A checkerboard ( North American English ) or chequerboard ( Commonwealth English except Canada ; see spelling differences ) is a game board of checkered pattern on which checkers (also known as English draughts ) is played. Most commonly, it consists of 64 squares (8×8) of alternating dark and light color, typically green and buff (official tournaments), black and red (consumer commercial), or black and white (printed diagrams). An 8×8 checkerboard is used to play many other games, including chess , whereby it is known as a chessboard . Other rectangular square-tiled boards are also often called checkerboards. In The Netherlands, however, a dambord (checker board) has 10 rows and 10 columns for 100 squares in total (see article International draughts).
48-550: Martin Gardner featured puzzles based on checkerboards in his November 1962 Mathematical Games column in Scientific American. A square checkerboard with an alternating pattern is used for games including: The following games require an 8×8 board and are sometimes played on a chessboard . Given a grid with m {\displaystyle m} rows and n {\displaystyle n} columns,
96-491: A petroleum geologist , and his wife, Willie Wilkerson Spiers, a Montessori-trained teacher. His mother taught Martin to read before he started school, reading him The Wizard of Oz , and this began a lifelong interest in the Oz books of L. Frank Baum . His fascination with mathematics started in his boyhood when his father gave him a copy of Sam Loyd 's Cyclopedia of 5000 Puzzles, Tricks and Conundrums . He attended
144-758: A Psi-Watcher") for Skeptical Inquirer , that organization's monthly magazine. These columns have been collected in five books starting with The New Age: Notes of a Fringe Watcher in 1988. Gardner was a critic of self-proclaimed Israeli psychic Uri Geller and wrote two satirical booklets about him in the 1970s using the pen name "Uriah Fuller" in which he explained how such purported psychics do their seemingly impossible feats such as mentally bending spoons and reading minds . Martin Gardner continued to criticize junk science throughout his life. His targets included not just safe subjects like astrology and UFO sightings , but topics such as chiropractic , vegetarianism , Madame Blavatsky , creationism , Scientology ,
192-409: A column for Scientific American called "Mathematical Games". It ran for over a quarter century and dealt with the subject of recreational mathematics . The "Mathematical Games" column became the most popular feature of the magazine and was the first thing that many readers turned to. In September 1977 Scientific American acknowledged the prestige and popularity of Gardner's column by moving it from
240-648: A copy of it. Over seven thousand requests came pouring in, some of them from other countries. This caused significant consternation in the US defense agencies and possible legal problems for Gardner himself. The National Security Agency (NSA) asked the RSA team to stop distributing the report and one letter to the IEEE suggested that disseminating such information might be violating the Arms Export Control Act and
288-459: A far greater debt to Martin Gardner than most conjurors realize. –Stephen Minch Martin Gardner held a lifelong fascination with magic and illusion that began when his father demonstrated a trick to him. He wrote for a magic magazine in high school and worked in a department store demonstrating magic tricks while he was at the University of Chicago. Gardner's first published writing (at
336-646: A free-standing article on hexaflexagons which ran in the December 1956 issue of Scientific American . Flexagons became a bit of a fad and soon people all over New York City were making them. Gerry Piel, the SA publisher at the time, asked Gardner, "Is there enough similar material to this to make a regular feature?" Gardner said he thought so. The January 1957 issue contained his first column, entitled "Mathematical Games". Almost 300 more columns were to follow. It ran from 1956 to 1981 with sporadic columns afterwards and
384-923: A function f ( m , n ) {\displaystyle f(m,n)} , f ( m , n ) = { black if m ≡ n ( mod 2 ) , white if m ≢ n ( mod 2 ) {\displaystyle \displaystyle {f(m,n)}={\begin{cases}{\text{black}}&{\text{if}}\ m\equiv n{\pmod {2}}\,,\\{\text{white}}&{\text{if}}\ m\not \equiv n{\pmod {2}}\\\end{cases}}} or, alternatively, f ( m , n ) = { black if m + n is even , white if m + n is odd {\displaystyle \displaystyle {f(m,n)}={\begin{cases}{\text{black}}&{\text{if}}\ m+n{\text{
432-538: A hub of this network helped facilitate several introductions that led to further fruitful collaborations. Mathematicians Conway, Berlekamp, and Guy, who met as a result of Gardner's influence, would go on to write Winning Ways for your Mathematical Plays , a foundational book in combinatorial game theory that Gardner subsequently championed. Gardner also introduced Conway to Benoit Mandelbrot because he knew of their mutual interest in Penrose tiles . Gardner's network
480-548: A lot of criticism from the advocates of alternative science and New Age philosophy . He kept up running dialogues (both public and private) with many of them for decades. In a review of Science: Good, Bad and Bogus , Stephen Jay Gould called Gardner "The Quack Detector", a writer who "expunge[d] nonsense" and in so doing had "become a priceless national resource." In 1976 Gardner joined with fellow skeptics philosopher Paul Kurtz , psychologist Ray Hyman , sociologist Marcello Truzzi , and stage magician James Randi to found
528-518: A major impact on mathematics in the second half of the 20th century. His column ran for 25 years and was read avidly by the generation of mathematicians and physicists who grew up in the years 1956 to 1981. His writing inspired, directly or indirectly, many who would go on to careers in mathematics, science, and other related endeavors. Gardner's admirers included such diverse individuals as W. H. Auden , Arthur C. Clarke , Carl Sagan , Isaac Asimov , Richard Dawkins , Stephen Jay Gould , and
SECTION 10
#1732771860971576-730: A monthly column on magic tricks called "Trick of the Month" in The Physics Teacher , a journal published by the American Association of Physics Teachers . In 1999 Magic magazine named Gardner one of the "100 Most Influential Magicians of the Twentieth Century". In 2005 he received a 'Lifetime Achievement Fellowship' from the Academy of Magical Arts . The last work to be published during his lifetime
624-634: A writer and editor at Humpty Dumpty magazine, where for eight years, he wrote features and stories for it and several other children's magazines. His paper-folding puzzles at that magazine led to his first work at Scientific American. For many decades, Gardner, his wife Charlotte, and their two sons, Jim and Tom, lived in Hastings-on-Hudson, New York , where he earned his living as a freelance author, publishing books with several different publishers, and also publishing hundreds of magazine and newspaper articles. In 1950, he wrote an article in
672-638: Is mathematical presentation aimed at a general audience. Sometimes this is in the form of books which require no mathematical background and in other cases it is in the form of expository articles written by professional mathematicians to reach out to others working in different areas. Some of the most prolific popularisers of mathematics include Keith Devlin , Rintu Nath , Martin Gardner , and Ian Stewart . Titles by these three authors can be found on their respective pages. The journals listed below can be found in many university libraries. Several museums aim at enhancing public understanding of mathematics: In
720-424: Is an anagram of "Mathematical Games". Virtually all of the games columns were collected in book form starting in 1959 with The Scientific American Book of Mathematical Puzzles & Diversions . Over the next four decades fourteen more books followed. Donald Knuth called them the canonical books . His depth and clarity will illuminate our world for a long time. – Persi Diaconis Martin Gardner had
768-656: Is even}},\\{\text{white}}&{\text{if}}\ m+n{\text{ is odd}}\\\end{cases}}} The element ( m , n ) = ( 0 , 0 ) {\displaystyle (m,n)=(0,0)} is black and represents the lower left corner of the board. Martin Gardner Martin Gardner (October 21, 1914 – May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic , scientific skepticism , micromagic , philosophy , religion , and literature – especially
816-495: Is unique – in its range, its insight, and understanding of hard questions that matter." Gardner repeatedly alerted the public (and other mathematicians) to recent discoveries in mathematics–recreational and otherwise. In addition to introducing many first-rate puzzles and topics such as Penrose tiles and Conway's Game of Life , he was equally adept at writing columns about traditional mathematical topics such as knot theory , Fibonacci numbers , Pascal's triangle ,
864-613: The Antioch Review entitled "The Hermit Scientist". It was one of Gardner's earliest articles about junk science , and in 1952 a much-expanded version became his first published book: In the Name of Science: An Entertaining Survey of the High Priests and Cultists of Science, Past and Present . The year 1960 saw the original edition of the best-selling book of his career, The Annotated Alice . In 1957 Gardner started writing
912-657: The Committee for the Scientific Investigation of Claims of the Paranormal (now called the Committee for Skeptical Inquiry). Intellectuals including astronomer Carl Sagan , author and biochemist Isaac Asimov , psychologist B. F. Skinner , and journalist Philip J. Klass became fellows of the program. From 1983 to 2002 he wrote a monthly column called "Notes of a Fringe Watcher" (originally "Notes of
960-780: The International Traffic in Arms Regulations . In the end the defense establishment could provide no legal basis for suppressing the new technology, and when a detailed paper about RSA was published in Communications of the ACM , the NSA’s crypto monopoly was effectively terminated. Martin Gardner is the single brightest beacon defending rationality and good science against the mysticism and anti-intellectualism that surround us. – Stephen Jay Gould Gardner
1008-648: The Laffer curve , Christian Science , and the Hutchins-Adler Great Books Movement . The last thing he wrote in the spring of 2010 (a month before his death) was an article excoriating the "dubious medical opinions and bogus science" of Oprah Winfrey – particularly her support for the thoroughly discredited theory that vaccinations cause autism ; it went on to bemoan the "needless deaths of children" that such notions are likely to cause. Skeptical Inquirer named him one of
SECTION 20
#17327718609711056-458: The Möbius strip , transfinite numbers , four-dimensional space , Zeno's paradoxes , Fermat's Last Theorem , and the four-color problem . Gardner set a new high standard for writing about mathematics. In a 2004 interview he said, "I go up to calculus, and beyond that I don't understand any of the papers that are being written. I consider that that was an advantage for the type of column I
1104-620: The University of Chicago where he studied history, literature and sciences under their intellectually-stimulating Great Books curriculum and earned his bachelor's degree in philosophy in 1936. Early jobs included reporter on the Tulsa Tribune , writer at the University of Chicago Office of Press Relations, and case worker in Chicago's Black Belt for the city's Relief Administration. During World War II , he served for four years in
1152-400: The game of Hex invented by Piet Hein and John Nash ; Tutte's account of squaring the square ; and many other topics. The wide array of mathematicians, physicists, computer scientists, philosophers, magicians, artists, writers, and other influential thinkers who can be counted as part of Gardner's mathematical grapevine includes: These new ciphers are not absolutely unbreakable in
1200-520: The rep-tiles and pentominos of Solomon W. Golomb; the space filling curves of Bill Gosper; the aperiodic tiles of Roger Penrose; the Game of Life invented by John H. Conway; the superellipse and the Soma cube of Piet Hein; the trapdoor functions of Diffie , Hellman , and Merkle ; the flexagons of Stone , Tuckerman , Feynman , and Tukey; the geometrical delights in a book by H. S. M. Coxeter;
1248-614: The Soma Cube . Charlotte died in 2000 and in 2004 Gardner returned to Oklahoma, where his son, James Gardner, was a professor of education at the University of Oklahoma in Norman . He died there on May 22, 2010. An autobiography – Undiluted Hocus-Pocus: The Autobiography of Martin Gardner – was published posthumously. I just play all the time and am fortunate enough to get paid for it. – Martin Gardner, 1998 The "Mathematical Games" column began with
1296-643: The Ten Outstanding Skeptics of the Twentieth Century. In 2010 he was posthumously honored with an award for his contributions in the skeptical field from the Independent Investigations Group . In 1982 the Committee for Skeptical Inquiry awarded Gardner its In Praise of Reason Award for his "heroic efforts in defense of reason and the dignity of the skeptical attitude", and in 2011 it added Gardner to its Pantheon of Skeptics. Card magic, and magic in general, owe
1344-574: The Twentieth Century". He was considered the doyen of American puzzlers. He was a prolific and versatile author, publishing more than 100 books. Gardner was best known for creating and sustaining interest in recreational mathematics —and by extension, mathematics in general—throughout the latter half of the 20th century, principally through his "Mathematical Games" columns. These appeared for twenty-five years in Scientific American , and his subsequent books collecting them. Gardner
1392-579: The U.S. Navy as a yeoman on board the destroyer escort USS Pope in the Atlantic . His ship was still in the Atlantic when the war came to an end with the surrender of Japan in August 1945. After the war, Gardner returned to the University of Chicago. He attended graduate school for a year there, but he did not earn an advanced degree. In the late 1940s, Gardner moved to New York City and became
1440-420: The advent of the personal computer." Colm Mulcahy described him as "without doubt the best friend mathematics ever had." Gardner's column introduced the public to books such as A K Dewdney ’s Planiverse and Douglas Hofstadter ’s Gödel, Escher, Bach . His writing was credited as both broad and deep. Noam Chomsky once wrote, "Martin Gardner's contribution to contemporary intellectual culture
1488-597: The age of fifteen) was a magic trick in The Sphinx , the official magazine of the Society of American Magicians . He focused mainly on micromagic (table or close-up magic) and, from the 1930s on, published a significant number of original contributions to this secretive field. Magician Joe M. Turner said, The Encyclopedia of Impromptu Magic , which Gardner wrote in 1985, "is guaranteed to show up in any poll of magicians' favorite magic books." His first magic book for
Checkerboard - Misplaced Pages Continue
1536-403: The back to the very front of the magazine. In 1979, Gardner left Scientific American . He and his wife Charlotte moved to Hendersonville, North Carolina . He continued to write math articles, sending them to The Mathematical Intelligencer , Math Horizons , The College Mathematics Journal , and Scientific American . He also revised some of his older books such as Origami, Eleusis, and
1584-569: The entire French literary group known as the Oulipo . Salvador Dalí once sought him out to discuss four-dimensional hypercubes . David Auerbach wrote: "A case can be made, in purely practical terms, for Martin Gardner as one of the most influential writers of the 20th century. His popularizations of science and mathematical games in Scientific American, over the 25 years he wrote for them, might have helped create more young mathematicians and computer scientists than any other single factor prior to
1632-460: The first chapter of his "best of" collection, The Colossal Book of Mathematics . In the 1980s "Mathematical Games" began to appear only irregularly. Other authors began to share the column, and the June 1986 issue saw the final installment under that title. In 1981, on Gardner's retirement from Scientific American , the column was replaced by Douglas Hofstadter 's " Metamagical Themas ", a name that
1680-417: The four Princeton University professors who had invented and investigated their mathematical properties. The subsequent article Gardner wrote on hexaflexagons led directly to the column. Gardner's son Jim once asked him what was his favorite puzzle, and Gardner answered almost immediately: " The monkey and the coconuts ". It had been the subject of his April 1958 Games column and in 2001 he chose to make it
1728-639: The general public, Mathematics, Magic and Mystery (Dover, 1956), is still considered a classic in the field. He was well known for his innovative tapping and spelling effects, with and without playing cards , and was most proud of the effect he called the "Wink Change". Many of Gardner's lifelong friends were magicians. These included William Simon who introduced Gardner to Charlotte Greenwald, whom he married in 1952, Dai Vernon , Jerry Andrus , statistician Persi Diaconis , and polymath Raymond Smullyan . Gardner considered fellow magician James Randi his closest friend. Diaconis and Smullyan like Gardner straddled
1776-540: The lost continents of Atlantis and Lemuria , Immanuel Velikovsky's Worlds in Collision , the reincarnation of Bridey Murphy , Wilhelm Reich's orgone theory , the spontaneous generation of life , extra-sensory perception and psychokinesis , homeopathy , phrenology , palmistry , graphology , and numerology . This book and his subsequent efforts ( Science: Good, Bad and Bogus , 1981; Order and Surprise , 1983, Gardner's Whys & Wherefores , 1989, etc.) provoked
1824-538: The material I got from them, so I owe them a big debt of gratitude." Gardner prepared each of his columns in a painstaking and scholarly fashion and conducted copious correspondence to be sure that everything was fact-checked for mathematical accuracy. Communication was often by postcard or telephone and Gardner kept meticulous notes of everything, typically on index cards. Archives of some of his correspondence stored at Stanford University occupy some 63 linear feet of shelf space. This correspondence led to columns about
1872-586: The sense of the one-time pad. but in practice they are unbreakable in a much stronger sense than any cipher previously designed for widespread use. In principle these new ciphers can be broken. but only by computer programs that run for millions of years! –Martin Gardner In his August 1977 column, "A new kind of cipher that would take millions of years to break", Gardner described a new cryptographic system invented by Ron Rivest , Adi Shamir and Leonard Adleman . The system, based on trapdoor functions ,
1920-463: The two worlds of mathematics and magic. Mathematics and magic were frequently intertwined in Gardner's work. One of his earliest books, Mathematics, Magic and Mystery (1956), was about mathematically based magic tricks. Mathematical magic tricks were often featured in his "Mathematical Games" column–for example, his August 1962 column was titled "A variety of diverting tricks collected at a fictitious convention of magicians." From 1998 to 2002 he wrote
1968-409: The writings of Lewis Carroll , L. Frank Baum , and G. K. Chesterton . He was a leading authority on Lewis Carroll; The Annotated Alice , which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies. He had a lifelong interest in magic and illusion and in 1999, MAGIC magazine named him as one of the "100 Most Influential Magicians of
Checkerboard - Misplaced Pages Continue
2016-509: Was a critic of fringe science . His book Fads and Fallacies in the Name of Science (1952, revised 1957) launched the modern skeptical movement. It debunked dubious movements and theories including Fletcherism , Lamarckism , food faddism , Dowsing Rods , Charles Fort , Rudolf Steiner , Dianetics , the Bates method for improving eyesight , Einstein deniers , the Flat Earth theory ,
2064-543: Was a magic trick in the May 2010 issue of Word Ways: The Journal of Recreational Linguistics . I am a philosophical theist. I believe in a personal God, and I believe in an afterlife, and I believe in prayer, but I don't believe in any established religion. This is called philosophical theism. ... Philosophical theism is entirely emotional. As Kant said, he destroyed pure reason to make room for faith. – Martin Gardner, 2008 Popular mathematics Popular mathematics
2112-484: Was also responsible for introducing Doris Schattschneider and Marjorie Rice , who worked together to document the newly discovered pentagon tilings. Gardner credited his network with generating further material for his columns: "When I first started the column, I was not in touch with any mathematicians, and gradually mathematicians who were creative in the field found out about the column and began corresponding with me. So my most interesting columns were columns based on
2160-771: Was doing because I had to understand what I was writing about, and that enabled me to write in such a way that an average reader could understand what I was saying. If you are writing popularly about math, I think it's good not to know too much math." John Horton Conway called him "the most learned man I have ever met." He had carried on incredibly interesting exchanges with hundreds of mathematicians, as well as with artists and polymaths such as Maurits Escher and Piet Hein. – AMS Notices Gardner maintained an extensive network of experts and amateurs with whom he regularly exchanged information and ideas. Doris Schattschneider would later term this circle of collaborators "Gardner's mathematical grapevine" or "MG . Gardner's role as
2208-434: Was known as RSA (after the three researchers) and has become a component of the majority of secure data transmission schemes. Since RSA is a relatively slow algorithm it is not widely used to directly encrypt data. More often, it is used to transmit shared keys for symmetric-key cryptography . Gardner identified the memorandum that his column was based on and invited readers to write to Rivest to request
2256-495: Was one of the foremost anti- pseudoscience polemicists of the 20th century. His 1957 book Fads and Fallacies in the Name of Science is a seminal work of the skeptical movement. In 1976, he joined with fellow skeptics to found CSICOP , an organization promoting scientific inquiry and the use of reason in examining extraordinary claims. Martin Gardner was born into a prosperous family in Tulsa, Oklahoma , to James Henry Gardner,
2304-411: Was the first introduction of many subjects to a wider audience, notably: Gardner had problems learning calculus and never took a mathematics course after high school. While editing Humpty Dumpty Magazine he constructed many paper folding puzzles. At a magic show in 1956 fellow magician Royal Vale Heath introduced Gardner to the intricately folded paper shapes known as flexagons and steered him to
#970029