Carnival of Mathematics 134

Sunday, May 15, 2016 · 6 min read

Welcome to the 134th Carnival of Mathematics, a monthly roundup of mathematics-related blog posts organized by The Aperiodical. The previous edition was hosted by Matthew at Chalkdust Magazine.

Tradition obliges me to begin with some facts about the number 134.

One of my best friends works backstage at my high school’s theater. He has a grueling and often thankless job, which involves staying backstage till midnight painting sets only to watch the actors get the applause the next evening. But he does it anyway, because he honestly loves it.

He reminds me of the number 134: the stage tech of numbers, enabling primes to shine without ever being recognized as one.

134 is the average of two consecutive odd primes, and both its double and its square are one less than a prime. The number 1344+1343+1342+1341+1340 is prime, as is 10134+7. The sum of the first 134 primes divides the product of the first 134 primes, and the sum of squares of 134’s prime factors… is prime.

Yet a number does not have to be prime to be special. 134 redeems itself by having the beautiful property,

\[ \binom{1+3+4}{1} + \binom{1+3+4}{3} + \binom{1+3+4}{4} \]

And now, the Carnival.

Writing this edition of the Carnival felt like writing a series of book reviews, but instead of dull Victorian literature, I got to review exciting mathematics written by fellow bloggers around the world. To celebrate this, I’d like to begin this roundup with a rather thought-provoking book review—of an actual book.

I received a book Really Big Numbers by Richard Schwartz for review. I was supposed to write the review a long time ago, but I’ve been procrastinating. Usually, if I like a book, I write a review very fast. If I hate a book, I do not write a review at all. With this book I developed a love-hate relationship.

The same blogger also writes about her feelings towards math, and I would like to mention one such article.

I remember I once bought a metal brainteaser that needed untangling. The solution wasn’t included. Instead, there was a postcard that I needed to sign and send to get a solution. The text that needed my signature was, “I am an idiot. I can’t solve this puzzle.” I struggled with the puzzle for a while, but there was no way I would have signed such a postcard, so I solved it. In a long run I am glad that the brainteaser didn’t provide a solution. …[T]here are no longer any discoveries. There is no joy. People consume the solution, without realizing why this puzzle is beautiful and counterintuitive.

Perhaps appropriately, the next entry is a puzzle, and its creator (who describes himself as a “stand-up mathematician”!) admits to not knowing the answer.

For completeness, I also include an article about solving a puzzle—about the joy of discovery and the frustration of exploring large search spaces.

I’m a software developer myself and so the most interesting part for me was finding shortcuts to the answer. While a computer can find the solution by brute force fairly quickly, it still takes a bit of time as there are 3,628,800 possible answers to check. … I read a little further, and found out that indeed there were shortcuts that could be made—this is the power of mathematical thinking in problem-solving: a monumental task can become considerably easier by simply applying a little bit of thought.

As we all know, mathematics isn’t just about solving brain-teasers. With that in mind, here are some practical applications of math.

What’s the most effective strategy for loading an airplane? Most airlines tend to work from the back to the front, accepting first the passengers who will sit in high-numbered rows (say, rows 25-30), waiting for them to find their seats, and then accepting the next five rows, and so on. Both the airline and the passengers would be glad to know that this is the most effective strategy. Is it?

On a less serious but equally practical note, we can apply math to almost any game you can think of.

From Risk to tic-tac-toe, popular games involve tons of strategic decisions, probability and math. So one happy consequence of being a data nerd is that you may have an advantage at something even non-data nerds understand: winning.

While applying math to games is practical (and perhaps even lucrative!) we all know that the proof of the pudding lies… in proofs. Proofs are there to reassure us that a statement really is true.

Pythagoras’s Theorem is perhaps the most famous theorem in maths. It is also very old, and for over 2500 years mathematicians have been explaining why it is true.

Thinking about it, perhaps I lied to you. Often, the most beautiful proofs are there, not to reassure us, but to explain why a seemingly ridiculous statement might actually be true. Here is one such outrageous statement and its proof.

This code, somewhat surprisingly, generates Fibonacci numbers.

def fib(n):
    return (4 << n*(3+n)) // ((4 << 2*n) - (2 << n) - 1) & ((2 << n) - 1)

In this blog post, I’ll explain where it comes from and how it works.

However, proofs are often written in scary, technical language with no intuition behind them. In reality, a proof is a narrative, and what better way to present a narrative than a story? The next entry takes on the challenge.

This may seem much harder than just counting the crossings, but it works well on an abstract level—that is, when we don’t know what the drawing looks like, but still have to prove things about it. This is what makes the theorem and its proof beautiful—we will prove there is something common to infinitely many different drawings of K_5. … This presentation is a first attempt at a response to the criticism: “Why don’t mathematicians or scientists ever write the way they think?” In fact, this “unnecessarily” long exposition does not mean that it is difficult; my hope is quite the opposite.

While an appreciation for proofs is healthy for any mathematician, some fascinating mathematics can emerge from seat-of-the-pants conjectures, calculations, and manipulations.

For most people it’d be bad news to end up with some complicated expression or long seemingly random number — because it wouldn’t tell them anything. But Ramanujan was different. Littlewood once said of Ramanujan that “every positive integer was one of his personal friends.” And between a good memory and good ability to notice patterns, I suspect Ramanujan could conclude a lot from a complicated expression or a long number. For him, just the object itself would tell a story.

While we’re on the topic of biographies, here’s another interesting character in the history of math.

…Shannon built a machine that did arithmetic with Roman numerals, naming it THROBAC I, for Thrifty Roman-Numeral Backward-Looking Computer. He built a flame-throwing trumpet and a rocket-powered Frisbee. He built a chess-playing automaton that, after its opponent moved, made witty remarks. Inspired by the late artificial-intelligence pioneer Marvin Minsky, he designed what was dubbed the Ultimate Machine: flick the switch to “On” and a box opens up; out comes a mechanical hand, which flicks the switch back to “Off” and retreats inside the box.

Often, the most brilliant mathematicians are not the most adept at solving equations, but the most insightful in finding math in places you may never have dreamed of.

Several students came to speak with me about the essay, including the second author. He told me that he just didn’t see any symmetry around him. At the time, we were standing in a building (the cafeteria) with sine-curve shaped ceiling beams arranged in parallel, supported by arms with 4-fold rotational symmetry. The light fixtures were hemispherical, with slits cut out in in an 8-fold rotational pattern. Parallel sliced kiwi fruits were being served, with their near rotational symmetry. I merely pointed these things out, saying that we are surrounded by symmetry, all we need to do is look and listen.

And, for a completely different perspective, here’s a lighthearted webcomic to bring a humorous end to this edition of the Carnival of Mathematics.

That’s all for this edition of the Carnival of Mathematics! Join us next time for the 135th edition, hosted by Gaurish4Math.

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