Can You Solve a Puzzle Unsolved Since 1996?

A challenge that the famous puzzler Martin Gardner put forward long ago remains open

For thousands of years people have played with magic squares—arrays of distinct numbers whose rows, columns and diagonals add to the same total.* A simple 3-by-3 array that sums to 15 every which way appears on the back of a turtle in the legend of Lo Shu, a Chinese tale from 650 B.C. Medieval mathematicians in the Middle East and India studied magic squares of varying sizes and Albrecht Dürer included a 4-by-4 magic square in his famous engraving, Melencolia I, in 1514. Today amateur and professional mathematicians continue to devise new magic squares, even adding extra dimensions to envision 3-D magic cubes and 4-D magic tesseracts.

Leonhard Euler, an 18th-century mathematician, puzzled over another type of exotic magic square, one made entirely of squared numbers. In 1770 he introduced the first 4-by-4 example of a magic square of squares (below), along with a formula for producing others.

On supporting science journalism

If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

Many 4-by-4 magic squares of squares are now known and about 10 years ago Christian Boyerreported the first examples of 5-by-5, 6-by-6 and 7-by-7 magic squares of squares. To date, though, no one has discovered a 3-by-3 magic square of squares nor has anyone proved it impossible.

In 1996 Martin Gardner, who had written Scientific American’s Mathematical Games column for some 25 years, offered a $100 prize to anyone who could devise a solution. A year later magic square expert Lee Sallows described a near miss (see below), with only one diagonal summing differently (going from the top left to the bottom right yields 38,307, not 21,609—the total in all other directions).

So far, that’s as close as anyone has gotten.Will you be the first to solve either problem? Post solutions and magic squares of your own design below. If enough interesting submissions come in, some will be featured in a future blog post at www.ScientificAmerican.com *Clarification (10/20/14): This sentence was edited after posting. The original did not specify that the numbers in each of the magic squares are different from one another.

You will need to enable JavaScript in order to use the Widget.

Subscribe to Support Independent Journalism

Great science journalism requires human expertise, time, effort and creativity. And it costs money. That’s why I and the journalists here at Scientific American hope you’ll join our community.

When you subscribe, you are supporting staff and freelance journalists who are passionate about telling science stories that are true, important and compelling. Our editors and reporters are often experts in their fields, which means they understand the nuances of big discoveries and can untangle the breakthroughs from the hype. With a subscription, you are also supporting rigorous fact-checking to ensure the words we publish are precise and accurate. And you’re supporting original illustrations, graphics and photos that bring you closer to an advanced laboratory, an ice sheet in Antarctica or a space mission in orbit. You’re helping us craft other types of high-quality journalism as well: Our newsletters are carefully written, edited and curated by staffers you have or will come to know and love. Our Science Quickly podcast is based on original reporting, collaboration with editors and scientists and exacting production.

Subscriptions keep this engine running so we can continue to deliver thoughtful, rigorous and independent science journalism to you. In an era of viral misinformation, this work is crucial. If you value what we do, I hope you’ll consider joining us as a subscriber

Thank you,

Jeanna Bryner, Editor in Chief, Scientific American