- 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.
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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.
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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."
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.
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Our hearts beat at a resting rate of 60 to 100 beats per minute. Lots of other things pulse, too. The colors we see and the pitches we hear, for example, are due to the different wave frequencies ("pulses") of light and sound waves.
Now, a study in the journal Geoscience Frontiers finds that Earth itself has a pulse, with one "beat" every 27.5 million years. That's the rate at which major geological events have been occurring as far back as geologists can tell.
A planetary calendar has 10 dates in red
Credit: Jagoush / Adobe Stock
According to lead author and geologist Michael Rampino of New York University's Department of Biology, "Many geologists believe that geological events are random over time. But our study provides statistical evidence for a common cycle, suggesting that these geologic events are correlated and not random."
The new study is not the first time that there's been a suggestion of a planetary geologic cycle, but it's only with recent refinements in radioisotopic dating techniques that there's evidence supporting the theory. The authors of the study collected the latest, best dating for 89 known geologic events over the last 260 million years:
- 29 sea level fluctuations
- 12 marine extinctions
- 9 land-based extinctions
- 10 periods of low ocean oxygenation
- 13 gigantic flood basalt volcanic eruptions
- 8 changes in the rate of seafloor spread
- 8 times there were global pulsations in interplate magmatism
The dates provided the scientists a new timetable of Earth's geologic history.
Tick, tick, boom
Credit: New York University
Putting all the events together, the scientists performed a series of statistical analyses that revealed that events tend to cluster around 10 different dates, with peak activity occurring every 27.5 million years. Between the ten busy periods, the number of events dropped sharply, approaching zero.
Perhaps the most fascinating question that remains unanswered for now is exactly why this is happening. The authors of the study suggest two possibilities:
"The correlations and cyclicity seen in the geologic episodes may be entirely a function of global internal Earth dynamics affecting global tectonics and climate, but similar cycles in the Earth's orbit in the Solar System and in the Galaxy might be pacing these events. Whatever the origins of these cyclical episodes, their occurrences support the case for a largely periodic, coordinated, and intermittently catastrophic geologic record, which is quite different from the views held by most geologists."
Assuming the researchers' calculations are at least roughly correct — the authors note that different statistical formulas may result in further refinement of their conclusions — there's no need to worry that we're about to be thumped by another planetary heartbeat. The last occurred some seven million years ago, meaning the next won't happen for about another 20 million years.
When I was a junior in college, my electromagnetism professor had an awesome idea. Apart from the usual homework and exams, we were to give a seminar to the class on a topic of our choosing. The idea was to gauge which area of physics we would be interested in following professionally.
Professor Gilson Carneiro knew I was interested in cosmology and suggested a book by Nobel Prize Laureate Steven Weinberg: The First Three Minutes: A Modern View of the Origin of the Universe. I still have my original copy in Portuguese, from 1979, that emanates a musty tropical smell, sitting on my bookshelf side-by-side with the American version, a Bantam edition from 1979.
Inspired by Steven Weinberg
Books can change lives. They can illuminate the path ahead. In my case, there is no question that Weinberg's book blew my teenage mind. I decided, then and there, that I would become a cosmologist working on the physics of the early universe. The first three minutes of cosmic existence — what could be more exciting for a young physicist than trying to uncover the mystery of creation itself and the origin of the universe, matter, and stars? Weinberg quickly became my modern physics hero, the one I wanted to emulate professionally. Sadly, he passed away July 23rd, leaving a huge void for a generation of physicists.
What excited my young imagination was that science could actually make sense of the very early universe, meaning that theories could be validated and ideas could be tested against real data. Cosmology, as a science, only really took off after Einstein published his paper on the shape of the universe in 1917, two years after his groundbreaking paper on the theory of general relativity, the one explaining how we can interpret gravity as the curvature of spacetime. Matter doesn't "bend" time, but it affects how quickly it flows. (See last week's essay on what happens when you fall into a black hole).
The Big Bang Theory
For most of the 20th century, cosmology lived in the realm of theoretical speculation. One model proposed that the universe started from a small, hot, dense plasma billions of years ago and has been expanding ever since — the Big Bang model; another suggested that the cosmos stands still and that the changes astronomers see are mostly local — the steady state model.
Competing models are essential to science but so is data to help us discriminate among them. In the mid 1960s, a decisive discovery changed the game forever. Arno Penzias and Robert Wilson accidentally discovered the cosmic microwave background radiation (CMB), a fossil from the early universe predicted to exist by George Gamow, Ralph Alpher, and Robert Herman in their Big Bang model. (Alpher and Herman published a lovely account of the history here.) The CMB is a bath of microwave photons that permeates the whole of space, a remnant from the epoch when the first hydrogen atoms were forged, some 400,000 years after the bang.
The existence of the CMB was the smoking gun confirming the Big Bang model. From that moment on, a series of spectacular observatories and detectors, both on land and in space, have extracted huge amounts of information from the properties of the CMB, a bit like paleontologists that excavate the remains of dinosaurs and dig for more bones to get details of a past long gone.
How far back can we go?
Confirming the general outline of the Big Bang model changed our cosmic view. The universe, like you and me, has a history, a past waiting to be explored. How far back in time could we dig? Was there some ultimate wall we cannot pass?
Because matter gets hot as it gets squeezed, going back in time meant looking at matter and radiation at higher and higher temperatures. There is a simple relation that connects the age of the universe and its temperature, measured in terms of the temperature of photons (the particles of visible light and other forms of invisible radiation). The fun thing is that matter breaks down as the temperature increases. So, going back in time means looking at matter at more and more primitive states of organization. After the CMB formed 400,000 years after the bang, there were hydrogen atoms. Before, there weren't. The universe was filled with a primordial soup of particles: protons, neutrons, electrons, photons, and neutrinos, the ghostly particles that cross planets and people unscathed. Also, there were very light atomic nuclei, such as deuterium and tritium (both heavier cousins of hydrogen), helium, and lithium.
Cosmic alchemy
So, to study the universe after 400,000 years, we need to use atomic physics, at least until large clumps of matter aggregate due to gravity and start to collapse to form the first stars, a few millions of years after. What about earlier on? The cosmic history is broken down into chunks of time, each the realm of different kinds of physics. Before atoms form, all the way to about a second after the Big Bang, it's nuclear physics time. That's why Weinberg brilliantly titled his book The First Three Minutes. It is during the interval between one-hundredth of a second and three minutes that the light atomic nuclei (made of protons and neutrons) formed, a process called, with poetic flair, primordial nucleosynthesis. Protons collided with neutrons and, sometimes, stuck together due to the attractive strong nuclear force. Why did only a few light nuclei form then? Because the expansion of the universe made it hard for the particles to find each other.
What about the nuclei of heavier elements, like carbon, oxygen, calcium, gold? The answer is beautiful: all the elements of the periodic table after lithium were made and continue to be made in stars, the true cosmic alchemists. Hydrogen eventually becomes people if you wait long enough. At least in this universe.
In this article, we got all the way up to nucleosynthesis, the forging of the first atomic nuclei when the universe was a minute old. What about earlier on? How close to the beginning, to t = 0, can science get? Stay tuned, and we will continue next week.
To Steven Weinberg, with gratitude, for all that you taught us about the universe.
