- 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.
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.
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."
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.
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.
- Paradoxes: 10 weird puzzles that will expand your mind - Big Think ›
- Strange eight-dimensional numbers could explain reality - Big Think ›
- The math problem that could change the world: Does P = NP? ›
Ash deposits of some of North America's largest volcanic eruptions.
The end of the world as we know it
Panoramic view of Yellowstone National Park
Image: Heinrich Berann for the National Park Service – public domain
Of the many freak ways to shuffle off this mortal coil – lightning strikes, shark bites, falling pianos – here's one you can safely scratch off your worry list: an outbreak of the Yellowstone supervolcano.
As the map below shows, previous eruptions at Yellowstone were so massive that the ash fall covered most of what is now the western United States. A similar event today would not only claim countless lives directly, but also create enough subsidiary disruption to kill off global civilisation as we know it. A relatively recent eruption of the Toba supervolcano in Indonesia may have come close to killing off the human species (see further below).
However, just because a scenario is grim does not mean that it is likely (insert topical political joke here). In this case, the doom mongers claiming an eruption is 'overdue' are wrong. Yellowstone is not a library book or an oil change. Just because the previous mega-eruption happened long ago doesn't mean the next one is imminent.
Ash beds of North America
Ash beds deposited by major volcanic eruptions in North America.
Image: USGS – public domain
This map shows the location of the Yellowstone plateau and the ash beds deposited by its three most recent major outbreaks, plus two other eruptions – one similarly massive, the other the most recent one in North America.
Huckleberry Ridge
The Huckleberry Ridge eruption occurred 2.1 million years ago. It ejected 2,450 km3 (588 cubic miles) of material, making it the largest known eruption in Yellowstone's history and in fact the largest eruption in North America in the past few million years.
This is the oldest of the three most recent caldera-forming eruptions of the Yellowstone hotspot. It created the Island Park Caldera, which lies partially in Yellowstone National Park, Wyoming and westward into Idaho. Ash from this eruption covered an area from southern California to North Dakota, and southern Idaho to northern Texas.
Mesa Falls
About 1.3 million years ago, the Mesa Falls eruption ejected 280 km3 (67 cubic miles) of material and created the Henry's Fork Caldera, located in Idaho, west of Yellowstone.
It was the smallest of the three major Yellowstone eruptions, both in terms of material ejected and area covered: 'only' most of present-day Wyoming, Colorado, Kansas and Nebraska, and about half of South Dakota.
Lava Creek
The Lava Creek eruption was the most recent major eruption of Yellowstone: about 640,000 years ago. It was the second-largest eruption in North America in the past few million years, creating the Yellowstone Caldera.
It ejected only about 1,000 km3 (240 cubic miles) of material, i.e. less than half of the Huckleberry Ridge eruption. However, its debris is spread out over a significantly wider area: basically, Huckleberry Ridge plus larger slices of both Canada and Mexico, plus most of Texas, Louisiana, Arkansas, and Missouri.
Long Valley
This eruption occurred about 760,000 years ago. It was centered on southern California, where it created the Long Valley Caldera, and spewed out 580 km3 (139 cubic miles) of material. This makes it North America's third-largest eruption of the past few million years.
The material ejected by this eruption is known as the Bishop ash bed, and covers the central and western parts of the Lava Creek ash bed.
Mount St Helens
The eruption of Mount St Helens in 1980 was the deadliest and most destructive volcanic event in U.S. history: it created a mile-wide crater, killed 57 people and created economic damage in the neighborhood of $1 billion.
Yet by Yellowstone standards, it was tiny: Mount St Helens only ejected 0.25 km3 (0.06 cubic miles) of material, most of the ash settling in a relatively narrow band across Washington State and Idaho. By comparison, the Lava Creek eruption left a large swathe of North America in up to two metres of debris.
The difference between quakes and faults
The volume of dense rock equivalent (DRE) ejected by the Huckleberry Ridge event dwarfs all other North American eruptions. It is itself overshadowed by the DRE ejected at the most recent eruption at Toba (present-day Indonesia). This was one of the largest known eruptions ever and a relatively recent one: only 75,000 years ago. It is thought to have caused a global volcanic winter which lasted up to a decade and may be responsible for the bottleneck in human evolution: around that time, the total human population suddenly and drastically plummeted to between 1,000 and 10,000 breeding pairs.
Image: USGS – public domain
So, what are the chances of something that massive happening anytime soon? The aforementioned mongers of doom often claim that major eruptions occur at intervals of 600,000 years and point out that the last one was 640,000 years ago. Except that (a) the first interval was about 200,000 years longer, (b) two intervals is not a lot to base a prediction on, and (c) those intervals don't really mean anything anyway. Not in the case of volcanic eruptions, at least.
Earthquakes can be 'overdue' because the stress on fault lines is built up consistently over long periods, which means quakes can be predicted with a relative degree of accuracy. But this is not how volcanoes behave. They do not accumulate magma at constant rates. And the subterranean pressure that causes the magma to erupt does not follow a schedule.
What's more, previous super-eruptions do not necessarily imply future ones. Scientists are not convinced that there ever will be another big eruption at Yellowstone. Smaller eruptions, however, are much likelier. Since the Lava Creek eruption, there have been about 30 smaller outbreaks at Yellowstone, the last lava flow being about 70,000 years ago.
As for the immediate future (give or take a century): the magma chamber beneath Yellowstone is only 5 percent to 15 percent molten. Most scientists agree that is as un-alarming as it sounds. And that its statistically more relevant to worry about death by lightning, shark, or piano.
Strange Maps #1041
Got a strange map? Let me know at strangemaps@gmail.com.
When the pandemic forced office employees into lockdown and cut them off from spending in-person time with their colleagues, they almost immediately realized that they favor remote work over their traditional office routines and norms.
As remote workers of all ages contemplate their futures – and as some offices and schools start to reopen – many Americans are asking hard questions about whether they wish to return to their old lives, and what they're willing to sacrifice or endure in the years to come.
Even before the pandemic, there were people asking whether office life jibed with their aspirations.
We spent years studying “digital nomads" – workers who had left behind their homes, cities and most of their possessions to embark on what they call “location independent" lives. Our research taught us several important lessons about the conditions that push workers away from offices and major metropolitan areas, pulling them toward new lifestyles.
Legions of people now have the chance to reinvent their relationship to their work in much the same way.
Big-city bait and switch
Most digital nomads started out excited to work in career-track jobs for prestigious employers. Moving to cities like New York and London, they wanted to spend their free time meeting new people, going to museums and trying out new restaurants.
But then came the burnout.
Although these cities certainly host institutions that can inspire creativity and cultivate new relationships, digital nomads rarely had time to take advantage of them. Instead, high cost of living, time constraints and work demands contributed to an oppressive culture of materialism and workaholism.
Pauline, 28, who worked in advertising helping large corporate clients to develop brand identities through music, likened city life for professionals in her peer group to a “hamster wheel." (The names used in this article are pseudonyms, as required by research protocol.)
“The thing about New York is it's kind of like the battle of the busiest," she said. “It's like, 'Oh, you're so busy? No, I'm so busy.'"
Most of the digital nomads we studied had been lured into what urbanist Richard Florida termed “creative class" jobs – positions in design, tech, marketing and entertainment. They assumed this work would prove fulfilling enough to offset what they sacrificed in terms of time spent on social and creative pursuits.
Yet these digital nomads told us that their jobs were far less interesting and creative than they had been led to expect. Worse, their employers continued to demand that they be “all in" for work – and accept the controlling aspects of office life without providing the development, mentorship or meaningful work they felt they had been promised. As they looked to the future, they saw only more of the same.
Ellie, 33, a former business journalist who is now a freelance writer and entrepreneur, told us: “A lot of people don't have positive role models at work, so then it's sort of like 'Why am I climbing the ladder to try and get this job? This doesn't seem like a good way to spend the next twenty years.'"
By their late 20s to early 30s, digital nomads were actively researching ways to leave their career-track jobs in top-tier global cities.
Looking for a fresh start
Although they left some of the world's most glamorous cities, the digital nomads we studied were not homesteaders working from the wilderness; they needed access to the conveniences of contemporary life in order to be productive. Looking abroad, they quickly learned that places like Bali in Indonesia, and Chiang Mai in Thailand had the necessary infrastructure to support them at a fraction of the cost of their former lives.
With more and more companies now offering employees the choice to work remotely, there's no reason to think digital nomads have to travel to southeast Asia – or even leave the United States – to transform their work lives.
During the pandemic, some people have already migrated away from the nation's most expensive real estate markets to smaller cities and towns to be closer to nature or family. Many of these places still possess vibrant local cultures. As commutes to work disappear from daily life, such moves could leave remote workers with more available income and more free time.
The digital nomads we studied often used savings in time and money to try new things, like exploring side hustles. One recent study even found, somewhat paradoxically, that the sense of empowerment that came from embarking on a side hustle actually improved performance in workers' primary jobs.
The future of work, while not entirely remote, will undoubtedly offer more remote options to many more workers. Although some business leaders are still reluctant to accept their employees' desire to leave the office behind, local governments are embracing the trend, with several U.S. cities and states – along with countries around the world – developing plans to attract remote workers.
This migration, whether domestic or international, has the potential to enrich communities and cultivate more satisfying work lives.
Rachael A. Woldoff, Professor of Sociology, West Virginia University and Robert Litchfield, Associate Professor of Business, Washington & Jefferson College
This article is republished from The Conversation under a Creative Commons license. Read the original article.
