100 Greatest Science Discoveries of All Time: Levers and Buoyancy

Year of Discovery: 260 B.C.

What Is It? The two fun da men tal prin ci ples un der ly ing all phys ics and
en gi neer ing.
Who Dis cov ered It? Ar chi me des
Why Is This One of the 100 Great est?
The con cepts of buoy ancy (wa ter pushes up on an ob ject with a force equal to the
weight of wa ter that the ob ject dis places) and of le vers (a force push ing down on one side of
a le ver cre ates a lift ing force on the other side that is pro por tional to the lengths of the two
sides of the le ver) lie at the foun da tion of all quan ti ta tive sci ence and en gi neer ing. They rep re
sent

hu man ity’s ear li est break throughs in un der stand ing the re la tion ships in the phys i cal
world around us and in de vis ing math e mat i cal ways to de scribe the phys i cal phe nom ena of
the world. Count less en gi neer ing and sci en tific advances have depended on those two
discoveries.
How Was It Dis cov ered?
In 260 B.C. 26-year-old Ar chi me des stud ied the two known sci ences—as tron omy and
ge om e try—in Syr a cuse, Sic ily. One day Ar chi me des was dis tracted by four boys play ing
on the beach with a drift wood plank. They bal anced the board over a waist-high rock. One
boy strad dled one end while his three friends jumped hard onto the other. The lone boy was
tossed into the air.
The boys slid the board off-cen ter along their bal anc ing rock so that only one-quar ter
of it re mained on the short side. Three of the boys climbed onto the short, top end. The
fourth boy bounded onto the ris ing long end, crash ing it back down to the sand and cat a pult ing
his three
friends
into the
air.
Ar chi me des was fas ci nated. And he de ter mined to un der stand the prin ci ples that so
eas ily al lowed a small weight (one boy) to lift a large weight (three boys).
Ar chi me des used a strip of wood and small wooden blocks to model the boys and their
drift wood. He made a tri an gu lar block to model their rock. By mea sur ing as he bal anced
dif fer ent com bi na tions of weights on each end of the le ver (le ver came from the Latin word
mean ing “to lift”), Ar chi me des re al ized that le vers were an ex am ple of one of Eu clid’s pro por
tions
at
work.
The force
(weight)
push ing
down on each
side
of the
le
ver
had
to be pro por
tional
to the lengths
of board on
each
side
of the bal
ance
point. He
had dis
cov
ered
the
math
e
mat
i
cal
con cept
of le vers,
the most
com
mon
and basic
lifting
system
ever
devised.

MYSTERY MAN

A trove of fossils found deep in a South African cave adds a baffling new branch to the human family tree. 

Conjured in clay and cast in silicone by paleoartist
John Gurche, Homo naledi is the newest addition to our genus.
MARK THIESSEN, NGM STAFF

BY JAMIE SHREEVE  •  PHOTOGRAPHS BY ROBERT CLARK (National Geographic - October 2015)

On September 13, 2013, two recreational cavers named Steven Tucker and Rick Hunter entered a dolomite cave system called Rising Star, some 30 miles northwest of Johannesburg. Rising Star has been a popular draw for cavers since the 1960s, and its filigree of channels and caverns is well mapped. Tucker and Hunter were hoping to find some less trodden passage.
In the back of their minds was another mission.
In the first half of the 20th century, this region produced so many fossils of our early ancestors that it  later became known as the Cradle of Humankind. Though the heyday of fossilhunting there was long past, the cavers knew that a scientist at the University of the Witwatersrand in Johannesburg was looking for bones. The odds of happening upon something were remote. But you never know.

Deep in the cave, Tucker and Hunter worked their way through a constriction called Superman’s
Crawl—because most people can fit through only by holding one arm tightly against the body and extending the other above the head, like the Man of Steel in flight. Crossing a large chamber, they climbed a jagged wall of rock called the Dragon’s Back. At the top they found themselves in a pretty
little cavity decorated with stalactites. Hunter got out his video camera, and to remove himself from the frame, Tucker eased himself into a fissure in the cave floor. His foot found a finger of rock, then another below it, then—empty space.

Dropping down, he found himself in a narrow, vertical chute, in some places less than eight inches  ide. He called to Hunter to follow him. Both men have hyper-slender frames, all ~on going~

Pigeons Make Flight Plans

Though their head-bobbing walk may be comical, pigeons in flight are no fools. They’re urban artful dodgers, threading their way among buildings and other obstacles. David Williams and his Harvard University colleagues studied the birds’ maneuvers to learn how they avoid collisions.
First Williams trained wild pigeons to fly through an empty corridor. Then he placed vertical poles at intervals in the corridor and videotaped the birds in the altered course. He expected them to use one evasive move consistently. Instead, the birds employed two moves, which researchers named: a “pause,” in which the wings stalled at the top of a stroke, and a “fold,” in which the wings were pulled back. Pausing was better for efficiently maintaining height, the study found, while folding
helped the birds fit through narrow gaps and remain stable in a collision. — Lindsay N. Smith

 source: National Geographic - October 2015

PUZZLE 001 - Arithmetic & Algebraic Problems

Source: 536 PUZZLES & CURIOUS PROBLEMS by Henry Ernest Dudeney
Introduction Henry Ernest Dudeney (the last name is pronounced with a long "u" and a strong accent on the first syllable, as in "scrutiny") was England's greatest maker of  puzzles. With respect to mathematical puzzles, especially problems of more than trivial mathematical interest, the quantity and quality of his out­ put surpassed that of any other puzzlist before or since, in or out of England. Dudeney was born at Mayfield, in Sussex, on April 10, 1857, the son of a local schoolmaster. His  father's father, John Dudeney, was well known in Sussex as a shepherd who had taught himself mathematics and astronomy while tending sheep on the downs above Lewes, a town fifty miles south of London. Later he became a schoolmaster in Lewes. Henry Dudeney, him­ self a self-taught mathematician who never went to college, was understand­ ably proud to be the grandson of this famous shepherd-mathematician.
Dudeney began his puzzle career by contributing short problems to news­ papers and magazines. His earliest work, published under the pseudonym of "Sphinx," seems to have been in cooperation with the American puzzlist, Sam Loyd. For a year and a half, in the late 1890's, the two men collaborated on a series of articles in Tit-Bits, an English penny weekly. Later, using his own name, Dudeney contributed to a variety of publications including The Week(y Dispatch, The Queen, Blighty, and Cassell's Magazine. For twenty years his puzzle page, "Perplexities," which he illustrated, ran in The Strand Magazine.
This was a popular monthly founded and edited by George Newnes, an enthusiastic chess player who had also started and formerly edited Tit-Bits. The Canterbury Puzzles, Dudeney's first book, was published in 1907. It was followed by Amusements in Mathematics (1917), The World's Best Word Puzzles (1925), and Modern Puzzles (1926). Two posthumous collections appeared: Puzzles and Curious Problems (1931) and A Puzzle-Mine (undated).
The last book is a mixture of mathematical and word puzzles that Dudeney had contributed to Blighty. With few exceptions, it repeats puzzles contained in his earlier books. The World's Best Word Puzzles, published by the London Daily News, contains nothing of mathematical interest. Dudeney's first two books have, since 1958, been available to American and British readers as paperback reprints. Modern Puzzles and Puzzles and Curious Problems, in many ways more interesting than the first two books because they contain less familiar puzzles, have long been out of print and are extremely hard to obtain. The present volume includes almost the entire contents of those two books.
Readers familiar with the work of Sam Loyd will notice that many of the same puzzles appear, in different story forms, in the books of Loyd and Dudeney. Although the two men never met in person, they were in frequent correspondence, and they had, Dudeney once said in an interview, an informal agreement to exchange ideas. Who borrowed the most? This cannot be answered with finality until someone makes a careful study of the newspaper and magazine contributions of both men, but it is my guess that most "Of the borrowing was done by Loyd. Dudeney never hesitated to give credits. He often gives the name or initials of someone who supplied him with a new idea, and there are even occasional references to Loyd himself. But Loyd almost never mentioned anyone. Mrs. Margery Fulleylove, Dudeney's only child, recalls many occasions on which her father fussed and fumed about the extent to which his ideas were being adapted by Loyd and presented in America as the other puzzlist's own. Loyd was a clever and prolific creator of puzzles, especially in his ability to dramatize them as advertising novelties, but when it came to problems of a more mathematically advanced nature, Dudeney was clearly his superior. There are even occasions on record when Loyd turned to Dudeney for help on difficult problems.
Geometrical dissections-cutting a polygon into the smallest number of pieces that can be refitted to make a different type of polygon-was a field in which Dudeney was unusually skillful; the present volume contains many surprising, elegant dissections that Dudeney was the first to obtain. He was also an expert on magic squares and other problems of a combinatorial nature, being the first to explore a variety of unorthodox types of magic squares, such as prime-number squares and squares magic with respect to operations other than addition. (There is an excellent article by Dudeney, on magic squares, in the fourteenth edition of the Encyclopaedia Britannica.) In recreational num­ber theory he was the first to apply "digital roots" -the term was probably coined by him-to numerous problems in which their application had riot been previously recognized as relevant. (For a typical example of how digital roots furnish a short cut to an answer otherwise difficult to obtain, see the answer to Problem 131 in this volume.)
Dudeney was tall and handsome, with brown hair and brown eyes, a slightly aquiline nose, and, in his later years, a gray mustache and short chin whiskers. As one would expect, he was a man of many hobbies. "He was naturally fond of, and skilled at games," his wife Alice wrote in a preface to Puzzles and Curious Problems, "although he cared comparatively little for cards. He was a good chess player, and a better problemist. As a young man he was fond of billiards, and also played croquet well." In his elderly days he enjoyed bowling every evening on the old bowling green within the Castle Precincts, an area surrounding the ruins of an old castle in Lewes. The Dudeneys owned a two hundred-year-old house in this area, where they were living at the time of Dudeney's death on April 24, 1930. (In Alice Dudeney's preface this date erroneously appears as 1931.)
Mrs. Fulleylove recalls, in a private communication, that her father's croquet lawn, "no matter how it was rolled and fussed over, was always full of natural hazards. Father applied his mathematical and logical skill to the game, with special reference to the surface of our lawn. He would infuriate some of our visitors, who were not familiar with the terrain, by striking a ball in what ap­peared to be the wrong direction. The ball would go up, down, around the hills and through valleys, then roll gaily through the hoop .... " Alice Dudeney speaks of her husband as a "brilliant pianist and organist," adding that, at different times, he was honorary organist of more than one church. He was deeply interested in ancient church music, especially plain song, which he studied intensively and taught to a choir at Woodham Church, Surrey. Mrs. Fulleylove tells me that her father, as a small boy, played the organ every Sunday at a fashionable church in Taunton, Somerset. He was a faithful Anglican throughout his life, attending High Church services, keenly interested in theology, and occasionally writing vigorous tracts in defense of this or that position of the Anglican church. As a little girl, Mrs. Fulleylove sometimes accompanied her father to his London club for dinner. She remembers one occasion on which she felt very proud and grown-up, hoping the waiter and other guests would notice her sophistication and good manners.To her horror, her father, preoccupied with some geometrical puzzle, began penciling diagrams on the fine damask tablecloth.
In his later life, Mrs. Fulleylove writes, her father lost interest in all composers except Richard Wagner. "He had complete transpositions for the piano of all Wagner's works, and played them unceasingly-to the great grief of my mother and myself, who preferred the gentler chamber music. "The house at Littlewick, in Surrey," Mrs. Fulleylove continues, "where we lived from 1899 to 1911, was always filled with weekend guests, mostly pub­lishers, writers, editors, artists, mathematicians, musicians, and freethinkers."
One of Dudeney's friends was Cyril Arthur Pearson, founder of the Daily Press and of C. Arthur Pearson, Ltd., a publishing house that brought out Dudeney's Modern Puzzles. Other friends included Newnes and Alfred Harrnsworth (later Lord Northcliffe), another prominent newspaper publisher. "Father provided me, by degrees, with a marvelous collection of puzzle toys, mostly Chinese, in ebony, ivory, and wood ... ," Mrs. Fulleylove recalls. "He was a huge success at children's parties, entertaining them with feats of legerdemain, charades, and other party games and stunts ....  "We had a mongrel terrier that I adored. His name, for some obscure reason, was Chance.
One day father fell over the dog's leash and broke his arm. His comment, made without anger, was a quotation: 'Chance is but direction which thou canst not see.' "
In an interview in The Strand (April, 1926) Dudeney tells an amusing stol)' about a code message that had appeared in the "agony column" of a London newspaper. A man was asking a girl to meet him but not to let her parents know about it. Dudeney cracked the code, then placed in the column a message to the girl, written in the same cipher, that said: "Do not trust him. He means no good. Well Wisher." This was soon followed by a code message from the girl to "Well Wisher," thanking him for his good advice. 
Alice Dudeney, it should be added, was much better known in her time than her husband. She was the author of more than thirty popular, romantic novels. A good photograph of her provides the frontispiece of her 1909 book, A Sense of Scarlet and Other Stories, and her biographical sketch will be found in the British Who Was Who. "A Sussex Novelist at Home," an interview with her that appeared in The Sussex County Magazine (Vol. I, No. I, December 1926, pp. 6-9), includes her picture and photographs of the "quaint and curious" Castle Precincts House where she and her husband then lived. Dudeney himself tried his hand on at least one short story, "Dr. Bernard's Patient," (The Strand, Vol. 13, 1897, pp. 50-55). Aside from his puzzle features, he also wrote occasional nonfiction pieces, of which I shall mention only two:
"The Antiquity of Modern Inventions" (The Strand, Vol. 45, 1913, p. 389 f) and "The Psychology of Puzzle Crazes" (The Nineteenth Century, a New York periodical, Vol. 100, December 1926, p. 868 D. I have rearranged and reclassified the puzzles that appear in this collection, but only minimally edited the text. British words such as "petrol" have been changed to their American equivalents; long paragraphs have been broken into shorter ones to make for easier reading; and in problems about money American currency has been substituted for British. Some of Dudeney's money problems, so dependent on the relationships between British coins that they cannot be formulated with American currency, have been omitted. In the few cases where duplicate problems, with only trivially different story lines, ap­peared in the two books I have chosen the version I considered best and left out the other. Titles for problems remain unaltered so that those who may wish to check back to the former appearance of a puzzle can do so easily. The illustrations reproduce the original drawings (some of them done by Mrs. Fulleylove when she was a young girl), enlarged and occasionally retouched to make them clearer. I have added several footnotes to the puzzles and in the answer section appended a number of comments that are bracketed and initialed. Some of these additions correct errors or point out how an answer has been improved or a problem extended by later puzzle enthusiasts. I hope no one will suppose that these comments reflect in any way on Dudeney's genius. The greatest of mathematicians build on the work of predecessors, and their work in turn is the foundation for the work of later experts. The mathematical-puzzle field is no exception. Dudeney was one of its greatest pioneers, perhaps the greatest, and it is a tribute to him that he was able to invent problems of such depth that decades would pass before others would find ways of improving his answers. It is Mrs. Fulleylove who is mainly responsible for the book now in the reader's hands.
We were in touch first by correspondence; then in 1966, when she took up residence in a New York City suburb, she informed me that she had obtained world reprint rights for Modern Puzzles and Puzzles and Curious Problems.
Would I be interested, she asked, in editing them into a single book? I replied that I would indeed. Enthusiasts of recreational mathematics will re­ joice in the appearance of this long inaccessible material, the cream of  Dudeney's later years. They will find the book a rich source of unusual problems, many of them leading into fascinating regions that have yet to be fully explored.  For much of the information in my notes I am indebted to Victor Meally, Dublin County, Ireland. Although he is mentioned often in the notes, there are many places where I followed his excellent and generously given advice without referring to him. I also wish to thank Harry Lindgren, Canberra, Australia; Thomas H. Q'Beirne, Glasgow; and C. C. Verbeek, the Hague, for other valuable suggestions. Martin Gardner
HASTINGS-aN-HUDSON, N.Y. 

Puzzle #001
1. CONCERNING A CHECK
A man went into a bank to cash a check. In handing over the money the cashier, by mistake, gave him dollars for cents and cents for dollars. He pocketed the money without examining it, and spent a nickel on his way home. He then found that he possessed exactly twice the amount of the check. He
had no money in his pocket before going to the bank. What was the exact amount of that check?

Answer: The amount must have been $31.63. He received $63.31. After he had spent a nickel there would remain the sum of $63.26, which is twice the amount of the check.