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Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

32 + 42 = 52

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=2 is a right triangle, but (1,1,2) is not a Pythagorean triple because 2 is not an integer. Moreover, 1and 2 do not have an integer common multiple because 2 is irrational.

Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10.

Plimpton 322, a Babylonian clay tablet from about 1800 BC
Plimpton 322, a Babylonian clay tablet from about 1800 BC

One example of a Pythagorean triple is a=3, b=4, and c=5: Ancient Egyptians used this group of Pythagorean triples to measure out right angles. They would tie knots in a piece of rope to create 3, 4, and 5 equal spaces. Three people would then hold each corner of the rope and form a right triangle!

3-4-5 triangle using rope in Egypt
3-4-5 triangle using rope in Egypt

right triangles during the construction process to help determine the slope of the pyramid. The Pythagorean Theorem states that given a right triangle with sides of length a and b respectively and a hypothenuse of length c, the lengths satisfy the equation a2 + b2 = c2.

Pyramids at Giza
All three of Giza’s famed pyramids and their elaborate burial complexes were built during a frenetic period of construction, from roughly 2550 to 2490 B.C.

When searching for integer solutions, the equation a2 + b2 = c2 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation. The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers. The solutions are described by the following theorem: This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b.

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old college try

Noun. old college try (plural old college tries) (informal) A vigorous, committed attempt or effort, often in the context of a nearly hopeless situation where failure is expected. Source


The expression give it the old college try came not from the college campus, but from the baseball diamond. At the turn of the century, a player was said to give it the old college try when attempting to make a play like a heroic attempt at catching a fly ball that was very far out of the player’s reach. Source

A wild and desperate attempt to make a play. Sometimes the term carries a hint of showboating.

Babe Ruth (Babe Ruth’s Own Book of Baseball, 1928) defined “giving it the old college try” as “playing to the grandstand or making strenuous effort to field a ball that obviously cannot be handled.”

The term was quickly applied to any effort with limited chances of success.

Old College Try | Family Guy

From The New Dickson Baseball Dictionary (1999) by Paul Dickson.

In a column that appeared in the Columbus (Ohio) Citizen (Nov. 26, 1927) and was quoted in American Speech (Apr. 1930), Billy Evans wrote that “I gave it the old college try” is a term “often used in big league baseball, when some player keeps on going after a fly ball, usually in foul territory, with the odds about ten to one he would never reach it. Teammates of such a player often beat him to it by shouting in unison with the thought of humor uppermost: ‘Well, kid, you certainly gave it the old college try,’ as he falls short of making the catch. When some player does something that a professional player might not ordinarily attempt, such as colliding with a fielder who had the ball ready to touch him out, in the hope that he might make him drop the ball, regardless of the danger he was courting, someone is sure to say, often ironically, if the speaker happens to be one of the players in the field: ‘That’s the old college spirit.'”
by akkartik April 10, 2009

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Alan Watts

Alan Watts
Alan Watts

“The only way to make sense out of change is to plunge into it, move with it, and join the dance.”

Alan Wilson Watts (6 January 1915 – 16 November 1973) was a British-born philosopher, writer, and speaker, best known as an interpreter and populariser of Eastern philosophy for a Western audience. Born in Chislehurst, he moved to the United States in 1938 and began Zen training in New York. Pursuing a career, he attended Seabury-Western Theological Seminary, where he received a master’s degree in theology. Watts became an Episcopal priest in 1945, then left the ministry in 1950 and moved to California, where he joined the faculty of the American Academy of Asian Studies.

Watts gained a large following in the San Francisco Bay Area while working as a volunteer programmer at KPFA, a Pacifica Radio station in Berkeley. Watts wrote more than 25 books and articles on subjects important to Eastern and Western religion, introducing the then-burgeoning youth culture to The Way of Zen (1957), one of the first bestselling books on Buddhism. In Psychotherapy East and West (1961), Watts proposed that Buddhism could be thought of as a form of psychotherapy and not a religion. He also explored human consciousness, in the essay “The New Alchemy” (1958), and in the book The Joyous Cosmology (1962).

Towards the end of his life, he divided his time between a houseboat in Sausalito and a cabin on Mount Tamalpais. His legacy has been kept alive by his son, Mark Watts, and many of his recorded talks and lectures are available on the Internet. According to the critic Erik Davis, his “writings and recorded talks still shimmer with a profound and galvanizing lucidity.

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Civilization | Bongo Bongo Bongo

Civilization” is an American traditional pop song. It was written by Bob Hilliard and Carl Sigman, published in 1947 and later included in the 1947 Broadway musical Angel in the Wings, sung by Elaine Stritch.[2] The song is sometimes also known as “Bongo, Bongo, Bongo (I Don’t Want to Leave the Congo)”, from its first line of the chorus. The sheet music gives the title as “Civilization (Bongo, Bongo, Bongo)”.


The song is a satire of modern society sung from the perspective of a native person whose village is visited by a “civilized person” and other “civilized” people whom the native refers to as “educated savages”. These visitors are trying to “civilize” the tribe. However, the native rejects them, and after listing the major flaws of civilization, ultimately decides that he will stay where he lives (presumably the Congo, as reflected in the song’s lyrics). Source: Wikipedia

Track Title: Civilization

Prime Artist: Frank Sinatra
Written by: Carl Sigman
Written by: Bob Hilliard


Each morning the missionary advertises on the neon sign,
It tells the native population that civilization is fine.
And pre-educated savages holler from the bamboo tree
That civilization is not for me to see.

Oh bongo bongo bongo, I don’t wanna leave the
Congo oh no, no, no, no, no,
Bingle bangle bongle, I don’t wanna leave jungle,
I refuse to go.
I don’t want landlords, junkyard, cocktails, caviar,
I’ll make it clear that no matter how they coax me,
I’ll stay right here.

I looked through a magazine a missionary’s wife concealed.
I see how people who are civilized bang you with automobile.
When they have two weeks’ vacation, they hurry to vacation grounds,
They swim and they fish, but that’s what I do all year round.

Oh bongo bongo bongo, I don’t wanna leave the
Congo oh no, no, no, no, no,
Bingle bangle bongle, I don’t wanna leave jungle,
I refuse to go.
I don’t want landlords, junkyard, cocktails, caviar,
I’ll make it clear that no matter how they coax me,
I’ll stay right here.

They have things like the atom bomb,
So I think I’ll stay here where I am,
Civilization, I’ll stay here.

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3 things I learned while my plane crashed

Ric Elias had a front-row seat on Flight 1549, the plane that crash-landed in the Hudson River in New York in January 2009. What went through his mind as the doomed plane went down? At TED, he tells his story publicly for the first time.

Ric Elias is the CEO and cofounder of Red Ventures, a portfolio of fast-growing digital businesses.

Why you should listen

Ric Elias was given the gift of a miracle: to face near-certain death, and then to come back and live differently.

Video 4m 45s

Ric Elias – Ted Talks

A native of Puerto Rico, Elias attended Boston College and Harvard Business School before starting his career as part of GE’s Financial Management program. He cofounded Red Ventures in 2000, just months before the dot-com bubble burst. The company weathered the storm; by 2007 it was ranked fourth on the Inc. 500 list, and in 2015 the company was valuated at more than $1 billion. Elias has cultivated an award-winning company culture, ranking as a “Best Place to Work” in Charlotte, North Carolina, for ten years in a row.

Elias’s leadership style and personal life are deeply influenced by his experience as a survivor of Flight 1549, also known as the “Miracle on the Hudson.” He is devoted to using his platform to “leave the woodpile higher than he found it” — spinning out multiple nonprofits from Red Ventures over the years, all of which are aimed at creating educational opportunity and economic mobility for under-served groups. In 2018, Elias launched Forward787, a social enterprise committed to raising and deploying $100 million to build businesses in Puerto Rico that compete with the world’s top companies. In 2019, he launched a podcast, 3 Things with Ric Elias, as a continuation of the learning journey he shared on the TED stage.

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The Three Laws of Recursion

Like the robots of Asimov, all recursive algorithms must obey three important laws:

  • A recursive algorithm must have a base case.
  • A recursive algorithm must change its state and move toward the base case.
  • A recursive algorithm must call itself, recursively.

Recursion is the process of defining a problem (or the solution to a problem) in terms of (a simpler version of) itself. For example, we can define the operation “find your way home” as: If you are at home, stop moving. Take one step toward home.

Let’s begin our discussion of recursion by examining the first appearance of fractals in modern mathematics. In 1883, German mathematician George Cantor developed simple rules to generate an infinite set:

Cantor’s rule for an infinite set

There is a feedback loop at work here. Take a single line and break it into two. Then return to those two lines and apply the same rule, breaking each line into two, and now we’re left with four. Then return to those four lines and apply the rule. Now you’ve got eight. This process is known as recursion: the repeated application of a rule to successive results. Cantor was interested in what happens when you apply these rules an infinite number of times.

George Cantor

Dichotomy paradox – Zeno’s

“That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”

— as recounted by Aristotle, Physics VI:9, 239b10

Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.

Zeno’s paradox was recursive by cutting the distance in half each time to the infinitesimal. This is also how the Tortoise beat the Hair by questioning time over distance.

Recursive Function Calls

The tortoise and the Hair – the paradox of time
int factorial(int n) 
{ if (n == 1) { return 1; }
else { return n * factorial(n-1); } }

A function that does call others is called a nonleaf function. … The factorial function can be rewritten recursively as factorial(n) = n × factorial(n – 1). The factorial of 1 is simply 1. The image shows an object trace of the factorial function written as a recursive function. Each call goes in the run time stack until the base case is reached, and the the stack is popped as the result is passed to each function on the stack.

Five Factorial (5!) in recursion

What Is a Fractal?

The term fractal (from the Latin fractus, meaning “broken”) was coined by the mathematician Benoit Mandelbrot in 1975. In his seminal work “The Fractal Geometry of Nature,” he defines a fractal as “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.”

Recursion in Nature

Looking closely at a given section of the tree, we find that the shape of this branch resembles the tree itself. This is known as self-similarity; as Mandelbrot stated, each part is a “reduced-size copy of the whole.”

The Three Laws of Robotics

Isaac Asimov was an American writer and professor of biochemistry at Boston University. During his lifetime, Asimov was considered one of the “Big Three” science fiction writers, along with Robert A. Heinlein and Arthur C. Clarke. A prolific writer, he wrote or edited more than 500 books.

  • A robot may not injure a human being or, through inaction, allow a human being to come to harm
  • A robot must obey the orders given it by human beings except where such orders would conflict with the First Law
  • A robot must protect its own existence as long as such protection does not conflict with the First or Second Laws
Partial sources:, Wikipedia, Google 
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Barth | Michael O’Donoghue

Michael O’Donoghue (January 5, 1940 – November 8, 1994) was an American writer and performer. He was known for his dark and destructive style of comedy and humor, was a major contributor to National Lampoon magazine, and was the first head writer of Saturday Night Live. He was also the first performer to utter a line on that series.

Lucy in the Sky with Diamonds

Look! Up in the sky! It’s a bird! It’s a plane! It’s …

Lucy in the Sky with Diamonds
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James Carville

“Pennsylvania is Pittsburgh and Philadelphia with Alabama in the middle”

James Carville
Pennsylvania | Nations Online Project

The actual quote, and I was present when he made it in 1991, was “Pennsylvania is bordered by two metropolises at either end with Alabama in the middle.”

Carrie Rickey | Quora

James Carville

Chester James Carville Jr. (born October 25, 1944) is an American political consultant and author who has strategized for candidates for public office in the United States, and in 23 nations abroad. A Democrat, he is also a media personality with expertise in U.S. elections who appears frequently on cable news programs, in podcasts, and in his public speeches. Source: Wikipedia

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Roll Models | Venti is large

After salesmen Danny (Paul Rudd) and Wheeler (Seann William Scott) trash a company truck, the court gives them a choice: jail time or community service in a mentoring program. Thinking to take the easy way out, the two overgrown adolescents find themselves paired with a teenager (Christopher Mintz-Plasse), who is experiencing the pangs of first love, and a foul-mouthed fifth-grader (Bobb’e J. Thompson), who needs an attitude adjustment.
Release date: November 7, 2008 (USA)

30 second video clip

Role Models clip | Venti

“Congratulations, you’re stupid in three languages”.

Paul Rudd | Role Models
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Alan Alda | The 3 Rules of 3

We Were Built to Connect with Other People — Here’s How to Be Better At It. Before you follow another “tip” or “trick,” there’s something Alan Alda wants you to know.

His best tip to become a better communicator is what he calls the three rules of three. Listen to his practical hints for becoming a communication pro but, as he remarks, try to get there organically through the process. Alan Alda’s most recent book is If I Understood You, Would I Have This Look on My Face?

“So the first rule is, I try only to say three important things when I talk to people”.

“The second rule is, if I have a difficult thing to understand, if there’s something I think is not going to be that easy to get, I try to say it in three different ways”.

“And the third tip, which I always forget, is that if I have a difficult thing that’s hard to get, I try to say it three times through the talk”.

—- Alan Alda
Alan Alda
Alan Alda

Alan Alda doesn’t want you to take “pro tips” from anyone-not even Alan Alda. When it comes to his area of expertise public speaking and empathetic communication there are no hacks or shortcuts; if you want to be a world class public speaker, you have to earn those stripes through the process of deeply understanding what it is to talk, listen, and connect.

Alda calls tips intellectual abstractions; it’s akin to the difference between information and knowledge, between parroting a few words in French and speaking the actual language. But, when pushed by yours truly at Big Think, Alda does give up the goods (willingly we promise no Alan Aldas were harmed in the making of this video).

5 min Video

Alan Alda | The 3 Rules of 3

Alan Alda has earned international recognition as an actor, writer and director. In addition to The Aviator, for which he was nominated for an Academy Award, Alda’s films include Crimes and Misdemeanours, Everyone Says I Love You, Flirting With Disaster, Manhattan Murder Mystery, And The Band Played On, Same Time, Next Year and California Suite, as well as The Seduction of Joe Tynan, which he wrote, and The Four Seasons, Sweet Liberty, A New Life and Betsy’s Wedding, all of which he wrote and directed. Recently, his film appearances have included Tower Heist, Wanderlust, and Steven Spielberg’s Bridge of Spies.


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Noise by Daniel Kahneman | 3 Distinctions

The Michael Shermer Show with Daniel Kahneman – Noise: A Flaw in Human Judgment


Imagine that two doctors in the same city give
different diagnoses to identical patients. Now
imagine that the same doctor making a different
decision depending on whether it is morning or
afternoon, or Monday rather than Wednesday.
This is an example of noise: variability in
judgments that should be identical.

Shermer speaks with Nobel Prize winning
psychologist and economist Daniel Kahneman
about the detrimental effects of noise and what
we can do to reduce both noise and bias, and
make better decisions in: medicine, law, economic
forecasting, forensic science, bail, child
protection, strategy, performance reviews, and
personnel selection.

Video clip – 3 minutes

Noise by Daniel Kahneman | 3 Distinctions

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Full Video – Noise by Daniel Kahneman

Full video at

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Michio Kaku: 3 mind-blowing predictions about the future

What lies in store for humanity? Theoretical physicist Michio Kaku explains how different life will be for your descendants—and maybe your future self, if the timing works out.

15 min

Michio Kaku

Michio Kaku: 3 mind-blowing predictions about the future
  1. We will become a space-faring species
  2. We will expand the brain’s capabilities
  3. We will defeat cancer


Michio Kaku (Japanese: カク ミチオ, 加来 道雄, born January 24, 1947) is an American theoretical physicist, futurist, and popularizer of science (science communicator). He is a professor of theoretical physics in the City College of New York and CUNY Graduate Center. Kaku is the author of several books about physics and related topics and has made frequent appearances on radio, television, and film. He is also a regular contributor to his own blog, as well as other popular media outlets. For his efforts to bridge science and science fiction, he is a 2021 Sir Arthur Clarke Lifetime Achievement Awardee.