.
"Few contributions to mathematics are still remembered after two centuries time, but Ben made his mark," Prof. Pasles wrote in the Franklin Gazette, published by the Friends of Franklin, a group dedicated to the study of the man on the U.S. $100 bill.
But the study of the magic square, he adds, is "purely theoretical and apparently useless aside from its role as an intellectual stimulant."
The problem-solving could be used to develop solutions to other problems and test other theories, Prof. Loly explained. It could also be used in cryptography to make information more secure. And the patterns magic squares generate have even been incorporated into architecture, he said.
