Month: October 2019

In sum, the sufficient condition (a.k.a. the “if” direction) allows you to get what you want. That is, if you assume the sufficient condition, you’ll obtain your desired conclusion. It’s enough. It’s sufficient.

On the other hand, the necessary condition (a.k.a. the “only if” direction) is the one you must assume in order to get what you want. In other words, if you don’t have the necessary condition then you can’t reach your desired conclusion. It is necessary.

Here’s a little graphic which summarizes this:

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Here’s one thing to impress your friends with the next time you order a takeaway: new and exotic ways to slice a pizza.

Most of us divide a pizza using straight cuts that all meet in the middle. But what if the centre of the pizza has a topping that some people would rather avoid, while others desperately want crust for dipping?

Mathematicians had previously come up with a recipe for slicing – formally known as a monohedral disc tiling – that gives you 12 identically shaped pieces, six of which form a star extending out from the centre, while the other six divide up the crusty remainder. You start by cutting curved three-sided slices across the pizza, then dividing these slices in two to get the inside and outside groups, as shown below.

Now Joel Haddley and Stephen Worsley of the University of Liverpool, UK, have generalised the technique to create even more ways to slice. The pair have proved you can create similar tilings from curved pieces with any odd number of sides – known as 5-gons, 7-gons and so on (shaded below) – then dividing them in two as before. “Mathematically there is no limit whatsoever,” says Haddley, though you might find it impractical to carry out the scheme beyond 9-gon pieces.

Haddley and Worsley went one further by cutting wedges in the corners of their shapes, creating bizarre, spikey pieces that still form a circle (the image below shows this happening with 5-gons). “It’s really surprising,” says Haddley.

As with many mathematical results, its usefulness isn’t immediately obvious. The same is true of another pizza theorem, which looks at what happens when a pizza is haphazardly cut off-centre.

“I’ve no idea whether there are any applications at all to our work outside of pizza-cutting,” says Haddley, who has actually tried slicing a pizza in this way for real (see below). But the results are “interesting mathematically, and you can produce some nice pictures”.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

A maths problem previously tackled with the help of a computer, which produced a proof the size of Wikipedia, has now been cut down to size by a human. Although it is unlikely to have practical applications, the result highlights the differences between two modern approaches to mathematics: crowdsourcing and computers.

Terence Tao of the University of California, Los Angeles, has published a proof of the Erdős discrepancy problem, a puzzle about the properties of an infinite, random sequence of +1s and -1s. In the 1930s, Hungarian mathematician Paul Erdős wondered whether such a sequence would always contain patterns and structure within the randomness.

One way to measure this is by calculating a value known as the discrepancy. This involves adding up all the +1s and -1s within every possible sub-sequence. You might think the pluses and minuses would cancel out to make zero, but Erdős said that as your sub-sequences got longer, this sum would have to go up, revealing an unavoidable structure. In fact, he said the discrepancy would be infinite, meaning you would have to add forever, so mathematicians started by looking at smaller cases in the hopes of finding clues to attack the problem in a different way.

Last year, Alexei Lisitsa and Boris Konev of the University of Liverpool, UK used a computer to prove that the discrepancy will always be larger than two. The resulting proof was a 13 gigabyte file – around the size of the entire text of Wikipedia – that no human could ever hope to check.

Helping hands

Tao has used more traditional mathematics to prove that Erdős was right, and the discrepancy is infinite no matter the sequence you choose. He did it by combining recent results in number theory with some earlier, crowdsourced work.

In 2010, a group of mathematicians, including Tao, decided to work on the problem as the fifth Polymath project, an initiative that allows professionals and amateurs alike to contribute ideas through SaiBlogs and wikis as part of mathematical super-brain. They made some progress, but ultimately had to give up.

“We had figured out an interesting reduction of the Erdős discrepancy problem to a seemingly simpler problem involving a special type of sequence called a completely multiplicative function,” says Tao.

Then, in January this year, a new development in the study of these functions made Tao look again at the Erdős discrepancy problem, after a commenter on his SaiBlog pointed out a possible link to the Polymath project and another problem called the Elliot conjecture.

Not just conjecture

“At first I thought the similarity was only superficial, but after thinking about it more carefully, and revisiting some of the previous partial results from Polymath5, I realised there was a link: if one could prove the Elliott conjecture completely, then one could also resolve the Erdős discrepancy problem,” says Tao.

“I have always felt that that project, despite not solving the problem, was a distinct success,” writes University of Cambridge mathematician Tim Gowers, who started the Polymath project and hopes that others will be encouraged to participate in future. “We now know that Polymath5 has accelerated the solution of a famous open problem.”

Lisitsa praises Tao for doing what his algorithm couldn’t. “It is a typical example of high-class human mathematics,” he says. But mathematicians are increasingly turning to machines for help, a trend that seems likely to continue. “Computers are not needed for this problem to be solved, but I believe they may be useful in other problems,” Lisitsa says.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

Felker prefaces the quote by saying,

Giving up on math means you don’t believe that careful study can change the way you think.

He further notes that writing, like math, “is also not something that anyone is ‘good’ at without a lot of practice, but it would be completely unacceptable to think that your composition skills could not improve.”

Friends, this is so true! Being ‘good’ at math boils down to hard work and perseverance, not whether or not you have the ‘math gene.’ “But,” you might protest, “I’m so much slower than my classmates are!” or “My educational background isn’t as solid as other students’!” or “I got a late start in mathematics!”* That’s okay! A strong work ethic and a love and enthusiasm for learning math can shore up all deficiencies you might think you have. Now don’t get me wrong. I’m not claiming it’ll be a walk in the park. To be honest, some days it feels like a walk through an unfamiliar alley at nighttime during a thunderstorm with no umbrella. But, you see, that’s okay too. It may take some time and the road may be occasionally bumpy, but it can be done!

This brings me to another point that Felker makes: If you enjoy math but find it to be a struggle, do not be discouraged! The field of math is HUGE and its subfields come in many different flavors. So for instance, if you want to be a math major but find your calculus classes to be a challenge, do not give up! This is not an indication that you’ll do poorly in more advanced math courses. In fact, upper level math classes have a completely (I repeat, completely!) different flavor than calculus. Likewise, in graduate school you may struggle with one course, say algebraic topology, but find another, such as logic, to be a breeze. Case in point: I loathed real analysis as an undergraduate** and always thought it was pretty masochistic. But real analysis in graduate school was nothing like undergraduate real analysis (which was more like advanced calculus), and now – dare I say it? – I sort of enjoy the subject. (Gasp!)

All this to say that although Felker’s article is aimed at folks who may be afraid to take college-level math, I think it applies to math majors and graduate students too. I highly recommend you read it if you ever need a good ‘pick-me-up.’ And on those days when you feel like the math struggle is harder than usual, just remember:

Even the most accomplished mathematicians had to learn HOW to learn this stuff!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*