Thursday, March 31, 2016

Public Service Announcement...

Just a public service warning today: TOMORROW IS APRIL 1st... otherwise known as APRIL FOOL'S DAY...

With that said, my recommendation is to NOT take too seriously any blog entries published tomorrow with the following phrases in the subject line:

1)  "the Riemann Hypothesis"
2)  "P vs. NP"
3)  "squaring the circle"
4)  "Goldbach Conjecture" 
5)  "finally solved"
6)   "Elvis Presley"
7)  "jet contrails"
8)  "A Nigerian Prince"
9)  "Your lottery winnings"
10)  "Donald Trump"

Otherwise, have a good weekend. I have to get back to writing out an exception I've discovered to the Collatz Conjecture; perhaps I can get it done and posted by tomorrow morning....

Wednesday, March 30, 2016

Of Marrieds and Singles...

In case you missed it, there was an Alex Bellos puzzle in The Guardian recently that almost 70% of responding readers missed (and supposedly 80+% of the general public usually miss) -- I was astounded by those numbers, since it doesn't appear terribly difficult, given just a modicum of thought. It ran as follows:

Jack is looking at Anne, but Anne is looking at George.
Jack is married, but George is not.
Is a married person looking at an unmarried person?

A)  Yes     B)  No     C)  Cannot be determined


James Grime covers it well here (and he also throws in a second, completely unrelated, problem worth seeing):

After this puzzle began making Twitter rounds, David Wees pointed out the very similar problem I've placed below (I advise NOT looking at it 'til you work the above puzzle), which interestingly seems much easier than the Bellos problem -- a strong indication of how words/language can get in the way of clearly understanding relationships or structure, or simply what is being asked.

"A coin is flipped 3 times. 1st flip comes up heads and 3rd flip is tails. Is heads ever followed directly by tails?"


Monday, March 28, 2016

Here's To Amateurs... and Professionals at Play

Astronomy is often considered the best science for amateurs because over the centuries amateur or backyard astronomers have contributed so many important findings to the field. The heavens are so expansive that they offer many niches for even backyard astronomers to make significant discoveries, or be involved in "citizen" science.

Mathematics... not so much.  With its highly-specialized, technical and abstract content, math is usually not seen as a playground for non-professionals. Indeed many who try (and there ARE many) end up classified as "crackpots," their ideas or approaches so off-base and unworthy of attention.
In recent times there is the famous case of "homemaker" Marjorie Rice who, playing around with tessellations, made important contributions to the geometry of tilings. And there are a few others... but the number is small. In the last couple decades, how many "proofs" have come along for the Riemann Hypothesis or other "Millennium" problems from people dabbling way outside their competency, largely wasting time and energy.

I mention all this because of my ongoing fascination with the recent findings regarding the "pattern" of consecutive prime last-digits. Robert Lemke Oliver and Kannan Soundararajan who discovered it are professional mathematicians, but it is the sort of thing that could have been discovered by amateurs just 'playing around' with prime numbers (as people often do).  Indeed, a common response to their finding has been, 'HOW did this go UNnoticed so long!?" In this day of Mathematica and similar programs it seems the door is wide open to all manner of analyses of prime number digits/succession/position that amateurs could imagine doing -- it might well take a trained number theorist or other specialist to explain a given outcome, but just generating that outcome might be do-able by brute-force amateurs.

I've been following Mike Lawler, inspired by Oliver/Soundararajan's work, play with prime triplets for the last week with results that, while beyond my comprehension, could hold significance for others (Mike is not an "amateur," as he has a math PhD., but he is not a number theorist or prime specialist, and what he is doing could be done by a non-math PhD.). Current posts for his work are here:

And he has been recording his results (looking at prime-last-digit-triples in billion increments) on an ongoing Google spreadsheet here:

This isn't for everyone, but for those mesmerized by the mysterious way prime numbers weave their way through our integer system, tantalizing us with their secrets and their non-random randomness(!), it can almost be addictive (beware). So here's to amateurs and professionals alike playing in that heady world of pure math, never quite knowing what they might find, where it might lead, or what it might mean.

...ADDENDUM:  after posting this, Mike put up another entry summarizing somewhat his experience thus far:

Sunday, March 27, 2016

The Contradiction of Primes

Primes been on my mind lately, so, for Sunday reflection:
"There are two facts about the distribution of prime numbers which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts.
"The first is that despite their simple definition and role as the building blocks of the natural numbers, the prime numbers... grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.
"The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision."
-- D. Zagier  in "The First 50 Million Prime Numbers" The Mathematical Intelligencer  (1977)

"One of the remarkable aspects of the distribution of prime numbers is their tendency to exhibit global regularity and local irregularity. The prime numbers behave like the 'ideal gases' which physicists are so fond of. Considered from an external point of view, the distribution is -- in broad terms -- deterministic, but as soon as we try to describe the situation at a given point, statistical fluctuations occur as in a game of chance where it is known that on average the heads will match the tail but where, at any one moment, the next throw cannot be predicted. Prime numbers try to occupy all the room available (meaning that they behave as randomly as possible), given that they need to be compatible with the drastic constraint imposed on them, namely to generate the ultra-regular sequence of integers."
-- G. Tenenbaum and M. Mendèfs France, in "The Prime Numbers and Their Distribution" (AMS, 2000)

[These come from a great collection of quotes about prime numbers from a page at Matthew Watkins' website: ]

Thursday, March 24, 2016

Calling All Number Theorists....

Those persnickety primes... Earlier today Mike Lawler posted some results from Mathematica for the patterns in last digits of prime-triples, essentially in intervals of a billion primes (not sure if I'm stating that very clearly, but read his post):

His columns are a bit hard to read, but you should be able to spot the "clustering" (or I would call it "coupling") he refers to which seems to largely hold for all 3 columns of the post. He notes in a comment that the "average" expected value for each entry would be about 15.6 million, so you can see how widely the values diverge from that, as well as see how they tend to pair up.
He is in the process of adding more columns at the below easier-to-read spreadsheet -- I assume he'll be going out to 10 billion primes, to restore data he originally had, but lost -- as I write this, columns for 4 billion primes are listed, and it appears to me (merely eyeballing it), that the numbers for the paired triplets are getting even closer(???):

 As each set of a billion primes is a somewhat independent and random-like group of integers, this pattern of the same ordered-triplet of last digits re-occurring in associated pairs seems, on the surface at least, rather odd and striking!?  What (if anything) is it about those pairs? Perhaps a number theorist can see through to a simple explanation for it (if so, I'm sure Mike would love to hear it). Or does this finding piggy-back in any way on to the peculiar result from a week prior of prime number last digits tending to avoid repetition in consecutive primes?
WHAT is going on here....?

ADDENDUM:  I should have included in this post that Mike has already recognized that the paired triplets involved are consistently of the form (a, b, c) and (-c, -b, -a) in mod 10. Now THAT surely must mean something! (Again, perhaps something obvious to a number theorist, but WHAT?)

Tuesday, March 22, 2016

Your Tax Money At Work (back when Erdös was around)

Somewhere in the Great Beyond, Paul Erdös may be saying, 'My file is open...*
A h/t to Nicholas Christakis for passing along the below article (via Twitter):

For years, the FBI maintained a surveillance file on eccentric, peripatetic genius Paul Erdös -- because, as you know, he was such a dangerous, menacing presence... in fact here he is in 1985, inculcating impressionable American youth** with heady (perhaps subversive) formulations and sly propaganda (notice too his frumpy apparel and deceptively simple-looking trademark coffee cup, all designed to lend Paul an innocent, innocuous appearance to the untrained eye):

Anyways, read the article below, which at the end actually links to Erdös' full, boring file:

Of course little of significance was ever found in tracking the incredible Erdös (...yo, FBI, you feel obliged to start a file? ...fill one on Mitch McConnell, Jeff Sessions, Karl Rove, Clarence Thomas, or other real dangers to democracy and the American way).

Perhaps J. Edgar Hoover's only true goal was to claim an Erdös number of 1: The final sentence of the piece reads, "His FBI surveillance, then, appears little more than a clever way to get J. Edgar Hoover's name on a paper with Paul Erdős."  (...yes, J. Edgar was a clever one... and, according to some, looked better in a dress than Paul ever would have! ...not that anyone even wants to imagine that).

The non-seditionary parts of Erdös' life (i.e., his entire life) have been well-chronicled in two biographies:
"The Man Who Loved Only Numbers" by Paul Hoffman
"My Brain Is Open" by Bruce Schecter

p.s...:  this coming Saturday would have been Erdös' 103rd birthday, so in case you're looking for a reason to party this weekend (now that your March Madness bracket is in a shambles) there you are! Have some coffee, dream up some theorems, and party hard.

p.p.s...:  I can only imagine what the FBI's file on Keith Devlin must look like by now ;-)

      *  a take-off on Erdös' signature phrase, "My brain is open" (or, more likely, "My  
          br-r-r-r-a-ain eeez o-o-open).

     **  lest anyone not be familiar with it, Paul Erdös pictured with 10-year-old Terence Tao,
         via Wikipedia.

Monday, March 21, 2016

The March of Primes

               ...1     ...3     ...7      ...9

Last week when an article reported looking at 'patterns' in the last digits of consecutive primes, I mentioned I thought it swung the door wide open to a plethora of prime digit data exploration one might do. Mike Lawler gets that ball rolling in this post (and I suspect others out there are looking at a variety of possibilities):

(specifically, -- Mike looks at final digits of prime triples, but one can imagine plenty of other possibilities -- having said that, it's also possible that the sheer act of looking over loads of data, now so easy to generate, will result in occasional pattern-like findings emerging... that may lack any real meaning, beyond "chance").
Anyway, could all make for a very interesting year ahead in primes and number theory....

Sunday, March 20, 2016

Messy Math...

 For Sunday reflection, Jim Holt commenting last year on Mochizuki's "proof" of the abc conjecture:
"...mathematics—which, in my cynical moods, I tend to regard as little more than a giant tautology, one that would be as boring to a trans-human intelligence as tic-tac-toe is to us—is really something weirder, messier, more fallible, and far more noble."

[p.s.... Math-Frolic Interview #36 is newly up over at MathTango]

Thursday, March 17, 2016

Smiles For Wiles

No great surprise, I suspect, that Andrew Wiles has won the 2016 Abel Prize for his untiring work on Fermat's Last Theorem... richly-deserved:

...coverage in Nature for the award:

It was almost 13 years ago that Wiles ended a Cambridge lecture to wild applause, with, "I think I'll stop here," having just concluded his monumental proof in public for the first time ever! (7+ years in-the-formulation, of a 350+year-old problem he'd been obsessed with since childhood... the proof required some later technical corrections).

Also inspired by Wiles' award, this brief, interesting post from Michael Harris:

I'd love to take a moment to explain Wiles' wonderful proof to you all, but I just don't quite have room for it in this post. ;-)
However, in a video of the actual Abel ceremony, starting at about the 10:20 mark, Alex Bellos offers a 15-min. explanation of Wiles' work):

Tuesday, March 15, 2016

Story of the Week! (and it's only Tuesday)

Like a good magician, prime numbers never quite reveal everything held up their sleeves.
As most have likely heard already the momentous story of the week (perhaps the month) for many of us, doesn't include Donald Trump, but rather a new 'pattern' or bit of non-randomness noted for prime numbers.

Of course primes aren't truly random to begin with (in their distribution throughout the number system), but this finding indicates that even their final individual digits appear for some reason skewed away from randomness, with a given prime being followed by another prime who's final digit differs from the first with greater-than-expected probability. As articles have mentioned, it is remarkable it's taken this long for anyone to notice. Like the magician who's sleight-of-hand distracts us from seeing what's right in front of our eyes. Whether this finding will have any practical application is difficult to see; for now it simply sits baldly and boldly in the rarefied domain of number theory awaiting further explication.

Significantly (I would think) it may open up a Pandora's Box of other questions to be looked at regarding individual digits of primes and their relationships in placement, order, succession... and will any variances from 'randomness' discovered be mere mathy statistical glitches or 'accidents', or do they hold some rich, deeper meaning not yet understood? I suspect most believe the latter.

Anyway, the story started in the popular press (so far as I'm aware) with Quanta Magazine and this fantastic piece by Erica Klarreich:

Evelyn Lamb covered the subject for Nature:

ADDENDUM:  Dr. Lamb now also has this fascinating followup post at her "Roots of Unity" blog

John Baez offers good coverage here:

And more technically, Terry Tao here:

(there are many other articles, but these give a great run-down)

Monday, March 14, 2016

Timing Is Everything...

Not sure when/if I'll get around to reading/reviewing either of these, but a couple of new popular books from Basic Books worth noting:

"Burn Math Class" by Jason Wilkes


"Fluke" by Joseph Mazur

Both books appear at a good time.  "Burn Math Class" seems to be another look at how math education ought to transpire (versus how it traditionally has), and arises right on the heels of renewed sound-and-fury from Andrew Hacker's book "The Math Myth."

The Mazur book is yet another entry on probability and coincidences and the like (and the public's mis-perception thereof), coming right at a moment when there has been tremendous attention, in both the professional and popular press, to the misuse of statistics.

Just from leafing through, the rebellious-looking Wilkes book looks the more interesting of the two to me, but both definitely warrant consideration, and hit bookstores at an excellent time.

Both books, by the way, get supportive blurbs from Jordan Ellenberg, among others...
Of Wilkes' book he writes: "spirited, hip-nerdy Burn Math Class is what high school math might look like if it were redesigned by people who loved math but hated high school."

And of Mazur's book, this: "Fluke walks the reader, hand in steady hand, through the weird and dangerous landscape of extreme probability, distinguishing cause from correlate, and phenomenon from mere coincidence."

Check 'em both out!

And lastly, yeah, I know it's Pi Day, so have at it with Vi Hart:

Sunday, March 13, 2016

Pure vs. Applied

For Sunday reflection, Paul R. Halmos, from his 1968 essay "Mathematics as a Creative Art":
"…a brief and perhaps apocryphal story about David Hilbert… When he was preparing a public address, Hilbert was asked to include a reference to the conflict between pure and applied mathematics, in the hope that if anyone could take a step toward resolving it, he could. Obediently, he is said to have begun his address by saying 'I was asked to speak about the conflict between pure and applied mathematics. I am glad to do so, because it is, indeed, a lot of nonsense -- there should be no conflict -- there is no conflict -- in fact the two have nothing whatsoever to do with one another!' "

Friday, March 11, 2016

Prepare To Submit... our Google Masters (perhaps):

I've never played "Go" in my life, but that hasn't stopped me from finding the current Man vs. Machine story of a human grand champion vs. Google's AlphaGo program fascinating... fascinating specifically, of course, because Google's AI program has already won the first 2 matches (as of this moment), of a game far more complex than chess, and needs only win one more to take the best-of-5 series (it could all be over later today -- ADDENDUM: AlphaGo won the 3rd match to take the series); much to the shock of the loser, 18-time world champion, grandmaster Lee Sedol (not to mention 1000s observing the matchup).

First, driverless cars, now a world champion level Go player; how much longer can it be before a Google computer proves the Riemann Hypothesis? ;-)

Anyway, read up on the remarkable milestone matchup to date:  (Slate)

And more basic information about AlphaGo here:

For those unfamiliar with this ancient game, a 15-min. YouTube tutorial on the basics:

Wednesday, March 9, 2016

Let's Get Real! (I'm a tad peeved)

"...there are real issues at stake here, and there’s nothing wrong—nothing wrong at all—with people arguing about the details while at the same time being aware of the big picture."   -- Andrew Gelman

Another great follow-up from Andrew Gelman today on the whole statistics debate I referenced yesterday:

Do read Gelman's more formal presentation, but here, my own further informal comment:

Some of those defensive about the "crisis in replication" in psychology are arguing that the so-called "replications" (that failed) weren't actually precise replications at all, but merely rough and poor approximations... well, DUHHHH, of course there are NO TRUE REPLICATIONS in psychology... get real!: when dealing with human behavior, if you use different samples at different times on different days in different places, under varying conditions, it is not going to be an exact replication (EVEN IF you were otherwise able to duplicate the same methodological steps).  And as others have noted, IF your finding requires absolute precise duplication in order to replicate, than you are NOT doing science, which seeks to find results that generalize more widely.

Moreover, Gilbert (and other psychologists) HAVE NEVER and WILL NEVER do an adequately controlled, well-defined, unbiased, highly-generalizable experiment in social psychology using a truly random sample... it IS NOT possible!... uncontrolled independent variables are far too many, complex, poorly-defined, and likely synergistic, to permit understanding of, or definitive conclusions about, any dependent variable(s) under study, AND no human sample is ever really random (...but, that won 't prevent them from publishing such studies). Why can't we just admit out-loud to such imperfections...

Nassim Taleb is fond of dismissing such studies as BS and moving on... I'm not that harsh -- social studies can lack rigor, yet still contain glints of value and interest for further exploration, SO LONG as their results aren't presumed reliable, valid, and widely-applicable from the get-go. GOOD (and meaningful) studies in psychology are very, very, very difficult to do, and so too, good replications.

Tuesday, March 8, 2016

Lies, Damned Lies, and...

yeah, you know what's coming....
If statistics talk leaves you yawning, followed by a major face-plant on the desktop, no reason to continue reading....

For a few years now controversy over the use of stats in social sciences in particular (but also in biomedicine and other research areas) has been brewing. It seems to have erupted Mt. Vesuvius fashion this week.

I was myself a psychology major long ago, but never held much credence in the methodologies or empiricism that was practiced. Indeed, I'm amazed it's taken this long for the controversy (in large part over two pillars: significance-testing and replication) to bubble over; there's so much weak, dodgy psyche research out there going back a century.

Below are a sampling of the many recent posts dealing with a new set of "six principles" (in regard to p-values) the American Statistical Association just recommended:

1)  First, this NY Times piece gives some of the background to the whole debate:

2) "Retraction Watch" interviews the Executive Director of the American Statistical Association, Ron Wasserstein, about the 'six new principles' for the use of p-values:  (a good read)

3)  Nature reports:

4)  from another British journal:

5)  Deborah Mayo adds some nuance to the discussion here:

6)  Andrew Gelman adds comments, with more emphasis on 'null hypothesis testing' and "the garden of forking paths":

7)  and from FiveThirtyEight blog, perhaps the most entertaining read:

8)  Some older work of Nassim Taleb, also pertinent to the discussion here:

These are just a smattering of what is out there right now on this matter. Much of what transpires in psychology (and particularly social psychology), I believe, would be better characterized as "social studies" and NOT "social science" -- and social studies, by the way, are very much worth doing (despite the variables, complexity, and ambiguities involved)... they just ought not be confused with good "science," especially in regards to the generalizability or extrapolation of the findings.

When I took statistics in grad school the professor warned us on the first day of class that he hated teaching these "statistics for social scientists" courses (but was required to) because there was no way to satisfactorily teach students what they needed to know for doing research, in a semester or two. And no matter what grade we got in the class our knowledge and understanding of statistics would be inadequate; we would be turned loose on the world thinking we knew more than we did.
I've often seen mathematicians write that no one really learns or understands calculus when they first take it (again no matter what grade you get); it is only after further years of mathematics study that a deeper understanding of the calculus finally emerges. So too statistics.

Anyway, these statistical debates entail hugely important issues that we need to attend to; issues going well beyond journal research articles. Currently, statistics and probability classes are increasingly a part of secondary education, and we need some assurance of getting it right... lest we send off a whole 'nother generation with wrong ways to think about and apply statistics.
This controversy will continue for a long awhile... because it is tangled and long overdue (and... it's putting a lot of practitioners on the defensive).

Monday, March 7, 2016

Old, Old Favorite

Below, a problem I gave to a local math meetup group some weeks back thinking it would be familiar to EVERYone (it is my favorite problem from geometry class that I had 50 years ago(!), and I've seen it on the Web multiple times). To my surprise, NO ONE was familiar with it, so I present it here in case anyone else has led such a sheltered life as to have missed this delightful puzzle ;-):

MNOP is a RECTANGLE inscribed in one quadrant of a circle ("O" being the origin/center of the circle).
PO = 10 cm.
SP= 3 cm.

What is the length of diagonal line PN?
(solution can be arrived at in seconds with no trig and very little geometry required)
answer below:
ANSWER:  as a diagonal, PN is EQUAL to MO (not drawn in). MO is a radius, as is SO. SO=13, thus PN=13.

Sunday, March 6, 2016

Essential Knowledge...

A Sunday reflection from K.C. Cole (in "The Universe and the Teacup;" from 1998, but perhaps appropriate to some of the debate happening today):
"Used correctly, math can expose the glitches in our perceptual apparatus that lead to common illusions -- such as our inability to perceive the difference between millions and billions -- and give us relatively simple ways of protecting ourselves from our own ignorance. As the physicist Richard Feynman once said: 'Science is a long history of learning how not to fool ourselves.' A knowledge of the mathematics behind our ideas can help us to fool ourselves a little less often, with less drastic consequences.
"In short, math matters -- a lot more than most people think. We have to make life-and-death decisions based on what numbers tell us. We cannot afford to remain dumb about mathematical ideas simply because we hated them in high school -- any more than we can remain dumb about computers, or AIDS. Mathematics is essential, not peripheral, knowledge."

Thursday, March 3, 2016

Some Picks From Pickover

via Wikipedia

h/t to Clifford Pickover for tweeting out a link to this Wikipedia page about mini- and micro chess -- games I was unfamiliar with, but that look interesting/fun:

And as long as I'm mentioning Cliff, here are a few other favorite/miscellaneous recent tweets (@pickover) from him:

...and his homepage:

Tuesday, March 1, 2016


So many cool mathy things crossing my screen lately I don't want to wait 'til Friday to pass some of them along, so a mini-potpourri today:

  With Andrew Hacker back in the NY Times and a new book out continuing his crusade for math education reform, Keith Devlin responds (in a very interesting, nuanced way) at the Huffington Post:

He ultimately concludes:
"I give his [Hacker's] essays an A for observation, C for background knowledge, and an F for drawing the wrong conclusions.
No, make that a D/F for his conclusions. His arguments did yield correct conclusions, he just did not realize they did, and claimed the opposite."

And Devlin's long-form review of Hacker's book at "Devlin's Angle" HERE.

2)  Speaking of books, the fascinating story of a state lottery beaten at its own game (from Adam Kucharski's new book, "The Perfect Bet"):

3)  A longish report from Tim Gowers on the launch of the new online journal "Discrete Analysis":

And finally this lovely official trailer from the new life of Ramanujan film: