Pumping the Primes

On your desktop is a black box. Actually it’s an orange box, because black boxes are usually painted “a highly visible vermilion colour known as international orange.” In any case, it’s an opaque box: You can’t see the whirling gears or the circuit boards or whatever else might be inside. There’s another, secret input: The tiny hole in the bottom right corner of the box. If you poke a paperclip into the hole, it will reset the mechanism.You have access only to the inputs and outputs. The input is a button labeled Next. The output is an adding machine tape.

Go ahead: Press the button. A number is printed on the tape. Press again and another number appears. Keep going. A few more. Notice anything special about those numbers? The sequence begins:

5, 3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, 5, 3, . . .oeis.org/A137613

Yep, they’re all primes. They are not in canonical order, and some of them appear more than once, but every number in the list is certifiably indivisible by any number other than 1 and itself. Does the pattern continue? Yes, there’s a proof of that. Do all primes eventually find a place in the sequence? The very first prime, 2, is missing. Whether all odd primes eventually turn up remains a matter of conjecture. On the other hand, it’s been proved that infinitely many distinct primes are included.


So what’s inside the box? Here’s the JavaScript function that calculates the numbers printed on the tape. There’s not much to it:

  var n = 2, a = 7;   // initial values
  
  function nextG() {
    var g = gcd(n, a);
    n = n + 1;
    a = a + g;
    return g;
  }

The function gcd(n, a) computes the greatest common divisor of n and a. As it happens, gcd is not a built-in function in JavaScript, but there’s a very famous algorithm we can easily implement:

  function gcd(x, y) {
    while (y > 0) {
      var rem = x % y;    // remainder operator
      x = y;
      y = rem;
    }
    return x;
  }

The value returned by nextG is not always a prime, but it’s always either \(1\) or a prime. To see the primes alone, we can simply wrap nextG in a loop that filters out the \(1\)s. The following function is called every time you press the Next button on the orange black boxSee the rest of the source code on GitHub.:

  function nextPrime() {
    var g;
    do g = nextG() while (g === 1);     // skip 1s
    return g;
  }

For a clearer picture of where those primes (and \(1\)s) are coming from, it helps to tabulate the successive values of the three variables n, a, and g.

n    2   3   4   5   6   7   8   9  10   11   12   13   14   15   16   17   18   19   20   21   22   23
a    7   8   9  10  15  18  19  20  21   22   33   36   37   38   39   40   41   42   43   44   45   46
g    1   1   1   5   3   1   1   1   1   11    3    1    1    1    1    1    1    1    1    1    1   23

From the given initial values \(n = 2\), \(a = 7\), we first calculate \(g = \gcd(2, 7) = 1\). Then \(n\) and \(a\) are updated: \(n = n + 1\), \(a = a + g\). On the next round the gcd operation again yields a \(1\): \(g = \gcd(3, 8) = 1\). But on the fourth iteration we finally get a prime: \(g = \gcd(5, 10) = 5\). The assertion that \(g\) is always either \(1\) or a prime is equivalent to saying that \(n\) and \(a\) have at most one prime factor in common.


This curious generator of primes was discovered in 2003, during a summer school exploring Stephen Wolfram’s “New Kind of Science.” A group led by Matthew Frank investigated various nested recursions, including this one:

\[a(n) = a(n-1) + gcd(n, a(n-1)).\]

With the initial condition \(a(1) = 7\), the sequence begins:

7, 8, 9, 10, 15, 18, 19, 20, 21, 22, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 69, . . .oeis.org/A106108

Participants noticed that the sequence of first differences — \(a(n) – a(n-1)\) — seemed to consist entirely of \(1\)s and primes:

1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, . . .oeis.org/A132199

Stripping out the \(1\)s, the sequence of primes is the same as that generated by the orange black box:

5, 3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, . . .oeis.org/A137613

During the summer school, Frank and his group computed 150 million elements of the sequence and observed no composite numbers, but their conjecture that the value is always \(1\) or prime remained unproved. One of the students present that summer was Eric S. Rowland, who had just finished his undergraduate studies and was about undertake graduate work with Doron Zeilberger at Rutgers. In 2008 Rowland took another look at the gcd-based prime generator and proved the conjecture.

The sequence beginning with \(a(1) = 7\) is not unique in this respect. Rowland’s proof applies to sequences with many other initial conditions as well—but not to all of them. For example, with the initial condition \(a(1) = 3765\), the list of “primes” begins:

53, 5, 57, 5, 9, 13, 7, 71, 3, 41, 3, 4019, 3, 8039, . . .

Neither 57 nor 9 is a prime.

A number of other mathematicians have since elaborated on this work. Vladimir Shevelev gave an alternative proof and clarified the conditions that must be met for the proof to apply. Fernando Chamizo, Dulcinea Raboso, and Serafín Ruiz-Cabello showed that even if a sequence includes composites, there is a number \(k\) beyond which all entries \(a(k)\) are \(1\) or prime. Benoit Cloitre explored several variations on the sequence, including one that depends on the least common multiple (lcm) rather than the greatest common factor; the lcm sequence is discussed further in a recent paper by Ruiz-Cabello.


Should we be surprised that a simple arithmetic procedure—two additions, a gcd, and an equality test—can pump out an endless stream of pure primality? I have been mulling over this question ever since I first heard about the Rowland sequence. I’m of two minds.

Part of the mystique of the primes is their unpredictability. We can estimate how many primes will be found in any given interval of the number line, and we can compile various other summary statistics, but no obvious rule or algorithm tells us exactly where the individual primes fall within the interval. For more on randomness and the primes, see the erudite essay of Terry Tao.There’s nothing random or indeterminate about the distribution of the primes, and yet, like a random sequence, the sequence of primes seems incompressible. The most concise way of describing the primes is just to list them. Thus a simple and compact formula that generates only primes seems a bit like magic.

But there’s another side to this story.See Underwood Dudley, “Formulas for Primes,” Mathematics Magazine, Vol. 56, No. 1 (Jan., 1983), pp. 17–22. It turns out the woods are full of prime-generating formulas. There’s this mysterious-looking formula for the nth prime, published in 1971 by J. M. Gandhi:

\[p_n = \left\lfloor 1 - \log_2 \left( -\frac{1}{2} + \sum_{d|P_{n-1}} \frac{\mu(d)}{2^d - 1} \right) \right \rfloor.\]

Here \(P_n\) is the primorial product \(p_{1}p_{2}p_{3} \ldots p_{n}\) and \(\mu\) is the Möbius function. (If you don’t know what the Möbius function is or why you should care, Peter Sarnak explains it all.)

Way back in 1947, W. H. Mills offered a formula with just three symbols and a pair of floor brackets. He proved that a real number \(A\) exists such that

\[\left \lfloor A^{3^{n}}\right \rfloor\]

is prime for all positive integers \(n\). One possible value oeis.org/A051021 of \(A\) lies somewhere in the neighborhood of 1.306377883863. The sequence of primes derived from that value begins:

2, 11, 1361, 2521008887, 16022236204009818131831320183, . . .oeis.org/A051254

A third example brings us back to the gcd function. For all \(n > 1\), \(n\) is prime if and only if \[\gcd((n - 1)!, n) = 1.\]

From this fact we can craft an algorithm that generates all the primes (and only the primes) in sequence.

The trouble with all these formulas is that they require prior knowledge of the primes, or else they have such knowledge already hidden away inside them. Solomon Golomb showed that Gandhi’s formula is just a disguised version of the sieve of Eratosthenes. The Mills formula requires us to calculate the constant \(A\) to very high accuracy, and the only known way to do that is to work backward from knowledge of the primes. As for \(\gcd((n – 1)!, n) = 1\), it’s really more of a joke than a formula; it just restates the definition that n is prime iff no integer greater than 1 divides it.

Underwood Dudley opined that formulas for the primes range “from worthless, to interesting, to astonishing.” That was back in 1983, before the Rowland sequence was known. Where shall we place this new formula on the Dudley spectrum?

Rowland argues that the sequence differs from the Gandhi and Mills formulas because it “is ‘naturally occurring’ in the sense that it was not constructed to generate primes but simply discovered to do so.” This statement is surely true historically. The group at the Wolfram summer school did not set out to find a prime generator but just stumbled upon it. However, perhaps the manner of discovery is not the most important criterion.

Let’s look again at what happens when the procedure NextG is invoked repeatedly, each time returning either \(1\) or a prime.

n    2   3   4   5   6   7   8   9  10   11   12   13   14   15   16   17   18   19   20   21   22   23
a    7   8   9  10  15  18  19  20  21   22   33   36   37   38   39   40   41   42   43   44   45   46
g    1   1   1   5   3   1   1   1   1   11    3    1    1    1    1    1    1    1    1    1    1   23

In the \(g\) row we find occasional primes or groups of consecutive primes separated by runs of \(1\)s. If the table were extended, some of these runs would become quite long. A good question to consider is how many times NextG must be called before you can expect to see a prime of a certain size—say, larger than 1,000,000. There’s an obvious lower bound. The value of \(gcd(n, a)\) cannot be greater than either \(n\) or \(a\), and so you can’t possibly produce a prime greater than a million until \(n\) is greater than a million. Since \(n\) is incremented by \(1\) on each call to NextG, at least a million iterations are needed. And that’s just a lower bound. As it happens, the Rowland sequence first produces a prime greater than 1,000,000 at \(n =\) 3,891,298; the prime is 1,945,649.

The need to invoke NextG at least \(k\) times to find a prime greater than \(k\) means that the Rowland sequence is never going to be a magic charm for generating lots of big primes with little effort. As Rowland remarks, “a prime \(p\) appears only after \(\frac{p – 3}{2}\) consecutive \(1\)s, and indeed the primality of \(p\) is being established essentially by trial division.”

Rowland also points out a shortcut, which is best explained by again printing out our table of successive \(n, a, g\) values, with an extra row for some \(a – n\) values:

n    2   3   4   5   6   7   8   9  10   11   12   13   14   15   16   17   18   19   20   21   22   23
a    7   8   9  10  15  18  19  20  21   22   33   36   37   38   39   40   41   42   43   44   45   46
g    1   1   1   5   3   1   1   1   1   11    3    1    1    1    1    1    1    1    1    1    1   23

a–n  5   5   5          11  11  11  11             23   23   23   23   23   23   23   23   23   23

Within each run of \(1\)s, \(a-n\) is a constant—necessarily so, because both \(a\) and \(n\) are being incremented by \(1\) on each iteration. What’s more, based on what we can see in this segment of the sequence, the value of \(a-n\) during a run of \(1\)s is equal to the next value of \(n\) that will yield a nontrivial gcd. This observation suggests a very simple way of skipping over all those annoying little \(1\)s. Whenever \(gcd(a, n)\) delivers a \(1\), set \(n\) equal to \(a-n\) and increment \(a\) by \(a – 2n\). Here are the first few values returned by this procedure:

5, 3, 11, 3, 23, 3, 47, 3, 95, 3, . . .

Uh oh. 95 is not my idea of a prime number. It turns out the shortcut only works when \(a-n\) is prime. To repair the defect, we could apply a primality test to each value of \(a-n\) before taking the shortcut. But if we’re going to build a primality test into our prime generator, we might as well use it directly to choose the primes.

It seems we are back where we began, and no closer to having a practical prime generator. Nevertheless, on Dudley’s scale I would not rank this idea as “worthless.” When you take a long hike around the shore of a lake, you eventually wind up exactly where you started, but that does not make the trip worthless. Along the way you may have seen something interesting, or even astonishing.

Posted in computing, mathematics | 8 Comments

Gazing into the CodeMirror

Marijn Haverbeke is a programmer based in Berlin whose book Eloquent JavaScript is freely available on the Web and published in various other formats by No Starch Press. It’s a fine book, and I warmly recommend it. Buying a copy would help support the author, but even Haverbeke might concede that the online version offers the superior experience. On the web, the code comes alive. You can run a program and see its output immediately. You can edit the program directly in the web page. You can write and run your own programs. It’s all frictionless.

The technology behind this magic trick is a JavaScript component called CodeMirror, which is also Haverbeke’s creation. It is a code editor that can be embedded in any web page, providing all the little luxuries we’ve come to expect in a modern programming environment: automatic indentation and bracket matching, syntax coloring, auto­completion. The program and all of its many addons are free and open source. By now it is very widely used but not as widely noticed, because it tends to get buried in the infrastructure of other projects. I am writing this brief note to bring a little attention to an underappreciated gem.

Haverbeke’s Eloquent JavaScript is not the only online book that relies on Codemirror. Another one that I find very engaging is Probabilistic Models of Cognition, by Noah D. Goodman and Joshua B. Tenenbaum, which introduces the Church programming language. There’s also The Design and Implementation of Probabilistic Programming Languages, by Goodman and Andreas Stuhlmüller, which provides a similar introduction to a language called WebPPL. And the Interactive SICP brings in-the-page editing to the Sussman-and-Abelson Dragon Wizard Book.

But CodeMirror has spread far beyond these pedagogic projects. It is built into the developer tools of both the Chrome and the Firefox web browsers. It is the editor module for the IPython notebook interface (a.k.a. Jupyter), which is also used by the Sage mathematics system. It’s in both Brackets and Light Table, two newish open-source code editors (which run as desktop applications rather than in a browser window). You’ll also find CodeMirror working behind the scenes at Bitbucket, JSFiddle, Paper.js, and close to 150 other placed.


I had only a vague awareness of CodeMirror until a few months ago, when the New England Science Writers put on a workshop for science journalists, Telling Science Stories with Code and Data. As part of that project I wrote an online tutorial, JavaScript in a Jiffy. CodeMirror was the obvious tool for the job, and it turned out to be a pleasure to work with. A single statement converts an ordinary textarea HTML element into fully equipped editor panel. Any text entered in that panel will automatically be styled appropriately for the selected programming language. The machinery allowing the user to run the code is almost as simple: Grab the current content of the editor panel, wrap it in a <script>...</script> tag, and append the resulting element to the end of the document. (Admittedly, this process would be messier with any language other than JavaScript.)

Just a screenshot, regrettably. You’ll need to go to JavaScript in a Jiffy to edit and run the code.CodeMirror editor window with a JavaScript fibonacci function, and the output in the panel below.

The trickiest part of the project was figuring out how to handle the output of the programs written in CodeMirror panels. Initially I thought it would be best to just use the browser’s JavaScript console, sending textual output as a series of console.log messages. This plan has the advantage of verisimilitude: If you’re actually going to create JavaScript programs, the console is where test results and other diagnostic information get dumped. You need to get used to it. But some of the workshop participants found the rigmarole of opening the browser’s developer tools cumbersome and confusing. So I went back and created pop-up panels within the page to display the output. (It still goes to the console as well.)

A project like this would have been beyond my abilities if I had had to build all the machinery myself. Having free access to such elegant and powerful tools leaves me with the dizzy sensation that I have stumbled into an Emerald City where the streets are paved with jewels. It’s not just that someone has taken the trouble to create a marvel like CodeMirror. They have also chosen to make it available to all of us. And of course Haverbeke is not alone in this; there’s a huge community of talented programmers, fiercely competing with one another to give away marvels of ingenuity. Who’d’ve thunk it?

Posted in computing | 2 Comments

AIpocalypse

The Murder by Robot in R.U.R. (Image from Wikipedia.)The robot that runs amok and turns on its maker has been a staple of fiction and film for at least a century. The plotline goes back to Karel ?apek’s 1921 play R.U.R., with earlier shadows of the same idea in Mary Shelley’s Frankenstein and the golem stories of Jewish folklore. Nowadays we have Arnold Schwarzenegger dressed up as The Terminator.

A number of thoughtful people (including Stephen Hawking, Nick Bostrom, Bill Gates) believe we should take the threat of AI insurrection seriously. They argue that in decades to come we could very well create some sort of conscious entity that might decide the planet would be a nicer place without us.

In the meantime there are lesser but more urgent threats—machines that would not exterminate our species but might make our lives a lot less fun. An open letter released earlier this week, at the International Joint Conference on AI, calls for an international ban on autonomous weapon systems.

Autonomous weapons select and engage targets without human intervention. They might include, for example, armed quadcopters that can search for and eliminate people meeting certain pre-defined criteria, but do not include cruise missiles or remotely piloted drones for which humans make all targeting decisions. Artificial Intelligence (AI) technology has reached a point where the deployment of such systems is — practically if not legally — feasible within years, not decades, and the stakes are high: autonomous weapons have been described as the third revolution in warfare, after gunpowder and nuclear arms.

When I last checked, the letter had 2,414 signers who identify themselves as AI/robotics researchers, and 14,078 other endorsers. I’ve added my name to the latter list.

A United Nations declaration, or even a multilateral treaty, is not going to totally prevent the development and use of such weapons. The underlying technologies are too readily accessible. The self-driving car that can deliver the kids to soccer practice can also deliver a bomb. The chip inside a digital camera that recognizes a smiling face and automatically trips the shutter might also recognize a soldier and pull the trigger. As the open letter points out:

Unlike nuclear weapons, they require no costly or hard-to-obtain raw materials, so they will become ubiquitous and cheap for all significant military powers to mass-produce. It will only be a matter of time until they appear on the black market and in the hands of terrorists, dictators wishing to better control their populace, warlords wishing to perpetrate ethnic cleansing, etc.


What would Isaac Asimov say about all this?

I was lucky enough to meet Asimov, though only once, and late in his life. He was in a hospital bed, recovering from heart surgery. He handed me his business card:

ISAAC ASIMOV

Natural Resource

No false modesty in this guy. But despite this braggadocio, he could equally well have handed out cards reading:

ISAAC ASIMOV

Gentle Soul

Asimov was a Humanist with a capital H, and he endowed the robots in his stories with humanistic ethics. They were the very opposite of killer machines. Their platinum-iridium positronic brains were hard-wired with rules that forbade harming people, and they would intervene to prevent people from harming people. Several of the stories describe robots struggling with moral dilemmas as they try to reconcile conflicts in the Three Laws of Robotics.

Asimov wanted to believe that when technology finally caught up with science fiction, all sentient robots and other artificial minds would be equipped with some version of his three laws. The trouble is, we seem to be stuck at a dangerous intermediate point along the path to such sentient beings. We know how to build machines capable of performing autonomous actions—perhaps including lethal actions—but we don’t yet know how to build machines capable of assuming moral responsibility for their actions. We can teach a robot to shoot, but not to understand what it means to kill.

Ever since the 1950s, much work on artificial intelligence and robotics has been funded by military agencies. The early money came from the Office of Naval Research (ONR) and from ARPA, which is now DARPA, the Defense Advanced Research Projects Agency. Military support continues today; witness the recently concluded DARPA Robotics Challenge. As far as I know, none of the projects currently under way in the U.S. aims to produce a “weaponized robot.” On the other hand, as far as I know, that goal has never been renounced either.

Posted in computing, modern life | 7 Comments

Meet the Julians

JuliaCon, the annual gathering of the community focused on the Julia programming language, convened last week at MIT. I hung out there for a couple of days, learned a lot, and had a great time. I want to report back with a few words about the Julia language and a few more about the Julia community.

the celebratory cake at the conference, inscribed JuliaCon 2015


It’s remarkable that after 70 years of programming language development, we’re still busily exploring all the nooks and crannies of the design space. A slide from a talk by David Beach, listing traits of various programming languages (C++, Java, Python, R, Matlab, and Fortran; Julia's trait is The last time I looked at the numbers, there were at least 2,500 programming languages, and maybe 8,500. But it seems there’s always room for one more. The slide at left (from a talk by David Beach of Numerica) sums up the case for Julia: We need something easier than C++, faster than Python, freer than Matlab, and newer than Fortran. Needless to say, the consensus at this meeting was that Julia is the answer.

Where does Julia fit in among all those older languages? The surface syntax of Julia code looks vaguely Pythonic, turning its back on the fussy punctuation of the C family. Other telltale traits suggest a heritage in Fortran and Matlab; for example, arrays are indexed starting with 1 rather than 0, and they are stored in column-major order. And there’s a strong suite of tools for working with matrices and other elements of linear algebra, appealing to numericists. Looking a little deeper, some of the most distinctive features of Julia have no strong connection with any of the languages mentioned in Beach’s slide. In particular, Julia relies heavily on generic functions, which came out of the Lisp world. (Roughly speaking, a generic function is a collection of methods, like the methods of an object in Smalltalk, but without the object.)

Perhaps a snippet of code is a better way to describe the language than all these comparisons. Here’s a fibonacci function:

function fib(n)
    a = b = BigInt(1)
    for i in 1:n
        a, b = b, a+b
    end
    return a
end

Note the syntax of the for loop, which is similar to Python’s for i in range(n):, and very different from C’s for (var i=0; i<n; i++). But Julia dispenses with Python’s colons, instead marking the end of a code block. And indentation is strictly for human readers; it doesn’t determine program meaning, as it does in Python.

For a language that emphasizes matrix operations, maybe this version of the fibonacci function would be considered more idiomatic:

function fibmat(n)
    a = BigInt[1 1; 1 0]
    return (a^n)[1, 2]
end

What’s happening here, in case it’s not obvious, is that we’re taking the nth power of the matrix \[\begin{bmatrix}1& 1\\1& 0\end{bmatrix}\] and returning the lower left element of the product, which is equal to the nth fibonacci number. The matrix-power version is 25 times faster than the loop version.

@time fib(10000)
elapsed time: 0.007243102 seconds (4859088 bytes allocated)

@time fibmat(10000)
elapsed time: 0.000265076 seconds (43608 bytes allocated)

Julia’s base language has quite a rich assortment of built-in functions, but there are also 600+ registered packages that extend the language in ways large and small, as well as a package manager to automate their installation. The entire Julia ecosystem is open source and managed through GitHub.

When it comes to programming environments, Julia offers something for everybody. You can use a traditional edit-compile-run cycle; there’s a REPL that runs in a terminal window; and there’s a lightweight IDE called Juno. But my favorite is the IPython/Jupyter notebook interface, which works just as smoothly for Julia as it does for Python. (With a cloud service called JuliaBox, you can run Julia in a browser window without installing anything.)


I’ve been following the Julia movement for a couple of years, but last week’s meeting was my first exposure to the community of Julia developers. Immediate impression: Youth! It’s not just that I was the oldest person in the room; I’m used to that. It’s how much older. Keno Fischer is now an undergrad at Harvard, but he was still in high school when he wrote the Julia REPL. Zachary Yedidia, who demoed an amazing package for physics-based simulations and animations, has not yet finished high school. Several other speakers were grad students. Even the suits in attendance—a couple of hedge fund managers whose firm helped fund the event—were in jeans with shirt tails untucked.

Four leaders of the Julia movement at JuliaCon 2015. From left: Stefan Karpinski, Viral B. Shah, Jeff Bezanson, Keno Fischer.

Four of the ringleaders of the Julia movement. From left: Stefan Karpinski, Viral B. Shah, Jeff Bezanson, Keno Fischer.

Second observation: These kids are having fun! They have a project they believe in; they’re zealous and enthusiastic; they’re talented enough to build whatever they want and make it work. And the world is paying attention. Everybody gets to be a superhero.


By now we’re well into the second generation of the free software movement, and although the underlying principles haven’t really changed, the vibe is different. Early on, when GNU was getting started, and then Linux, and projects like OpenOffice, the primary goal was access to source code, so that you could know what a program was doing, fix it if it broke, customize it to meet your needs, and take it with you when you moved to new hardware. Within the open-source community, that much is taken for granted now, but serious hackers want more. The game is not just to control your own copy of a program but to earn influence over the direction of the project as a whole. To put it in GitHub terminology, it’s not enough to be able to clone or fork the repo, and thereby get a private copy; you want the owners of the repo to accept your pull requests, and merge your own work into the main branch of development.

GitHub itself may have a lot to do with the emergence of this mode of collective work. It puts everything out in public—not just the code but also discussions among programmers and a detailed record of who did what. And it provides a simple mechanism for anyone to propose an addition or improvement. Earlier open-source projects tended to put a little more friction into the process of becoming a contributor.

In any case, I am fascinated by the social structure of the communities that form around certain GitHub projects. There’s a delicate balance between collaboration (everybody wants to advance the common cause) and competition (everyone wants to move up the list of contributors, ranked by number of commits to the code base). Maintaining that balance is also a delicate task. The health of the enterprise depends on attracting brilliant and creative people, and persuading them to freely contribute their work. But brilliant creative people bring ideas and agendas of their own.

The kind of exuberance I witnessed at JuliaCon last week can’t last forever. That’s sad, but there’s no helping it. One reason we have those 2,500 (or 8,500) programming languages is that it’s a lot more fun to invent a new one than it is to maintain a mature one. Julia is still six tenths of a version number short of 1.0, with lots of new territory to explore; I plan to enjoy it while I can.


Quick notes on a few of the talks at the conference.

Zenna Tavares discusses probabilistic programming with Sigma.jl at JuliaCon 2015

Zenna Tavares described sigma.jl, a Julia package for probabilistic programming—another hot topic I’m trying to catch up with. Probabilistic programming languages try to automate the process of statistical modeling and inference, which means they need to build things like Markov chain Monte Carlo solvers into the infrastructure of the programming language. Tavares’s language also has a SAT solver built in.

Chiyuan Zhang gave us mocha.jl, a deep-learning/neural-network package inspired by the C++ framework Caffe. Watching the demo, I had the feeling I might actually be able to set up my own multilayer neural network on my kitchen table, but I haven’t put that feeling to the test yet.

Finally, because of parallel sessions I missed the first half of Pontus Stenetorp’s talk on “Suitably Naming a Child with Multiple Nationalities using Julia.” I got there just in time for the big unveiling. I was sure the chosen name would turn out to be “Julia.” But it turns out the top three names for the offspring of Swedish and Japanese parents is:

Best Sweedish-Japanese names, according to Pontus Stenetorp's Julia program 4573

Steneport wants to extend his algorithm to more language pairs. And he also needs to tell his spouse about the results of this work.

Posted in computing | 3 Comments

Fast money, slow data

The other day I bought lunch from a food truck on the Harvard campus, paying with a debit card that my server swiped through one of those little plastic doodads attached to an iPhone. Ten minutes later, while I ate my báhn mì in a shady corner of Harvard Yard, I opened my laptop and logged on to my bank’s website. The $7 transaction had already been posted to my account.

So here’s a question to think about: If the credit card networks and the banks can track the movement of money minute by minute, why does it take months to calculate the overall level of economic activity in the U.S.?

The Department of Commerce reports on gross domestic product (GDP) at quarterly intervals. This number (or its first derivative) is often cited as a kind of benchmark of economic wellbeing. The initial announcement comes a month after the close of each quarter; for Q1 2015, the news was released April 29. That news wasn’t cheering: a measly 0.2 percent rate of growth. The New York Times coverage began, “Repeating an all-too-familiar pattern, the American economy slowed to a crawl in the first quarter of 2015….”

But that wasn’t the last word on the first quarter. At the end of May, a revised report came out, saying the winter months had been even bleaker than first thought: not +0.2 percent but –0.7 percent. Then yesterday a third and “final” estimate was released (PDF). It was a Goldilocks number, –0.2 percent, pretty near the mean of the two earlier figures. Said the Times: “While nothing to brag about, the economy’s performance in early 2015 was not quite as bad as the number-crunchers in Washington had thought.”

If we take it for granted that the end-of-June GDP estimate is somewhere near correct, then the first two reports were worse than useless; they were misleading. Taking action on the basis of those numbers—making an investment, say, or setting an interest rate—would have been foolish. It seems that if you want to know how the economy is doing in the first quarter, you have to wait until the second quarter ends. And of course we’re about to repeat the cycle. How did business fare this spring? Check back at the end of September, when my $7 food-truck sandwich may finally register in the government’s books.


Why so slow? To answer this, I thought I ought to learn a little something about how the Department of Commerce calculates GDP. Here’s where I learned that little something:

Landefeld, J. Steven, Eugene P. Seskin, and Barbara M. Fraumeni. 2008. Taking the pulse of the economy: Measuring GDP. Journal of Economic Perspectives 22(2):193–216. PDF.

It’s worse than I had guessed. The baseline for all these “national accounts estimates” is an economic census conducted every five years (in years congruent to 2 mod 5). For the 19 quarters between one economic census and the next, numbers are filled in by extrapolation, guided by a miscellany of other data sources. Landefeld et al. write:

The challenge lies in developing a framework and methods that take these economic census data and combine them using a mosaic of monthly, quarterly, and annual economic indicators to produce quarterly and annual GDP estimates. For example, one problem is that the other economic indicators that are used to extrapolate GDP in between the five-year economic census data—such as retail sales, housing starts, and manufacturers shipments of capital goods—are often collected for purposes other than estimating GDP and may embody definitions that differ from those used in the national accounts. Another problem is some data are simply not available for the earlier estimates. For the initial monthly estimates of quarterly GDP, data on about 25 percent of GDP— especially in the service sector—are not available, and so these sectors of the economy are estimated based on past trends and whatever related data are available. For example, estimates of consumer spending for electricity and gas are extrapolated using past heating and cooling degree data and the actual temperatures, while spending for medical care, education, and welfare services are extrapolated using employment, hours, and earnings data for these sectors from the Bureau of Labor Statistics.

Is this methodology really the best way to measure the state of the economy in the age of Big Data?

Actually, there’s a lot to be said for the quinquennial economic census. It goes to a large sample (four million businesses), and the companies are compelled by law to respond, which mitigates selection bias. The data series goes back to 1934; maintaining continuity with past measurements is valuable. Furthermore, the census and other survey-based instruments probe much more than just transactional data. They try to quantify what a company manufactures (or what services it provides), what labor and material inputs are consumed, and where the company’s revenue comes from. The analysis includes not just income and expenditures but also depreciation and amortization and other scary abstractions from the world of accounting. You can’t get all that detail just by following the money as it sloshes around the banking system.

Still, can we afford to wait five years between surveys? Three months before we get a reliable (?) guess about what happened in the previous quarter? Consider the predicament of the Federal Reserve Board, trying to walk the narrow line between encouraging economic growth and managing inflation. This is a problem in control theory, where delayed feedbacks lead to disastrous instabilities. (Presumably Janet Yellin and her colleagues have access to more timely data than I do. I hope so.)

Could we really create an up-to-the-minute measure of national economic health by mining credit card data, bank accounts, supermarket inventories, and food-truck receipts? I really don’t know. But I’ll quote a 2014 Science review by Liran Einav and Jonathan Levin:

Whereas the popular press has focused on the vast amount of information collected by Internet companies such as Google, Amazon, and Facebook, firms in every sector of the economy now routinely collect and aggregate data on their customers and their internal businesses. Banks, credit card companies, and insurers collect detailed data on household and business financial interactions. Retailers such as Walmart and Target collect data on consumer spending, wholesale prices, and inventories. Private companies that specialize in data aggregation, such as credit bureaus or marketing companies such as Acxiom, are assembling rich individual-level data on virtually every household….

One potential application of private sector data is to create statistics on aggregate economic activity that can be used to track the economy or as inputs to other research. Already the payroll service company ADP publishes monthly employment statistics in advance of the Bureau of Labor Statistics, MasterCard makes available retail sales numbers, and Zillow generates house price indices at the county level. These data may be less definitive than the eventual government statistics, but in principle they can be provided faster and perhaps at a more granular level, making them useful complements to traditional economic statistics.

I suppose I should also mention worries about giving government agencies access to so much personal financial data. But that horse is out of the barn.

Posted in social science, statistics | 5 Comments

Traffic Jams in Javascript

Building bigger roads can make traffic worse. The usual explanation for this counter­intuitive and counterproductive outcome is that bigger roads attract bigger shopping malls, which in turn attract more cars. But that’s not the whole story. In the 1960s Dietrich Braess discovered a theoretical configuration of roadways where building a new connecting road can make everyone’s trip slower even when the number of vehicles is held constant. Conversely, closing a road in the Braess network can allow everyone to get home quicker. The behavior is weird enough to justify the term “Braess’s paradox.”

A few years ago Joel Cohen suggested to me that Braess’s paradox might make a good topic for my “Computing Science” column. I hesitated. There were already plenty of published discussions of the paradox, including excellent ones by Cohen himself, as well as a book by Tim Roughgarden (which I had reviewed for American Scientist). I didn’t think I had anything more to add to the conversation.

Lately, though, I began to consider the problem of visualizing Braess’s paradox—presenting it in such a way that you can watch individual cars navigating through the road network, rather than merely calculating average speeds and travel times. Having an opportunity to play with the model—fiddling with the knobs and switches, trying different routing algorithms—might lead to a clearer understanding of why well-informed and self-interested drivers would choose a route that winds up making everybody late for dinner.

I now have a working JavaScript model of something resembling Braess’s paradox, which I urge you to go take for a test drive. There’s an accompanying “Computing Science” column in the latest issue of American Scientist, available on the magazine website in HTML or PDF. If you’re interested in the source code, it’s on Github. Here I’m going to say a few words about the implementation of the model and what I’ve learned from it.


Adapting Braess’s mathematical model to a more mechanistic and visual medium was trickier than I had expected. The original formulation is quite abstract and not very physical; it’s closer to graph theory than to highway engineering. In the diagrams below the wide, blue roads labeled A and B never become congested; no matter how much traffic they carry, the transit time for these links is always one hour. The narrow red roads a and b offer a transit time of zero when they’re empty, but traffic slows as the load increases; if everyone crowds onto a single red link, the transit time is again one hour. As for the gold route X, this magical thoroughfare offers instantaneous transport for any number of vehicles.

diagrams of the Braess diamond road network

The presence or absence of the X road is what triggers the paradox. Without the golden crosslink (left diagram), traffic splits equally between the Ab and aB routes, and all cars take 90 minutes to complete the trip. When the X link is opened (right diagram) all drivers favor route aXb, and everyone spends a full two hours on the road.

The essential supposition behind the paradox is that everyone follows a selfish routing policy, choosing whichever route offers the quickest trip, and ignoring all factors other than travel time. And, ironically, it’s the insistence that no one else may have a shorter trip that leaves all the drivers in a bumper-to-bump jam on route aXb, while AXB sits empty. Why? If any driver decided to defect to AXB, the departure would marginally lesson the load on aXb, giving the latter route a slight speed advantage. Under the selfish policy, the defecting driver would then have to return to aXb. It’s a stalemate.


Cars moving at infinite speed and roads with infinite capacity may be all right in a mathematical model, but they are troublesome in a simulation that’s supposed to look something like real highway traffic. Seeking a more realistic model, I settled on the road layout shown below, inspired by a diagram in a 1997 paper by Claude M. Pinchina (Braess Paradox: maximum penalty in a minimal critical network. Transportation Research A 31(5):379–388).

Road layout of the JavaScript traffic simulation.

The topology is the same as that of Braess’s original network, but the geometry is different, and so is the relation between congestion and speed. The aim is still to get from start to end—or, rather, from Origin to Destination. The A and B road segments are again wide and insensitive to traffic congestion. The a and b roads are straighter and shorter but also narrower. With zero traffic, vehicle speed is the same on a and b as on A and B, but as traffic gets heavier, speed falls off. The analogue of the “golden road” is a short bridge at the center of the map, which has the same speed properties as A and B. In the initial configuration, the bridge is blocked off, but a mouseclick opens it. In the snapshot of the running model shown above, the bridge is open and carrying traffic.

Cars, represented by colored dots, enter the system at the Origin. At the moment of launching, each car chooses one of the available routes. If the bridge is closed, there are just two possibilities: Ab and aB. With the bridge open, drivers also have the option of the ab shortcut or the longer AB. The cars then advance along the selected roadways, governed by the speed restrictions on each segment, until they reach the Destination.

This scheme differs in several important ways from the original Braess formulation. Does it still exhibit the paradox? In other words, does the travel time increase when the bridge is opened, allowing traffic to flow over routes ab and AB? The answer is yes, for a wide range of parameter values, as shown in these outputs:

Braess paradox timings

The tables show the number of vehicles choosing each route and the average time they spend on the road. (The times are measured in units of the quickest possible trip: taking the shortest route ab with zero traffic.) Note that opening up the bridge slowed down travel on all four routes. Even though the Ab and aB routes carried about 37 percent less traffic when the bridge was open, cars on those routes took 9 to 15 percent longer to complete their journeys. The ab and AB routes were even slower.


But the numbers don’t tell the whole story—that was the first thing I learned when I got the simulation running. In the bridge-closed state, where the total traffic splits into two roughly equal streams, you might imagine successive cars alternating between Ab and aB, so that the system would reach a statistical steady state with equal numbers of cars on each of the two routes at any given moment. That’s not at all what happens! The best way to see what does happen is to go run the simulation, but the graph below also gives a clue.

number of cars on routes Ab and aB as a function of time

Instead of settling into a steady state, the system oscillates with a period of a little less than 500 times steps, which is roughly half the time it takes a typical car to make a complete trip from Origin to Desitination. The two curves are almost exactly 180 degrees out of phase.

It’s not hard to understand where those oscillations come from. As each car enters the road network, it chooses the route with the shortest expected travel time, based on conditions at that moment. The main determinant of expected travel time is the number of cars occupying the a and b segments, where speed decreases as the roads get more crowded. But when cars choose the less-popular route, they also raise the occupany of that route, making it appear less favorable to the cars behind them. Furthermore, on the Ab route there is a substantial delay before the cars reach the congestion-sensitive segment. The delay and the asymmetry of the network create an instability—a feedback loop in which overshooting and overcorrection are inevitable.

When the connecting bridge is opened, the pattern is more complicated, but oscillations are still very much in evidence:

Four route car census

There seem to be two main alternating phases—one where Ab and ab dominate (the Christmas phase, in this color scheme), and the other where ab and AB take over (the Cub Scout phase). The period of the waves is less regular but mostly longer.

I am not the first to observe these oscillations; they are mentioned, for example, by Daniele Buscema and colleagues in a paper describing a NetLogo simulation of Braess-like road networks. Overall, though, the osccilations and instabilities seem to have gotten little attention in the literature.

The asymmetry of the layout is crucial to generating both the oscillations and the paradoxical slowdown on opening the central connecting link. You can see this for yourself by running a symmetric version of the model. It’s quite dull.


One more bug/feature of the dynamic model is worth a comment. In the original Braess network, the A and B links have unlimited capacity; in effect, the model promises that the time for traversing those roads is a constant, regardless of traffic. In a dynamic model with discrete vehicles of greater-than-zero size, that promise is hard to keep. Consider cars following the Ab route, and suppose the b segment is totally jammed. At the junction where A disgorges onto b, cars have nowhere to go, and so they spill back onto the A segment, which can therefore no longer guarantee constant speed.

In implementing the dynamic model, I discovered there were a lot of choices to be made where I could find little guidance in the mathematical formulation of the original Braess system. The “queue spillback” problem was one of them. Do you let the cars pile up on the roadway, or do you provide invisible buffers of some kind where they can quietly wait their turn to continue their journey? What about cars that present themselves at the origin node at a moment when there’s no room for them on the roadways? Do you queue them up, do you throw them away, do you allow them to block cars headed for another road? Another subtle question concerns priorities and fairness. The two nodes near the middle of the road layout each have two inputs and two outputs. If there are cars lined up on both input queues waiting to move through the node, who goes first? If you’re not careful in choosing a strategy to deal with such contention, one route can be permanently blockaded by another.

You can see which choices I made by reading the JavaScript source code. I won’t try to argue that my answers are the right ones. What’s probably more important is that after a lot of experimenting and exploring alternatives, I found that most of the details don’t matter much. The Braess effect seems to be fairly robust, appearing in many versions of the model with slightly different assumptions and algorithms. That robustness suggests we might want to search more carefully for evidence of paradoxical traffic patterns out on real highways.

Posted in computing, mathematics, modern life | 12 Comments

Whales of the Web

The average web site has links connecting it with 29 other sites. I came up with this number in the following way. A data set I’ve been playing with for a few weeks lists 43 million web sites and 623 million links between sites; the quotient of those numbers is about 14.5. Since each link has two ends, the per-site total of inbound plus outbound links is double the quotient.

Twenty-nine links is the average, but by no means is it a typical number of links. Almost half of the sites have four or fewer links. At the other end of the spectrum, the most-connected web site (blogspot.com) has almost five million links, and there are six more sites with at least a million each. The distribution of link numbers—the degree sequence—looks like this (both scales are logarithmic, base 2):

Degree sequence of the WWW

I want to emphasize that these are figures for web sites, not web pages. The unit of aggregation is the “pay-level domain”—the domain name you have to pay to register. Examples are google.com or bbc.co.uk. Subdomains, such as maps.google.com, are all consolidated under the main google.com entry. Any number of links from pages on site A to pages on site B are recorded as a single link from A to B.

The source of these numbers is the Web Data Commons, a project overseen by a group at the University of Mannheim. They extracted the lists of domains and the links between them from a 2012 data set compiled and published by the Common Crawl Foundation (which happens to be the subject of my latest American Scientist column). The Common Crawl does essentially the same thing as the big search engines—download the whole Web, or some substantial fraction of it—but the Common Crawl makes the raw data publicly available.

There are interesting questions about both ends of the degree sequence plotted above. At the far left, why are there so many millions of lonely, disconnected web sites, with just one or two links, or none at all? I don’t yet feel I know enough to tell the story of those orphans of the World Wide Web. I’ve been focused instead on the far right of the graph, on the whales of the Web, the handful of sites with links to or from many thousands of other sites.

From the set of 43 million sites, I extracted all those with at least 100,000 inbound or outbound links; in other words, the criterion for inclusion in my sample was \(\min(indegree, outdegree) \ge 100,000\). It turns out that just 112 sites qualify. In the diagram below, they are grouped according to their top-level domain (com, org, de, and so on). The size of the colored dot associated with each site encodes the total number of links; the color indicates the percent of those links that are incoming. Hover over a site name to see the inbound, outbound and bidirectional links between that site and the other members of this elite 112. (The diagram was built with Mike Bostock’s d3.js framework, drawing heavily on this example.)

Patience, please . . .

The bright red dots signify a preponderance of outgoing links, with relatively few incoming ones. Many of these sites are directories or catalogs, with lists of links classified by subject matter. Such “portal sites” were popular in the early years of the Web, starting with the World Wide Web Home at CERN, circa 1994; another early example was Jerry and David’s Guide to the World Wide Web, which evolved into Yahoo. Search engines have swept aside many of those hand-curated catalogs, but there are still almost two dozen of them in this data set. Curiously, the Netherlands and Germany (nl and de) seem to be especially partial to hierarchical directories.

Bright blue dots are rarer than red ones; it’s easier to build a site with 100,000 outbound links than it is to persuade 100,000 other sites to link to yours. The biggest blue dot is for wordpress.org, and I know the secret of that site’s popularity. If you have a self-hosted WordPress blog (like this one), the software comes with a built-in link back to home base.

Another conspicuous blue dot is gmpg.org, which mystified me when I first noticed that it ranks fourth among all sites in number of incoming links. Having poked around at the site, I can now explain. GMPG is the Global Multimedia Protocols Group, a name borrowed from the Neal Stephenson novel Snow Crash. In 2003, three friends created a real-world version of GMPG as a vehicle for the XHTML Friends Network, which was conceived as a nonproprietary social network. One of the founders was Matt Mullenweg, who was also the principal developer of WordPress. Hence every copy of WordPress includes a link to gmpg.org. (The link is in the <head> section of the HTML file, so you won’t see it on the screen.) At this point GMPG looks to be a moribund organization, but nonetheless more than a million web sites have links to it.

Networkadvertising.org is the web site of a trade group for online advertisers. Presumably, its 143,863 inbound links are embedded in ads, probably in connection with the association’s opt-out program for behavioral tracking. (To opt out, you have to accept a third-party cookie, which most people concerned about privacy would refuse to do.)

Still another blue-dot site, miibeian.gov.cn, gets its inward links in another way. If I understand correctly, all web sites hosted in China are required to register at miibeian.gov.cn, and they must place a link back to that site on the front page. (If this account is correct, the number of inbound links to miibeian.gov.cn tells us the number of authorized web sites in China. The number in the 2012 data is 289,605, which seems low.)

One final observation I find mildly surprising: Measured by connectivity, these 112 sites are the largest on the entire Web, and you might think they would be reasonably stable over time. But in the three years since the data were collected, 10 percent of the sites have disappeared altogether: Attempts to reach them either time out or return a connection error. At least a few more sites have radically changed their character. For example, serebella.com was a directory site that had almost 700,000 outbound links in 2012; it is now a domain name for sale. Among web sites, it seems, nobody is too big to fail.

The table below lays out the numbers for the 112 sites. It’s sortable: Click on any of the column headers to sort on that field; click again to reverse the ordering. If you’d like to play with the data yourself, download the JSON file.

site inlinks outlinks total links % inbound




Posted in computing | 7 Comments

A taxing algorithm

My first encounter with the term algorithm did not come from Don Knuth. I learned it from the Internal Revenue Service. Sometime in the late 1960s or early 70s the IRS introduced a redesigned Form 1040 based on a principle borrowed from the world of computing: the algorithm. What this meant, I soon discovered, was that the wording of the instructions would be procedural rather than declarative. Instead of “Your tax is 15 percent of the amount on line 42,” we had “Multiply the amount on line 42 by 0.15 and write the result on line 43.” I had expected something more revolutionary, but at least I expanded my vocabulary.

I’ve filled out a lot of 1040s since then, but until yesterday I had never become acquainted with Schedule D (Capital Gains and Losses). What a treat I had waiting for me! Tucked inside the Schedule D instruction book, I found a marvel of labyrinthine arithmetic and logic. The tax worksheet on pages D-15 and D-16 might well be the most complex algorithm that ordinary people are ever asked to perform.

Below is my attempt to represent the worksheet as a data-flow diagram. Inputs (mostly dollar amounts copied from elsewhere in the tax return) are in blue; the eventual output (tax owed) is in red; the flow is mostly from bottom to top. The numbers in the boxes correspond to line numbers in the worksheet.

data-flow diagram for IRS Schedule D tax worksheet

The directed graph has 82 nodes and 96 edges. Twelve subtractions, seven additions, four multiplications, ten comparisons, and two table lookups. Now that’s an algorithm! It’s gnarlier than calculating the date of Easter.

What are the chances that I correctly followed all the steps of this algorithm? What are the chances that the published algorithm correctly implements the intent of the tax code?

Posted in computing, modern life | 12 Comments

Mrs. Nguyen’s prestidigitation

From a set of 1 through 9 playing cards, I draw five cards and get cards showing 8, 4, 2, 7, and 5. I ask my 6th graders to make a 3-digit number and a 2-digit number that would yield the greatest product. I add, “But do not complete the multiplication — meaning do not figure out the answer. I just want you to think about place value and multiplication.”box model of 3 x 2 digit multiplication

The problem above comes from Fawn Nguyen, who teaches math at a public school in southern California and writes a blog called Finding Ways. To her students she is “Mrs. Win,” an English approximation to the Vietnamese pronunciation.

It’s clear that much care and craftsmanship went into formulating this problem. Why did Nguyen choose a two-digit-by-three-digit product, rather than two-by-two or three-by-three? The asymmetry ensures a unique solution. Why did she use playing cards to select the digits, rather than simply asking the kids to call out numbers? The cards produce a set of distinct digits, without duplicates, and they also rule out the possibility of choosing zero. Repeated digits or zeros would not ruin the problem, but they would needlessly complicate it. Nguyen’s classroom procedure eliminates these distractions without even mentioning them. This is the kind of subtle indirection you find in the performance of a first-rate stage magician.

I’ve had some serious fun with Nguyen’s problem. Finding the answer is not difficult—especially if you cheat and set aside her boldfaced injunction forbidding arithmetic. But of course the answer itself isn’t the point, as Nguyen makes clear in her blog post describing the students’ search for a solution. What we really care about is why one particular arrangement of those digits yields a larger product than all other arrangements. We want an explanatory pattern or principle, a rule that works not just for this one set of digits but for any instance of the max-product problem. I had many stumbles on the path to finding such a rule. Here are some notes I made along the way. But before reading on you might want to write down your own best guess at the answer.


  1. First, some notation. Let’s label the digits like so:

    \[\begin{align}
    x_{2}\; x_{1}\; x_{0}&{}\\
    \underline{\times \quad y_{1} \; y_{0}}&{},
    \end{align}\]

    where the subscripts encode the power of 10 associated with each digit. The object is to find a one-to-one mapping between the sets \(\{x_{2}, x_{1}, x_{0}, y_{1}, y_{0}\}\) and \(\{8, 4, 2, 7, 5\}\) that maximizes the product.

  2. Nguyen’s sixth graders were quick to perceive one important principle: Larger digits should generally go to the left, and smaller digits to the right, in order to get the most benefit from place values in decimal notation. No one proposed 258 × 47 as the maximum product; that combination is obviously beaten by 852 × 74.
  3. If we wanted to solve this problem by exhaustive search, how exhausting would the search be? There are \(5! = 120\) permutations of the five digits, and each permutation corresponds to a different multiplication problem. But in view of the observation made in item 2, the number of plausible candidates is much smaller: just \(\binom{5}{2} = \binom{5}{3} = 10\). You could check them all in a matter of minutes. (But Mrs. Nguyen would not approve.)
  4. Again following the logic of item 2, we can stipulate that the 8—the largest of the five digits—must wind up either in position \(x_2\) or in position \(y_1\). The question is which. My first thought was that surely the 8 goes in the three-digit number, because position \(x_2\) gives the 8 a value of 800, whereas it’s only worth 80 as \(y_1\).
  5. My second thought undermined my first thought. Suppose the problem had been formulated without Mrs. Nguyen’s clever card trick, and we had to work with the digits 8, 7, 0, 0, 0. Then the arrangement yielding the maximum product is surely either 800 × 70 or 700 × 80. But of course those two products are both equal to 56,000. In other words, if we consider just the leading digits, it doesn’t matter which goes in the multiplier and which in the multiplicand.
  6. Let’s write out the full six-term polynomial defining the product:\[1000 x_{2}y_{1} + 100 x_{1}y_{1} + 10 x_{0}y_{1} + 100 x_{2}y_{0} + 10 x_{1}y_{0} + x_{0}y_{0}\]The leading term provides another way of understanding why the largest digit can be either \(x_2\) or \(y_1\). It’s just the commutative law. We get \(1000 x_{2}y_{1}\) with either ordering. Thus we need to consider the smaller terms to decide which placement is better. Hint: \(y_1\) appears in three terms but \(x_2\) turns up in only two.
  7. Maybe it would help to look at a smaller version of the problem? For example, maximize the product \(x_{1}\, x_{0} \times y_{0}\) with digits drawn from the set \(\{1, 2, 3\}\). Where does the largest digit belong?
  8. Here’s a vague, daydreamy notion: The task of maximizing the product of these numbers looks something like an isoperimetric problem. If you have a rectangle of perimeter \(2(h + w)\), you maximize the area \(hw\) by making \(h=w\), so that the rectangle becomes a square. Maybe, by analogy, we should make the three-digit and the two-digit numbers as close to equal as possible, while at the same time making both of them as large as possible.
  9. At this point I was ready to commit. I was persuaded by the arguments in items 6 and 7, and 8 that the largest digit should be bound to variable \(y_1\), and the next largest to \(x_2\). Then the same rule could be applied recursively to the remaining digits and variables, yielding this assignment:

    \[\begin{align}
    7\, 4\, 2&{}\\
    \underline{\times \quad 8 \, 5}&{}\\
    6\, 3\, 0\, 7\, 0&{}
    \end{align}\]

  10. Of course you already know I was wrong. Even if you haven’t yet solved the problem for yourself or looked up the answer elsewhere, it’s a well-established principle of proper storytelling that no one ever stumbles on the right answer on the first try.

    My error was revealed by a program running an exhaustive-search algorithm. It showed that exchanging the positions of the 7 and the 8 yields a larger product:

    \[\begin{align}
    8\, 4\, 2&{}\\
    \underline{\times \quad 7 \, 5}&{}\\
    6\, 3\, 1\, 5\, 0&{}
    \end{align}\]

    But that isn’t the right answer either. Instead of switching the 7 and 8, you can exchange the 5 and the 4 to get the following result, which does turn out to be optimal:

    \[\begin{align}
    7\, 5\, 2&{}\\
    \underline{\times \quad 8 \, 4}&{}\\
    6\, 3\, 1\, 6\, 8&{}
    \end{align}\]


So that’s it—the answer we’ve been seeking, the maximum product, the solution to Mrs. Nguyen’s problem. There’s no arguing with arithmetic.

But it’s hardly the end of the trail. Why does that peculiar permutation of the digit set gives the biggest product? Does the same pattern work for other two-digit-by-three-digit products? What about problems of other shapes and sizes? And what is the pattern, anyway? How would you succinctly state the rule for placing digits in Mrs. Nguyen’s boxes?

In trying to make sense of what’s going on here, I’ve found a certain graphic device to be helpful. I call it a lacing diagram, because it reminds me of the various schemes for lacing up a pair of sneakers. The patterns are easier to perceive with larger numbers of digits (i.e., high-top sneakers), so the examples given below show sets of 10 digits arranged to form a five-by-five multiplication problem.

In a lacing diagram the red arrows trace a path that threads its way through all the digits, ordered from largest to smallest. Each segment of this path can either cross from one factor to the other (a trans step) or stay within the same factor (a cis step). The particular lacing shown here is made up of alternating trans and cis segments. The sequence of t’s and c’s below the diagram serves as a linearized encoding of the pattern; the number below the encoding is the product of the two factors.

Tctctctct 200

Here is the lacing diagram corresponding to the pattern that I initially believed would prevail in all Nguyen products:

Ttttttttt 200

In this case, all the steps are trans, as the arrows zigzag from one side to the other. The linear encoding consists of nine t’s.

And finally here is the lacing diagram for the winning permutation, the arrangement that gives the largest product. The pattern is the same as the second lacing diagram above—the zigzag—except for a single twist: The leftmost digits of the two factors have been switched, and so there is one cis step in the path.

lacing pattern tcttttttt

After a bit of puzzling, I was able to work out why that single twist yields a better score than the plain zigzag. It’s because \((9 \times 7) + (8 \times 6) = 111\), whereas \((9 \times 6) + (8 \times 7) = 110\). Likewise \((9 \times 5) + (8 \times 4) = 77\), whereas \((9 \times 4) + (8 \times 5) = 76\), and so on. Note the difference between the twisted and the zigzag products: \(8439739020\, – 8428629020 = 11110000\). Each of the four pairings of the leftmost digits with those to their right contributes a 1 in that difference.

If one twist is a good thing, maybe more twists would be even better? For example, if we invert the 5 and the 4, we get \((5 \times 3) + (4 \times 2) = 23\) instead of \((5 \times 2) + (4 \times 3) = 22\), again for a net gain. But of course there’s a flaw in this strategy. Flipping the 5 and the 4 increases their products with neighboring digits to the right, but lowers those with digits to the left. Flipping the leftmost digits doesn’t run afoul of this rule because there are no neighbors to the left.

For a more explicit definition of the zigzag-with-a-twist arrangement, here is a Python implementation. The basic idea is to deal out the digits alternately to the \(x\) and \(y\) factors, starting with \(x\) and working through the digits in descending order. When \(y\) (the smaller factor) runs out of room, any remaining digits are tacked onto the end of \(x\). Finally—this is the twist—the leftmost \(x\) and \(y\) digits are swapped. This procedure works in any base.

def twisted_zigzag(digit_set, s):
    s = min(s, len(digit_set) - s)    # len of smaller factor
    digits = sorted(list(digit_set))  # smallest to largest
    x = []
    y = []
    while digits:                     # pop from end of list
        x.append(digits.pop())        # start with x
        if len(y) < s:                # zigzag until y is full
            y.append(digits.pop())
    x[0], y[0] = y[0], x[0]           # swap first digits
    return x, y

Does the zigzag-with-a-twist arrangement give the maximum product for all possible Nguyen-type problems? I can offer substantial evidence supporting that proposition. For base-10 numbers formed without repeated digits there are 4,097 two-factor multiplication problems with digits sorted in descending order. The zigzag-twist pattern gives the correct result for all of them.For digit sets that include 0, the solution is not always unique. Indeed, the algorithm works for all problems in bases between 2 and 15. It’s hardly a daring conjecture to suggest that it works in all bases, but I have not produced a proof. Maybe Mrs. Nguyen’s sixth graders will do so.

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A quasirandom talk

I’ll be giving a talk at Harvard this Friday, February 6: “Orderly Randomness: Quasirandom Numbers and Quasi–Monte Carlo.” If you’re in the neighborhood, please stop by. Lunch at 12:30; quasirandomness at 1:00 pm.

Update 2015-02-08: Slides online. And video.

Posted in computing, mathematics | 1 Comment