Easy-to-See, Hard-to-Solve Problems

This has been around for awhile, but still contains some interesting problems to play with... Responses to the Quora question, "What are some unsolved problems in math which seem easy at first glance?":

http://www.quora.com/What-are-some-unsolved-problems-in-math-which-seem-easy-at-first-glance

Meanwhile, ICYMI, I've posted a bit of a year-end blog retrospective over at MathTango:
http://mathtango.blogspot.com/2015/01/proverbial-look-back.html

...and hope before month's end to have 1-2 new interviews over there as well.

Gardner Reflects...

What better way to begin a new year of Sunday reflections than with some words of reflection from Martin Gardner (from his autobiographical "Undiluted Hocus Pocus"):

"Writing the column for more than twenty-five years was one of the greatest joys of my life. If you look over all my columns you'll find that they steadily become more sophisticated mathematically. That was because I was learning math. I had not taken a single math course in college, although I loved the low-level math I learned in high school. And I had always been fond of recreational math ever since I was introduced to it as a boy by Sam Loyd's mammoth Cyclopedia of puzzles.
"One of the pleasures in writing the column was that it introduced me to so many top mathematicians, which of course I was not. Their contributions to my column were far superior to anything I could write, and were a major reason for the column's growing popularity. The secret of success was a direct result of my ignorance. Even today my knowledge of math extends only through calculus, and even calculus I only dimly comprehend. As a result, I had to struggle to understand what I wrote, and this helped me write in ways that others could understand."


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

Teach the Children Well...

Fitting I s'pose that I should start a new year with Keith Devlin, who has probably graced my posts more often than any other individual. His blog-year started off (yesterday) with another piece about the educational system in his ongoing attempt to persuade Americans of the need for Common Core. I'm afraid by now he's preaching to the choir -- views on this topic are so hardened. Either you believe in Common Core (perhaps with some reservations, but nonetheless support it), or you think it the product of overly-liberal educators and incompetent, intrusive government... and not a lot of wiggle-room in-between.
Keith writes succinctly at one point:

"The fact is, any parent who opposes adoption of the CCSS is, in effect, saying, 'I do not want my child prepared for life in the Twenty-First Century.' They really are. Not out of lack of concern for their children, to be sure. Quite the contrary. Rather, what leads them astray is that they are not truly aware of how the huge shifts that have taken place in society over the last thirty years have impacted educational needs."

Common Core opponents focus on the past, which somehow they think was just fine, while Keith is focused on the future, especially in terms of needed job skills for rising generations.
He also links to some great TEDTalks (Sugata Mitra and Ken Robinson) that I suspect most readers here have seen... but if you haven't, by all means, watch.

Read Keith's entire piece here:
http://devlinsangle.blogspot.com/2015/01/your-fathers-mathematics-teaching-no.html

A central part of his post is a chart showing the "skills" most sought-after in new workers by Fortune 500 companies back in 1970 versus 1999 (is there not an even more up-to-date list?)
"Problem-solving," which didn't appear in the top 10 in 1970, is #2 in 1999 (and certainly one of the impetuses for education change). In fact, the top 3 sought-after skills had completely changed by 1999, and I have to give some cynical, non-mathematical commentary about the other two:

1)  #3 on the list is "interpersonal skills," non-existent in the top 10 from 1970! I can't help but think that this is partly the result of today's world becoming a courser, less-civil place than the world of 1970. Perhaps in 1970 it was presumed that if you made it to adulthood and were seeking employment, then you had sufficient interpersonal skills for a workplace -- today employers must seek out people with such interpersonal skills! Also, on an even more cynical note, I suspect there are more jobs today (sales, marketing, public relations, management etc.) that require a person to be persuasive and controlling of others, than in 1970, when "honesty" or "sincerity" may have been greater virtues than the ability to manipulate and sway people.

2)  Meanwhile, #1 on the 1999 list is "Teamwork" (it was #10 in 1970) -- this troubles me a bit! One of the most common questions prospective employees hear these days is along the lines of "Are you a team player?" I've felt for some time now that this is often code for, "will you do whatever the company asks you to, regardless of laws and ethics?" Employers don't want workers with consciences or personal ethics (who might object to something or be whistleblowers), so much as company drones who will 'look the other way' when needed and march to the company anthem. That's a broad generalization, but from observation of big business behavior over the last 3 decades (yes, there are some ethical businesses out there; their numbers just seem in steady decline).

I usually agree with Dr. Devlin's views, but was once troubled by a response he gave me to a question I posed to him last year. He wrote:
"...the only possible answer to the provision of good education in this country is by private enterprise. The state system is a century out of date and broken beyond repair."

Wow!, "broken beyond repair"... As a proponent of public education and critic of private industry, that gloomy reply stung! The thought of private companies in charge of our education system (and it's already happening in some places) disturbs me (I continue to believe it possible for public education, and for that matter, big government, to work successfully). Keith is part of a small entrepreneurial company (BrainQuake, Inc.) and I suppose his optimism stems in part from that experience. I'm not concerned with enterprises of that size... but am worried by the companies that will, if Keith's company is successful, gobble up his enterprise and strictly subordinate whatever its good works and intentions are, to a bottom-line (...and then burp, ignominiously).

No doubt in my mind that Dr. Devlin is right and the educational system badly needs reforming... but it's so much broader than that... the Fortune 500 itself is also in desperate need of reform! And, unfortunately, I'm not convinced either will happen anytime soon.

Keith and I are close to the same age, and he once again closed out his post with some classic 60's music... so once again he inspires me to do the same ;-):

HAPPY 11111011111

(via WikimediaCommons)

YES, in binary,  2015  is a palindromic year!  Lookin' forward (...and backward?) to it ;-)

...And wishing everyone a multitude of merry math moments in the many months ahead!!

Two Cultures (EXCELLENT read)

Almost two weeks ago I posted about the obituary for Alexander Grothendieck that was rejected by the journal Nature. Many math/science sites covered that little news story, and I sort of assumed by now it was over-and-done-with.  But even in death, as in life, Grothendieck seems to spread ongoing controversy!
Today, launching off that rejected obituary, mathematician/biologist Lior Pachter of UC Berkeley has posted a remarkable, really incredible and rich post I think, about the two "cultures" of mathematics and molecular biology, which he straddles, but finds little common ground on for its participants.
It's a long, and often technical post, but I think all should have a go at reading it (it may well require more than one sitting, and don't expect to follow all parts).  As a layperson myself, I'm more interested in the broad strokes he is painting than many of the technical arguments that I can't grasp. His "list of specific differences" between mathematicians and biologists is especially interesting, and Pachter is pessimistic about the relationship between the "two cultures," writing at one point, "The relationship between biology and mathematics is on the rocks and prospects are grim," and "The extent to which the two cultures have drifted apart is astonishing." As I implied in my original post (linked to above) I'm not so sure we really have a 'two culture' problem anymore (the term, as most know, comes originally from C.P. Snow over fifty years ago), so much as a fiefdom problem, with intense specialization having subsumed pretty much every field of technical study.

Anyway, if you're a working biologist or mathematician (or really, a scientist of any stripe) READ this piece!:

https://liorpachter.wordpress.com/2014/12/30/the-two-cultures-of-mathematics-and-biology/

As I post this, there are 3 comments to Pachter's article; I suspect there will be many more over time.
Agree or disagree with him, there's LOTS to chew on.

Puzzletime...

To ease into the week, a problem I adapted from one seen over at the 7puzzle blog site:

From the numbers 1-37, find the five integers that remain when you eliminate the following:

1)  any integers containing a 1
2)  prime numbers
3)  factors of 72
4)  numbers divisible by 3 or 5

Once you have the five 'finalists,' eliminate those number pairs that add up to 60.  Then, what number is left?
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ANSWER:  22

The Density of Numbers

Sunday reflection on rationals and irrationals....

"It is possible to show that both the rationals and the irrationals are densely distributed along the number line in the following sense: Between any two rational numbers, there lie infinitely many irrationals and, conversely, between any two irrationals are to be found infinitely many rationals. Consequently, it is easy to conclude that the real numbers must be evenly divided between the two enormous, and roughly equivalent, families of rationals and irrationals.
"As the nineteenth century progressed, mathematical discoveries came to light indicating, to the contrary, that these two classes of numbers did not carry equal weight. The discoveries often required very technical, very subtle reasoning. For instance, a function was described that was continuous (intuitively, unbroken) at each irrational point and discontinuous (broken) at each rational point; however it was also proved that no function exists that is continuous at each rational point and discontinuous at each irrational point.  Here was a striking indicator that there was not a symmetry or balance between the set of rationals and the set of irrationals.  It showed that, in some fundamental sense, the rationals and irrationals, were not interchangeable collections, but to the mathematicians of the day, it was unclear exactly what was going on."

-- William Dunham in "Journey Through Genius"

"Trip The Light"

Shalom... Namaste... Noel... Gratis... Wonder... Joy... Amity... Equanimity. . . . . .

Last posting 'til after Christmas day.  Per usual, there will be a "Sunday Reflection" here on the forthcoming Sunday, and probably a potpourri list over at MathTango on same day).
After a trying year for the world in so many ways, may everyone find reasons for cheer & celebration in the days ahead... and beyond.



(...or, for another treat, watch the fun Matt Harding version HERE.)

A Big Family (puzzle)

Another problem to kick off the week, once again adapted from Henry Dudeney:

Max, who already has some children from a prior marriage, marries the widow Wilma who also has some prior children. A dozen years later their family has a total of 12 children, including all prior children and the new ones resulting from their marriage. Each partner, Max and Wilma, have 9 children (out of the 12) that they are direct parents of.  How many children have been born to Max and Wilma together in the last 12 years?:
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[...And alternatively, if you want a little meatier puzzle, Mike Lawler just walked his kids (and readers) through one from MIT yesterday: http://tinyurl.com/njkkepd ]
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ANSWER:  6 of their own  (having 3 each from prior marriages)

A Mathematical Tension

The unexplained mystery...

"I, for one, find Gödel's incompleteness theorems rather comforting. It means that mathematicians will never be complete. There will always be something else which is undecidable with the current axioms. Should the human species survive another few million years and continue churning out mathematics at the rate we've done for the past few thousand years, we still won't have considered it all. There will always be work for all of the future mathematicians. As always, some of that work will go on to be incredibly useful for the rest of civilization, and much of it will remain the pointless but endlessly amusing plaything of academics.
"There's an unexplained mystery behind all of this, which I've been delicately avoiding throughout the book. If maths is the consequence of games and puzzles, the result of pure intellectual thought, why does it end up being so practically useful?  I keep promoting maths as a bit of fun, yet no one can ignore that mathematical techniques are the workhorse of modern technology. In reality, mathematics is a serious industrial endeavor. There's a tension between what I claim to be the origin of maths and where it ends up being used."

-- Matt Parker from "Things To Make and Do In the Fourth Dimension"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

Honoring Grothendieck...

We live in a day of extraordinary and over-riding science specialization....

H/T to Jordan Ellenberg for pointing to this post about an obituary for Alexander Grothendieck that was rejected by Nature -- a fascinating read, even if Nature didn't find it so for their obit. purposes. David Mumford, one of the authors, finds it "very depressing" that a STEM publication would judge this piece unsuitable for its readers, but I'd opt for a different view... namely, that Grothendieck was simply too far advanced beyond the minds that run (or read) generalist journals like Nature and Science (which are far from the bastions they once were, before such modern-day field-of-study specialization took hold):

http://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html

Early on, the piece reads as follows:

"His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed themselves -- and then, like a magician, show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed. His strength and intensity were legendary. He worked long hours, transforming totally the field of algebraic geometry and its connections with algebraic number theory. He was considered by many the greatest mathematician of the 20th century."

Surely there is a far more appropriate (specialist) math journal out there that would love to run Mumford and John Tate's wonderful tribute piece for an appreciative audience....

A Li'l More On That Wily Matt Parker....

I already wrote a blurb at MathTango about Matt Parker's fantastic book, "Things To Make and Do In the Fourth Dimension," but now that I've finished reading it, just want to add a few quick notes:

1)  First, I'll reiterate it's a wonderful volume -- I enjoyed the second half (which touched on several of my favorite topics, and also told perhaps the most fun story of Tartaglia's rivalry with Fior over algebraic/cubic equations) even more than the first half.
2)  Do note however, that at least parts of the volume may require slightly more math sophistication, or interest, or just persistence, than some of the other volumes I included on my Holiday gift list; i.e. while Matt's book is a fun and educational read, not every chapter is an easy read. 
3)  Also, one small complaint:  the book lacks an index, which because of the sheer number and diversity of topics/information included, would've been helpful.
4)  Finally, (and the REAL reason for this additional posting), BE SURE to read the "Acknowledgements" section at the very conclusion of the book! (...a section readers often skip over). Not only is the section entertaining to read, BUT in it are buried these innocuous, cryptic lines:

"Oh yeah, and there is a competition hidden somewhere in this book. If anyone wins it, I'll think of a suitable prize. Beware of the traps."

Leave it to Matt to concoct such a ploy! And I assume by "competition," he is not referencing proving the Riemann hypothesis! ;-).  (The book poses various questions and problems at points, but I'm not sure what is being referred to as "a competition" or "the traps" -- could be fun going back through the pages trying to figure out what it's all about.)
Anyway, have at it, and, with this heads-up, may one of my readers win the prize!

Triangle Puzzler

To start the week, a quick and simple (...or, not so simple) problem:

You have a triangle. In some particular order, the three sides and height of the triangle are four consecutive integers. What is the area of the triangle?
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ANSWER:  84  (a 13-14-15 triangle with height 12)

An Epiphany

Sunday reflection via Steven Strogatz....

"The teacher, Mr. diCurcio, said, 'I want you to figure out a rule about this pendulum.' He handed each of us a little toy pendulum with a retractable bob. You could make it a little bit longer or shorter in clicks in discrete steps. We were each handed a stopwatch and told to let the pendulum swing ten times, and then click, measure how long it takes for ten swings, and then click again, repeating the measurement after making the pendulum a little bit longer. The point was to see how the length of the pendulum determines how long it takes to make ten swings. The experiment was supposed to teach us about graph paper and how to make a relationship between one variable and another, but as I was dutifully plotting the length of time the pendulum took to swing ten times versus its length it occurred to me, after about the fourth or fifth dot, that a pattern was starting to emerge. These dots were falling on a particular curve I recognized because I'd seen it in my algebra class. It was a parabola, the same shape that water makes coming out of a fountain.
"I remember having an enveloping sensation of fear. It was not a happy feeling but an awestruck feeling. It was as if this pendulum knew algebra. What was the connection between the parabolas in algebra class and the motion of this pendulum? There it was on the graph paper. It was a moment that struck me, and was my first sense that the phrase 'law of nature' meant something. I suddenly knew what people were talking about when they said there could be order in the universe and that, more to the point, you couldn't see it unless you knew math. It was an epiphany that I've never really recovered from."

-- Steven Strogatz from "Who Cares About Fireflies"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]
 

A Friday Puzzle

To end the week, a simple-to-state puzzle that I've re-written/adapted from an old Henry Dudeney volume:

In the course of a year, the cats (and there are more than one) on Mr. Schlobotnik's farm killed 999,919 mice. If every cat killed exactly the same number of mice (and more than 1), then how many cats reside on the farm, given that the total number of cats is LESS than the number of mice killed per cat?
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Answer:
there were 991 cats, who each killed 1009 mice

NOT To Be Missed... on number theory/prime gaps

"After a while, these things taunt you".... (T. Tao)

FANTASTIC piece from Erica Klarreich and Quanta Magazine today on another obvious, but deep question from number theory (how LARGE can prime gaps be? ...sort of the reverse of the twin-prime question):

https://www.quantamagazine.org/20141210-prime-gap-grows-after-decades-long-lull/

Includes a "favorite joke" of number theorists that I'd not heard before :-) and also perhaps my favorite photo from all of mathematics: Paul Erdös and Terence Tao (as a child) together.
Seriously, with mentions of Yitang Zhang, Erdös, Tao, James Maynard, prime gaps, a crazy-ass log formula, and $10,000 prize, what is there not to love!

"Mathematical Mystery Tour"

The BBC has done some great hour-long mathematical presentations. Recently, Cliff Pickover tweeted one of the old Horizon episodes (I've linked to before), called "Mathematical Mystery Tour." It's interesting with its discussion of "proof" in mathematics (in light of the post I did a bit ago over at MathTango), as well as other subjects. And though it's rather dated, still a worthwhile 50 mins. if you've missed it, and have some time:



[In other news, I've now posted a blurb on Matt Parker's recent volume, "Things To Make and Do In the Fourth Dimension," over at MathTango.]

A Book Recommended

I mentioned last week (at MathTango) that Richard Elwes' 2013 book, "Chaotic Fishponds and Mirror Universes," was one of my very favorite reads of 2014. So this morning just a blurb from its Introduction, as the Sunday reflection, in hopes of encouraging you to check it out further:

"Of all the subjects studied, debated and fought over in the course of human history, I happen to believe that the most fascinating is mathematics. That's a bold claim -- perhaps mystifying to readers who were bored or baffled by the subject at school. Well, of course fascination is in the eye of the beholder, and certainly there will be those who need some persuading. I hope this book will go some way towards doing that.
"What is irrefutable, however, is that in modern life mathematics is both important and ever-present. Even the most entrenched maths-hater has an awareness that it plays a central role in today's world, touching our lives in more ways than ever before. But that is where the details are liable to become hazy... yes, important, but where exactly is it used, and in what ways?
"In response, I present in the pages that follow a selection of 35 diverse applications of mathematics. I attempt to unravel some of the principles that underlie aspects of our daily lives, as well as those that inform today's boldest thinkers....

"I hope that, by the end of this book, readers will have a more precise sense of where mathematics fits into modern life -- and, en route that some doubters become devotees of the subject that I find endlessly, gloriously, fascinating."

Deja Vu: Revisiting The Flash Mind Reader

About 9 months ago, in a tweet, math teacher Fawn Nguyen casually mentioned "The Flash Mind Reader," a delightful Web-based puzzle, that I was unfamiliar with, though she apparently has known of it for close to a dozen years. Go here to check it out, if perchance you've not seen it:
http://www.cs.nyu.edu/~dodis/magic-ball.swf

What a great game/puzzle for younguns, but it also stumped me for awhile before I figured it out, and wrote a post alluding to it. So I was delighted to be reading "The Best Writing On Mathematics 2014" recently and come upon this great teaser once again (pgs. 171-5). It comes up in a selection aptly entitled "Wondering About Wonder In Mathematics" by Dov and Rina Zazkis.  It's one of my favorites of many great selections in this year's anthology. The authors pinpoint "surprise" as the underlying component of "wonder" in mathematics, and then list four types of "mathematical surprise":

1) perceived "magic"
2) counterintuitive results
3) variation on a known result or procedure
4) paradoxes

"The Flash Mind Reader" falls under the 'perceived magic' category, and they write this about it:

"We have used this activity [the Mind Reader] several times with both elementary school and university students. It's not uncommon for members of both groups to try to cover the webcams on their computers or face away from the screen, as if the Mind Reader was determining what number was in their head using some elaborate eye-tracking mechanism. Obviously, these actions do not prevent the Mind Reader from working.  However, these reactions serve both to illustrate some rudimentary theory testing -- 'Is this website tapping into the webcam?' -- and to demonstrate students' need to understand how this 'Mind Reader' works, which is catalyzed by their curiosity."

I'm heartened to know that university students can be as duped by this little gem as I was at first blush ;-) Of course a lot of number and card tricks are based on pure mathematics; in some ways, Flash Mind Reader takes the element of 'distraction,' which is often a component of such "magic," to another subtle level, which helps make it so effective. [In the event you don't see how the puzzle works, you'll have to buy the book, or google for the answer, I won't give it away here!]
The rest of the chapter looks at some other classic and interesting examples from mathematics, placing them in the four categories above. The Mandelbrot set, Platonic solids, the Monty Hall problem, and Simpson's paradox, are among standards mentioned in the chapter.

Anyway, I encourage folks to get a hold of this year's "Best Writing On Mathematics," as I think it the best edition yet (and unfortunately most expensive) of a series that I hope maintains interest and support. It was recently reviewed by Alexander Bogomolny:  http://tinyurl.com/p28acrz
[I included it on my recent list of books for the Holidays at MathTango.]


On a sidenote, thinking there might be some interesting back-story here, I attempted to find information about web designer Andy Naughton, who created The Flash Mind Reader, to include in this post, and was surprised that though his name and the game are found MANY times in Google searches, I couldn't actually find any background info on him... is he alive??? is he very private? Is Naughton his real name (both "Andy Naughton" and "Andy Wolfe" seem to be associated with "FlashLight Creative" -- are they 2 different people or one-and-the-same?) Is there some mystery to all this? Does anyone happen to know much about the fellow?  Just curious what the history to the Mind Reader might be, and how, if at all, its success affected Andy's life??? (...if I could locate him, and he was willing, I might be interested in doing one of my Math-Frolic interviews with him, as well). ...maybe if I just hone my own mind-reading skills I can find him.

Math, Women, Tessellation, Intuition

(image: WikimediaCommons)

A lot of discussion around the Web these days about women in STEM, and at Math-Frolic I'm even more interested in women in math, so thought it would be fun/timely to recount the unusual story of Marjorie Rice -- worth repeating, even if most are familiar with it, as a rare instance of someone becoming involved with math almost by accident.
[Most of this information was reported over a year ago in a MathMunch piece on Marjorie here:
http://mathmunch.org/2013/02/25/marjorie-rice-inspired-by-math-and-subways/  also see Ivars Peterson's 2010 piece here: http://mathtourist.blogspot.com/2010/06/tiling-with-pentagons.html ]

Marjorie discovered her senior year in high school that she found math interesting, but by then it was too late to do much with it. She went on to marry, have children, be a housewife; i.e. she took NO mathematics past high school. But after getting a subscription to Scientific American for her son, she began reading the Mathematical Games column of Martin Gardner, including a 1975 column concerning "pentagon tessellations," i.e. pentagon forms that could cover an entire plane, repeating themselves with no gaps, like a jigsaw puzzle. At one time mathematicians believed there were only five such pentagon shapes that achieved tessellation, but in 1968 three more were discovered, and a fourth new one had just been added in 1975 that Gardner was reporting on.

Marjorie was intrigued. And playing with different pentagons, with different internal angles, she finally found a fresh one that accomplished the feat of tessellation. Inventing her own unconventional notation to describe her work she wrote to Gardner showing the result. And he sent her correspondence on to another female mathematician, Doris Schattschneider, who confirmed Marjorie's success and translated her work into more standard mathematical format.  Marjorie went on to find yet three more successful pentagon tessellations, and also DISproved a conjecture made by Doris.
Successful amateurs have made significant contributions to astronomy, but in most sciences, and particularly in mathematics, it is rare for an academically-untrained amateur to accomplish something missed by professionals... but apparently Marjorie didn't know that! Her own website on her work is here:
https://sites.google.com/site/intriguingtessellations/home

She is now over ninety, and remains an inspiration, not just to women, but to amateur enthusiasts everywhere. What I love most though about the Marjorie Rice story isn't that she was a female in mathematics, nor even that she was an amateur contributing to a technical field, but rather what the story says about the role of intuition and insight in math. Despite mathematics' image of being cold, dry, and rigid, and despite its abstractness in advanced study, scrape below the surface and there remains, on occasion, a powerful substrate of intuition and mental imagery, accessible to many.

Below is a video segment (from about the 31:50 point to 35:45) talking about Marjorie's work (again h/t to MathMunch for this):





We now know of 14 tessellating pentagon forms! Are there more?