Monday, December 22, 2014

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)


Sunday, December 21, 2014

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.
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-- 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.]

Thursday, December 18, 2014

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....


Wednesday, December 17, 2014

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!


Monday, December 15, 2014

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)

Sunday, December 14, 2014

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.
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-- 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.]
 

Friday, December 12, 2014

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

Wednesday, December 10, 2014

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!


Tuesday, December 9, 2014

"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.]


Sunday, December 7, 2014

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.
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Wednesday, December 3, 2014

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.

Monday, December 1, 2014

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?


Sunday, November 30, 2014

Flashes, Pitches, Models, etc.


"We think that if, say, two variables are causally linked, then a steady input in one variable should always yield a result in the other one. Our emotional apparatus is designed for linear causality. For instance, if you study everyday, you expect to learn something in proportion to your studies. If you feel that you are not going anywhere, your emotions will cause you to become demoralized. But modern reality rarely gives us the privilege of a satisfying, linear, positive progression: you may think about a problem for a year and learn nothing; then, unless you are disheartened by the emptiness of the results and give up, something will come to you in a flash."
-- Nassim Taleb from "The Black Swan"

...and from Edward Tufte, this:

People and institutions cannot keep their own score accurately. Metrics soon become targets and then pitches, and are thereby gamed, corrupted, misreported, fudged…
"Examples: premature revenue recognition, Libor rates, beating the quarterly forecast by a single penny, terrorist attacks prevented, Weapons of Mass Destruction, number of Twitter followers, all body counts (crowd sizes, civilians blown up). Sometimes call the Principle of Lake Woebegone, where all children are above average.

...and a couple more quickies:

"If you torture the data long enough, it will confess to anything." -- Ronald Coase (British economist)

"Essentially, all models are wrong, but some are useful."  -- statistician George Box


[…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.]


Wednesday, November 26, 2014

Happy Thanksgiving Holiday To All

(via WikimediaCommons)



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"If the only prayer that you say 

your whole life is 'thank you'

that would suffice."

~~ Meister Eckhart

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Tuesday, November 25, 2014

Tuesday Tidbits


Ed Frenkel has a stirring tribute to "visionary" Alexander Grothendieck in the NY Times today, hitting on some of the highlights of the reclusive mathematician's life, and ending as follows:
"A party of one, he was unafraid to be himself and to speak his truth. The man who had advanced mathematics in the most profound ways did not believe that math was the answer to everything. He taught us that life is more valuable than any equation."
And at MathTango yesterday I (aided by Keith Devlin) considered the nature of "proof":
http://mathtango.blogspot.com/2014/11/proofiness.html

BTW, I'll probably be posting a 2014 book wrap-up for the popular math volumes of the year sometime after Thanksgiving. It's pretty much already written, but if you have volumes you're partial to for people's Holiday lists, feel free to mention them in the comments here, or in the later post to come.
And for my own interest, curious if anyone is directly familiar with the book "Gödel's Mistake" by Ashish Dalela? If so, do you recommend it, or, not so much (for an interested layreader, not academic logician)?


Sunday, November 23, 2014

Mathematics: "A Growing Organism"... "A Connected Web"


This Sunday's Reflection:

"Mathematics is a living and growing organism; within it are intricate and delicate structures of strong aesthetic appeal.  It offers opportunities for surprise as unexpected vistas open the mind to new lines of thought...
"Mathematics was created by all manner of people. There were religious bigots and atheists, political reactionaries and wild revolutionaries, snobs and egalitarians; some were people of great charm, some odious. If there is any common denominator, it is a driving curiosity, a desire to understand, a need to build, even if the structures be abstract. Admirably suited though mathematics is to modelling the real world, it can be developed totally without dependence on anything outside itself. Parts of it are simply mind creations, owing nothing to the physical world. It is a playground for the mind...

"Regrettably, many of us have never been allowed to see what mathematics is. It has been obscured by pointless emphasis on routines rather than ideas. This failure to distinguish what is important has led many people to see mathematics as a collection of totally arbitrary rules which have to be learnt by rote, and performed with the exactness and precision of a religious rite. Ask a person if there is much to be remembered in mathematics; if they speak of an overwhelming mass of material, their education in this area has been counter-productive; not merely neutral. Mathematics, properly seen, is a connected web; grasp at one piece and all the surrounding region comes to mind.
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-- Laurie Buxton from the Introduction to  "Mathematics For Everyone" (1984)