'Magic square' math puzzle has gone unsolved since 1996

Think you can solve it? One mathematician has already offered about $1,000 and a bottle of champagne to whoever cracks it first.

'Magic square' math puzzle has gone unsolved since 1996
pxfuel.com
  • The puzzle involves a particularly complicated type of magic square.
  • Magic squares are square arrays containing distinct numbers, and the sums of the numbers in the columns, rows and diagonals must be equal.
  • In 1996, the recreational mathematics writer Martin Gardner offered $100 to whoever could solve a 3x3 magic square — but using squared numbers.

Magic squares have fascinated mathematicians for thousands of years, with the earliest known example dating back to 2,800 B.C.E., in China. The idea behind magic squares is simple, though the puzzles can get mind-numbingly complex.

First, take a square array — say, a 3x3 grid divided into 9 squares — and put a unique number in each square. But you must arrange the numbers such that the sums of the numbers in each row, column and diagonal add up to the same number.

Here's an example of a partially completed magic square. Try to figure out which numbers you'd need to put in the blank spaces in order to complete it.

Magic square

docdroid.net

Given that you need each column, row and diagonal to add up to 15, you'd need to fill in the empty squares with a 9, 7 and 8.

Magic square

docdroid.net

That may be easy enough. But magic squares become far more difficult when they use squared numbers, a concept first exemplified by the 18th-century mathematician Leonhard Euler.

Since, mathematicians have generated various configurations of 4x4 magic squares of squares, including 5x5, 6x6 and 7x7 versions. But nobody has yet proven that a 3x3 magic square of squares is possible — or impossible, for that matter.

To date, there have been at least two prizes offered to whoever can solve this longstanding puzzle. Martin Gardner, a science and mathematics writer who was perhaps best known for devising recreational mathematics games that appeared for 25 years in a column published by Scientific American, offered a prize of $100 in 1996 to whoever could crack the code first.

    "So far no one has come forward with a "square of squares"—but no one has proved its impossibility either," Gardner wrote in 1998 in Scientific American. "If it exists, its numbers would be huge, perhaps beyond the reach of today's fastest supercomputers."

Magic square

Melancholia I. (A 4x4 magic square is depicted in the top right of the painting.)

Dürer's

In 2005, the mathematician Christian Boyer raised the stakes by offering €1,000 plus a bottle of champagne to anyone who could complete a 3x3 magic square of squares — using seven, eight or nine distinct squared integers. (Boyer also offered a prize for anyone who can show the puzzle is impossible, and he lists smaller prizes for other unsolved puzzles on his website.)

While both prizes remain unclaimed, some people have come close to solving the 3x3 magic square of squares, like this configuration listed on Christian Boyer's website.

Magic square

Christian Boyer

To those unfamiliar with high-level mathematics, it may come as a surprise that there's no shortage of well-known unsolved math problems, from the inscribed square problem in Euclidean geometry, to the Bombieri–Lang conjecture in algebra. Solving some of the these puzzles could lead to useful applications in the real world. But cracking the magic square of squares problem? Not so much.

Still, that's unlikely to deter mathematicians from seeking solutions.

"Such a magic square would probably not have any practical use," Gardner wrote in Scientific American. "Why then are mathematicians trying to find it? Because it might be there."

Not to mention the champagne.

COVID-19 amplified America’s devastating health gap. Can we bridge it?

The COVID-19 pandemic is making health disparities in the United States crystal clear. It is a clarion call for health care systems to double their efforts in vulnerable communities.

Willie Mae Daniels makes melted cheese sandwiches with her granddaughter, Karyah Davis, 6, after being laid off from her job as a food service cashier at the University of Miami on March 17, 2020.

Credit: Joe Raedle/Getty Images
Sponsored by Northwell Health
  • The COVID-19 pandemic has exacerbated America's health disparities, widening the divide between the haves and have nots.
  • Studies show disparities in wealth, race, and online access have disproportionately harmed underserved U.S. communities during the pandemic.
  • To begin curing this social aliment, health systems like Northwell Health are establishing relationships of trust in these communities so that the post-COVID world looks different than the pre-COVID one.
Keep reading Show less

Decades of data suggest parenthood makes people unhappy

Decades of studies have shown parents to be less happy than their childless peers. But are the kids to blame?

(Photo by Alex Hockett / Unsplash)
Sex & Relationships
  • Folk knowledge assumes having children is the key to living a happy, meaningful life; however, empirical evidence suggests nonparents are the more cheery bunch.
  • The difference is most pronounced in countries like the United States. In countries that support pro-family policies, parents can be just as happy as their child-free peers.
  • These findings suggest that we can't rely on folk knowledge to make decisions about parenting, on either the individual or societal levels.
Keep reading Show less

Lonely? Hungry? The same part of the brain worries about both

MRI scans show that hunger and loneliness cause cravings in the same area, which suggests socialization is a need.

Credit: Dương Nhân from Pexels
Mind & Brain
  • A new study demonstrates that our brains crave social interaction with the same areas used to crave food.
  • Hungry test subjects also reported a lack of desire to socialize, proving the existence of "hanger."
  • Other studies have suggested that failure to socialize can lead to stress eating in rodents.
Keep reading Show less

A Chinese plant has evolved to hide from humans

Researchers document the first example of evolutionary changes in a plant in response to humans.

Credit: MEDIAIMAG/Adobe Stock
Surprising Science
  • A plant coveted in China for its medicinal properties has developed camouflage that makes it less likely to be spotted and pulled up from the ground.
  • In areas where the plant isn't often picked, it's bright green. In harvested areas, it's now a gray that blends into its rocky surroundings.
  • Herbalists in China have been picking the Fritillaria dealvayi plant for 2,000 years.
Keep reading Show less
Scroll down to load more…
Quantcast