.

Can you figure out a way to prove the following cute theorem?

If evenly divides , then evenly divides .

(Incidentally, the existence of this theorem constitutes good evidence that the “correct” definition of is , not .)

For example, evenly divides , and sure enough, evenly divides . evenly divides , and sure enough, evenly divides (in particular, ).

I know of two different ways to prove it; there are probably more! Neither of the proofs I know is particularly obvious, but they do not require any difficult concepts.

Welcome to the 86th Carnival of Mathematics! is semiprime, nontotient, and noncototient. It is also happy since and . In fact, it is the smallest happy, nontotient semiprime (the only smaller happy nontotient is 68—which is, of course, 86 in reverse—but 68 is not semiprime).

However, the most interesting mathematical fact about 86 (in my opinion) is that it is the largest known integer for which the decimal expansion of contains no zeros! In particular, . Although no one has proved it is the largest such , every up to (which is quite a lot, although still slightly less than the total number of integers) has been checked to contain at least one zero. The probability that any larger power of 2 contains no zeros is vanishingly small, given some reasonable assumptions about the distribution of digits in base-ten expansions of powers of two.

“Eighty-six” is also apparently some sort of slang term in American English, but it really has nothing to do with math, so who cares? Onward to the carnival! I had a lot of fun reading all the submissions, and have decided to organize them somewhat thematically—though they don’t always fit perfectly, so don’t assume you won’t be interested in a post just because of my categorization!

Art

Christian Perfect has started a series of posts on the theme of “Arty Maths”, with links to artistic images and videos with a mathematical bent. Above is a cool example, some sort of stellated polyhedron made out of money by Kristi Malakoff (you can find more here).

Katie Steckles submitted a link to Robby Ingebretsen’s blog post First Digital 3D Rendered Film (from 1972) and My Visit to Pixar. Katie says,

This is possibly the earliest example of a computer animation, and one of its two creators, Edwin Catmull, who went on to found Pixar, is credited with “having work[ed] out [the] math to handle things like texture mapping, 3D anti-aliasing and z-buffering”. Fascinating to think he had to invent all of that in order to do this!

Robby’s blog post (and the extensive comments on it) give a lot more context and fascinating details. And, of course, you can watch the video itself!

Mike Croucher of Walking Randomly writes about some cool mathematically-themed stained glass windows, and wonders whether anyone knows of any others.

Statistics/data analysis

Arthur Charpentier of Freakonometrics writes about Nonconvexity, and playing indoor paintball: if a bunch of people in a nonconvex playing area are all holding water pistols and shoot at the closest person, who doesn’t get wet?

Katie Steckles submitted a link to Data: it’s how stores know you’re pregnant, an article by Matthew Lane of Math Goes Pop! Ever wonder how companies can predict various things about you (such as whether you are pregnant!) based on your browsing habits and other publicly available data? This article explains some of the basic math underlying this sort of “data mining”.

John Cook of The Endeavour answers the question: What is randomness? in Random is as random does. It turns out that the best answer might just be to avoid answering at all!

Geometry

Augustus Van Dusen of thinkingmachineblog submitted a post titled Superellipse, saying

I read an article about Sergels torg, a plaza in Stockholm, being an example of a superellipse. When I looked up superellipse on Wolfram math world, I noticed that the area formula involved gamma functions. I then decided to derive the result myself to see if it could be simplified and how it would reduce to the familiar formula for the area of an ellipse.

Frederick Koh of White Group Mathematics shares a geometric solution to an optimization problem that doesn’t initially seem like it has anything to do with geometry.

Zachary Abel of Three-Cornered Things has written a series of three “excursions into the miraculous and interconnected workings of the humble triangle”: Many Morley Triangles, Several Sneaky Circles, and Three-Cornered Deltoids. These are some of my personal favorites from this month’s Carnival: chock-full of surprising mathematics and beautiful illustrations and animations!

Teaching

Colin Wright writes The Trapezium Conundrum: how should a trapezium (aka trapezoid if you’re from the US) be defined—with exactly one pair of parallel sides or at least one pair of parallel sides? More generally, how are definitions arrived at and agreed upon? The answer may depend on the audience.

Karen G. of school•a•rama muses upon the relationship between language and learning place value in her post Looking to Asia.

On her blog Math Mama Writes…, Sue VanHattum writes about Linear Algebra: Leading into the Eigen Stuff. Sue says, “I’m teaching linear algebra for the first time in over a decade. That has meant relearning it—a delightful experience.”

Paul Salomon of Lost In Recursion writes Exponents and the Scale of the Universe – a 21st Century math lesson, a fun story about how an initially dry lesson on exponents turned into a remarkable learning experience.

Fun

Alistair Bird submitted a link to Enormous Integers, saying,

It’s still a common enough misconception that pure mathematics research must be about larger and larger numbers, but it’s still nice to sometimes play up to this stereotype, as John Baez’s blogpost on Azimuth about ‘Enormous Integers’ does. Comments are worth a look too.

Pat Ballew writes on Pat’sBlog about Pandigital Primes: “exploring pandigital primes and finding out how handy Computer programming skills might be”.

Quick, what comes next in the series ? The answer, as explained by Steven Landsburg on his blog, The Big Questions, may surprise you! (Thanks to Katie Steckles for the submission, via Alexandre Borovik.)

Paul Salomon of Lost In Recursion displays The “Lost in Recursion” Recursion. Can you figure out what’s going on without getting lost in the “The ‘Lost in Recursion’ Recursion” recursion?

Stuff That Did Not Fit In Any Other Category But Is Still Awesome

Colin Beveridge of Flying Colours Maths submitted Secrets of the Mathematical Ninja: The surprising integration rule you don’t get taught in school, and writes,

When I stumbled across this rule, my reaction was ‘whoa.’ It’s quick, it’s extremely dirty, and it’s surprisingly accurate. The kind of thing the mathematical ninja dreams of.

Andrew Taylor writes a guest post, Electoral Reforms and Non-Transitive Dice, on The Aperiodical, explaining Why Choosing a Voting System is Hard in terms of a set of nontransitive dice.

Peter Rowlett, of Travels in a Mathematical World, opines in his post, What a nice job you have, that a popular ranking listing “mathematician” as one of the top ten best jobs shouldn’t just be accepted and repeated uncritically.

In her article How culture shaped a mathematician, Carol Clark gives a glimpse into the life and background of mathematician Skip Garibaldi. She writes:

Mathematicians see the world differently than me. I interviewed a mathematician to get a glimpse of that view, and learned how everything from fine art to popular films and books played a role in shaping that view.

The previous Carnival of Mathematics was hosted at Travels in a Mathematical World; next month, the 87th Carnival will be hosted by Mr. Chase at Random Walks, so start getting your submissions ready now! For lists of past and future carnivals, instructions on submitting, and answers to frequently asked questions, see the main Carnival of Mathematics site. The next Math Teachers at Play carnival is also coming up soon, with a submission deadline of this Friday.

As mathematical problems go, the “traveling salesman problem” (TSP) is a rare gem: it is simultaneously of great theoretical, historical, and practical interest. On the theoretical front, it is a well-known example of the class of “NP-complete” problems, which lie at the heart of the million-dollar “P vs NP” question (which I still intend to explain at some point). Historically, it has been studied for almost 200 years (given a sufficiently inclusive definition of “study”), and has occupied a place in the public consciousness for at least the last 50. And this great historical interest is partly due to the problem’s practical significance.

So, what is it? Given a set of points in the plane (or, more generally, a set of points with “distances” specified somehow between each pair of points), the problem is to determine the shortest path which visits every point exactly once and then returns to the starting location. Of course, in one sense this is “easy”: just list all possible paths and compute the length of each. Note, however, that for a set of points, there are (that is, ) possible paths that visit every point once and then return to the start. Even with only points, that’s a whopping possible paths—if you could compute the length of one trillion paths every second, it would still take 280 million millenia (that’s years, slightly longer than I’ve been alive) to check all of them! And that’s only points—in practice, people want to solve the TSP for sets of points much larger than . So the point is not just to “solve” the problem; the real question is, can it be solved efficiently?

Amazingly, no one knows! But that hasn’t stopped people from coming up with extremely clever algorithms that seem to work well in practice, on very large sets of points (i.e. thousands, or even tens of thousands of points)—even though there are “pathological” inputs for which these algorithms do essentially no better than just trying every path. So these algorithms constitute a good solution to the TSP from a practical point of view but not a theoretical one!

William Cook’s new book, In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation, does a wonderful job presenting the history and significance of the TSP and an overview of cutting-edge research. It’s a beautiful, visually rich book, full of color photographs and diagrams that enliven both the narrative and mathematical presentation. And it includes a wealth of information (perhaps even a bit too much at times; I got lost in a few places). But it actually bills itself, partly, as an introduction to cutting-edge ideas in TSP research—and I think overall it succeeds admirably, explaining ideas in ways that are accessible but not patronizing. Read this book if you want a fun, beautifully illustrated introduction to (this one fascinating piece of) the edge of human knowledge!

I’ll be hosting Carnival #86 here, so please submit something! The deadline is May 1st, and I’ll post the carnival sometime the week after that.

We’re still trying to find a proof of the equation

which expresses the fact that a certain arithmetic procedure always seems to result, strangely enough, in the factorial of . Last time I introduced the idea of using a combinatorial proof, and gave a simple example involving a binomial coefficient identity.

In order for this idea to yield any fruit, we need a way to interpret the various pieces of the equation as counting something. Let’s go over the pieces one by one and discuss some ways to interpret them combinatorially.

that is,

Now, it’s possible (probable, in fact) that this can be proved by purely algebraic means. If you come up with such a proof I would love to see it! But I must confess that I spent several hours banging my head against it (algebraically speaking) without making any progress. Eventually I turned to the idea of a combinatorial proof.

What do I mean by that? Combinatorics is the subfield of mathematics concerned with counting. The essence of a combinatorial proof is to show that two different expressions are just two different ways of counting the same set of objects—and must therefore be equal. I’ve described some combinatorial proofs before, in counting the number of ways to distribute cookies.

As another simple example, consider the binomial coefficient identity

It’s certainly possible to prove this algebraically, by expanding out the binomial coefficients (using ), but we can give a more elegant proof, based on the fact that is the number of ways to choose a subset of things out of a set of things. For example, here are the ways to choose three things out a set of five:

Consider the first element of the size- set. Every subset of size either includes this first element, or it doesn’t. The number of size- subsets which do not include the first element is , since that’s the number of ways to choose things from the remaining elements. The number of size- subsets which do include the first element is , because they correspond to choosing of the remaining things. Therefore .

Here’s an illustration of how this works in the particular case when and :

Notice how the ten subsets from above have been split into two groups: the first group, on the left, are those that don’t include the first element; you can see that each of them corresponds to one of the size- subsets of the remaining four elements. The second group, on the right, are those that do include the first element; each corresponds to one of the size- subsets of the remaining four elements.

So that’s the idea of a combinatorial proof. And we want to do something similar for the identity we are trying to prove—although, of course, it’s going to be a bit more difficult!

You might have fun trying to think about what a combinatorial proof of our target equation might look like; although if you don’t have much experience with combinatorics you may have trouble coming up with what sorts of things the two sides of the equation might be counting! That’s what I’ll talk about in my next post.

For example:

Several commenters figured this out as well. Does this always happen? If so, can we prove it?

Let’s start by thinking about what happens when we do the successive-differencing procedure. If we start with the list , then we get . (I want to keep the letters in order, which is why I wrote instead of . Instead of subtracting the first value from the second, we can think of it as adding the negation of the first value to the second.) If we start with , we get

.

(The negation of is ; adding this to yields .) From we get

Do you see any patterns yet? Let’s do one more. From the above calculation we can already see that doing four iterations on will result in (do you see why?). Doing one final iteration gives us

.

Hmm. Let’s make a table.

	Result
1
2
3
4
5

I included one more row (which you can verify if you like). Now do you see a pattern? The coefficients seem to be taken from Pascal’s triangle, but with alternating signs!

In fact, it’s actually not too hard to see why this happens. At each step we take two offset copies of the previous row (by "offset" I mean that the letters are shifted by one, like and ) and add the negation of the first to the second. Since the signs are alternating, we really end up adding absolute values of the coefficients. Probably the best way to really see this is through an example:

Notice how we flip all the signs in the first row, so that they match the signs in the second row. But this is exactly how Pascal’s triangle is generated—each row is the sum of the previous row with itself, offset by one.

Now, in the real problem, we don’t start with , but with the th powers of consecutive integers. Let’s call the first integer , so the sequence of consecutive integers is . Given this, we can now write down an expression for what we end up with after doing the iterated difference procedure:

Let’s break this down a bit. We know that we get a sum of terms; that’s the part (you can read more about sigma notation here). We’ll use to index the terms. We also know that the terms alternate sign, so we need to include raised to some power involving ; the last term is always positive, so we use , which is equal to when . The binomial coefficient gives us the th entry on the th row of Pascal’s triangle. Finally, of course, is the th power of one of the integers we started with.

The claim, therefore, is that

And I will prove it to you, with pretty pictures, as promised!

As just announced by Bill Gasarch, this is a grid which has been four-colored (that is, each point in the grid has been assigned one of four colors) in such a way that there are no monochromatic rectangles, that is, no four grid points forming the corners of an axis-aligned rectangle are all of the same color. For example, if we change the top-left grid point to red, we can see several monochromatic rectangles pop up:

Or here’s another version where I randomly picked a grid point in the middle, changed its color, and sure enough, more monochromatic rectangles result:

As you can try verifying for yourself (and as I also verified using a computer program), there are no such monochromatic rectangles in the four-coloring at the top of this post! (If you want to play with the four-coloring yourself, here it is in a simple data format.)

For several years no one knew if this was possible, and Bill had offered a prize of $289 (that’s , of course) to anyone who could find such a four-coloring. The prize will be collected by Bernd Steinbach and Christian Posthoff—you can find more details in Bill’s post. No one yet knows exactly how they found their four-coloring, but apparently they will be presenting a paper about it in May. I’ll try to write more about it then (if I understand it at all)!

If you’re interested in reading more about the history and math behind this problem (and to get some intuition for why it is difficult), take a look at these posts by Brian Hayes on his blog, bit-player: The 17×17 challenge and 17 x 17: A nonprogress report. I also remember seeing Here’s a fun interactive applet where you can play around with the problem, created by Martin Schweitzer.

Nine Algorithms that Changed the Future: the Ingenious Ideas that Drive Today’s Computers, by John MacCormick. Princeton University Press, 2012.

I’m often wary of books written for general audiences on technical topics. It’s quite difficult to write in a way that is both accessible to a wide audience and technically accurate. Many such books end up sacrificing accuracy in the name of accessibility, trying to convey just the “intuition” or “general sense” of some topic, but often end up giving people the wrong idea instead.

I was quite happy to find, therefore, that John MacCormick nails it: “9 Algorithms that Changed the Future” is technically right on the money, but manages to explain things in ways that are both understandable and fun. Want to understand how Google ranks search results? Or how Amazon manages to never lose or mess up your order information, even though they get hundreds of thousands of orders each day and (as we all know) networks and hard drives are unreliable? Ever wonder how you can order something over the internet without your credit card number being stolen? Or how “zip” is able to make your files smaller, seemingly by magic? Even if you have never wondered about these things, perhaps I have made you wonder about them just now. And that’s exactly the point of this book: there are quite a few ingenious algorithmic ideas that most of us rely on every day that we rarely—or never—even stop to wonder about.

For example, I actually learned something new: I knew about public-key cryptography but had never really known much about Diffie-Hellman key exchange, which is what allows your web browser to talk to, say, Amazon’s servers securely even though they have never communicated before. It’s like having a secret conversation in code with a pen-pal whom you’ve never met, even though lots of people are reading your mail. How can you ever agree on a secret code in the first place without the people reading your mail finding out (and hence being able to read all your subsequent coded messages)? Sounds impossible, doesn’t it? But it turns out that it is possible, with some clever ideas, which MacCormick skillfully explains using a fun metaphor about mixing colors of paint.

Each chapter starts out very simply, gradually building up more complex examples until you reach a full understanding of the algorithm being explained. Along the way MacCormick introduces the “tricks”—the clever, central ideas—that make each algorithm work. The writing is excellent: clear, precise, and fun. I highly recommend this book to anyone curious about the ingenious mathematical and algorithmic ideas underlying some of today’s most ubiquitous technology.