'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
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  • 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

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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

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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.

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Urban foxes self-evolve, exhibiting Darwin’s domestication syndrome

A new study finds surprising evidence of the self-evolution of urban foxes.

A fox at the door of 10 Downing Street on Janurary 13, 2015.

Photo by JUSTIN TALLIS/AFP via Getty Images
Surprising Science
  • A study from the University of Glasgow finds urban foxes evolved differently compared to rural foxes.
  • The skulls of the urban foxes are adapted to scavenging for food rather than hunting it.
  • The evolutionary changes correspond to Charles Darwin's "domestication syndrome."

How much can living in the city change you? If you were an urban fox, you could be evolving yourself to a whole new stage and becoming more like a dog, according to a fascinating new study.

Researchers compared skulls from rural foxes around London with foxes who lived inside the city and found important variations. Rural foxes showed adaptation for speed and hunting after quick, small prey, while urban fox skulls exhibited changes that made it easier for them to scavenge, looking through human refuse for food, rather than chasing it. Their snouts were shorter and stronger, making it easier to open packages and chew up leftovers. They also have smaller brains, not meant for hunting but for interacting with stationary food sources, reports Science magazine.

Interestingly, there was much similarity found between the male and female skulls of the urban foxes.

The observed changes correspond to what Charles Darwin called the "domestication syndrome," comprised of traits that go along with an animal's transition from being wild, to tamed, to domesticated.

The study was led by Kevin Parsons, an evolutionary biologist at the University of Glasgow.

"What's really fascinating here is that the foxes are doing this to themselves," Parsons told the BBC. "This is the result of foxes that have decided to live near people, showing these traits that make them look more like domesticated animals."

The researchers are not suggesting you should go out and get a fox as a house-pet just yet. But they are seeing the evolutionary process taking place that's moving the urban foxes along the path towards becoming more like dogs and cats, explained the study's co-author Dr. Andrew Kitchener from National Museums Scotland.

A fox beneath a tree in Greenwich park, south east London

A fox beneath a tree in Greenwich park, south east London on May 14, 2020.

Photo by Glyn KIRK / AFP

"Some of the basic environmental aspects that may have occurred during the initial phases of domestication for our current pets, like dogs and cats, were probably similar to the conditions in which our urban foxes and other urban animals are living today," said Kitchener. "So, adapting to life around humans actually primes some animals for domestication."

The specimen came from the National Museum Scotland's collection of around 1,500 fox skulls.

You can read the study in the Proceedings of the Royal Society B.

fox sleeping beneath stadium seats

A fox at the LV County Championship, Division two match between Surrey and Derbyshire at The Brit Oval on April 9, 2010 in London, England.

Photo by Clive Rose/Getty Images

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