test 1 review: The Puzzle Instinct

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University of Toronto St. George
Victoria College Courses
Marcel Danesi

11/1/2012 10:13:00 AM The Puzzle Instinct – VIC211 Why puzzles? ~ chapter 1  human beings require that things be mysterious  horror/ detective narrative genre “The Castle of Otranto” – Horace Walpole (1764) and “The Murders in the Rue Morgue” – Edgar Allan Poe (1841)  prior, medieval times re-enacted miracles performed by saints  ancient world, dramas of Aeschylus (525-456 B.C), Sophocles (c. 496-406 B.C), and Euripides (c. 480-406 B.C) – fateful actions of legendary hero and heroines  scholars believe drama genre may have developed from rituals performed by secret mystery cults of ancient Greece – potentially w/ Pythagoras and Plato  central purpose – questions about the mystery of existence  plato – no purpose, arise simply out of superstitious traditions which put obstacles in the path of true science  proclaimed that dialetic reasoning is the only useful method of gaining knowledge – Socratic practice of examining ideas logically by means of a sequence of questions and answers – ignoring that “supersititious traditions” were also dialetic reasoning  puzzles arrive around the same time as mystery cults  appeal to humans for the same reason  catharsis – used by Aristotle to describe the sense of emotional relief that results from watching a tragic drama on stage  mental catharisis – senses of relief from answer to puzzle or mystery  Lewis Carroll saw puzzles as structures born of the imagination – “Alice in the Wonderland” and “ What Alice Found There” w/ personified puzzles concept e.g. Humpty Dumpty, Tweedledee and Tweedledum, Cheshire cat Puzzle land Puzzle-Making Through the Ages  “The Book of Games” – one of the best selling books of the entire medieval period ( Mohr 1993:11) o commissioned by King Alfonso X of Castile and Leon o chess, checkers, various card and board games  puzzles are one of the most profitable commercial product o e.g. over 2 mil rubric cubes sold in earl 1980s  “puzzles fixation” and “puzzle depression” syndromes found by psychologists for fierce, irrational craving for puzzles  first “intelligence test” – riddle of Sphinx (Grimal 1963: 324) o spinx was a monster w/ head and breasts of a woman, body of a lion, wings of a bird, accosted all who dared enter city of Thebes w/ the riddle: what is it that has four feet in the morning, two at noon, and three at twilight” o Oedipus answered: “Man, who crawls on four limbs as a baby, wals upright on two as an adult, and gets around with the aid of a stick in old age” o Sphinx became a statue and Oedipus married Jocasta and became king  labyrinths – physical building of puzzles  path to enlightenment and true knowledge  e.g. funeral temple constructed by Amenemhet III in Egypt- 3000 chambers  prison on island of Crete (only in myth) – Androgeus, son of King Minos Crete was killed by Athenians, minos hired craftsman daudalus to build a dungeon to house Minotaur “a creature half human half bull at centre” and send 14 Athenians into maze every year. Athens were saved by Theseus, son of King Aegeus and his love Ariadne Mino‟s daughter o hero killed Minotaur and found way out w/ thread o similar to a place in Cretan city of Knossos  some scholar speculate that around 11000 years ago a tribe near st Lake Edward in modern-day Zaire, invented 1 math game, consisting of two dice – bones w/ notches  one of the earliest surviving manuscripts of human civilization is a collction of puzzles – “ahmes Papyrus” or “rhind papyrus” o 84 challenging math problems, tables for calculation of areas, conversion of fraction, elementary sequences, linear equations and extensive information about measurements  mystery, wisdom and puzzle-solving were intrinsically intertwined in the ancient world  many first books of wisdom turn out to be elaborate puzzle creations – “I Ching” – used for divination organized around 65 symbolic hexagrams are miniature puzzles explained by texts that can be characterized as “cryptic poems”  Rind Papyrus contain concepts that have surfaced in other areas of the world o E.g. papyrus‟s problem number 79 o House 7, cats 49, mice 343, sheaves if e=wheat 2,401, hekat of grain 16,807, estate 19,607 o 1 5 numbers are successive power of 7 o same question in the book of the Abacus - by medieval Italian mathematician Lenonardo Fibonacci o “ seven old women are on the road to Rome. Each woman has seven mules, each mule carries seven sacks, each sack contains seven loaves, to slice each loaf there are seven knives, and for each knife there are seven sheaths to hld it. How many are there altogether: women, mules, sacks, loaves, knives, sheaths? o Fibonacci could not have known about Ahmes‟ puzzle but the similarity is unmistakable, the same puzzle appeared as a popular nursery rhyme in eighteenth-century England \  Why are they so similar? – o Could be number 7 has a strange appeal to people, commonly imbued with mystical connotations  7 gods of good fortune in Japanese lore  7 chieftains in Greek mythology who undertook an ill- fated expedition against the city of Thebes  7 deadly sins  the ubiquity can perhaps be explain from the unconscious symbolic appeal that 7 seems universally to possess o Gillings believe “ the number 7 often presents itself in Egyptian multiplication because, by regular doubling, the first three multipliers are always 1, 2,4, which add to 7” o But Maor argue “it would equally apply to 3 (=1_2), to 15 (= 1+2+4+8) – “7 was chosen because a larger number would have made the calculation too long, while a smaller one would not have illustrated the rapid growth of the progression” Numerological symbolism  Numerological symbolism Not unusual in creation of puzzles  Also characterizes the emerging science of mathematics  E.g. C2=a2+b2, where c is the length of the hypotenuse and a and b the lengths of the other two sides Pythagoras  Believed divinities allowed him, a mere mortal, to catch a glimpse into the raison d’etre of one of the numerical laws governing the cosmos –good array in greek or “universe”  Supposedly offered a sacrifice for granting him “entrance into the knowledge of an obscure secret”  Demanded followers take an oath not to revewl proof  But same theory appeared approx. 600 years before Pythagoras in various other regions o E.g. in the Arithmetic Classic of the Gnomon and the Circular Paths of Heaven - chinese treaties around 1100 B.C Motives  Motives for puzzle makers and mathematicians were often less commendable and more mundane o E.g. Sicilian mathematician and scientist Archimedes invented arithmetical puzzles to befuddle his foes o Cattle Problem was a way to take revenge on one of his adversaries – to dumbfound w/ his computing dexterity Nonetheless, that puzzles has been the source to many mathematical insight scholars  Heron of Alexandria ( C. A.D. 20-c.62) o Known for mechanical inventions, also contrived puzzles as frameworks for investigating square and cube roots  Alexandrian mathematician Diophantus o Constructed puzzles to illustrate methods of solving algebraic equations o One type of puzzle known as Diophantine: an algebraic problem in which there are more unknowns than there are equations, thus the solution is theoretically infinite o When 100 byshels of grain are distributed aong 100 persons so that each man receives 3 bushels, each woman 2 bushels, and each child ½ bushel, how many men, women, and children are here?  Some of the most popular works of medieval era were puzzle collections, e.g. Greek Anthology o Contains many same puzzle concepts found in Rhind Papyrus o I desire my two sons to receive the thousand staters of which I am possessed, but let the fifth part of the legitimate one’s share exceed by ten the fourth part of what falls to the illegitimate one o Easily solved with algebra today, but not in medieval times Charlegmagne (A.D 742-814)  One of the first “puzzle addicts” of history  Founder of the holy Roman Empire  Hired English scholar Alcuin as fulltime job to create puzzles  Allowed Alcuin to establish an effective education program among the Franks, which has a lasting influence on the intellectual life of the western world  Put together 56 of the puzzles he invited as an instructional manual problem to Sharpen the Young Diophantus  Spread practice of using symbols for unknown number (variables)  Often called “the father of algebra”  Ideas were developed throughout the edieval ages primarily by Arab scholars, Al-Khwarizimi, teach in Baghdad, and Persian astronomer Omar Khayyam Fibonacci‟s Book of the Abacus  Published in 1202,  One of the most significant anthologies in the 13 thcentury  Fibonacci‟s real name is Leonardo Pisano  Designed book as a practical, user-friendly intro to hindu-Arabic number system  Solved the intellectual puzzle of the zero concept  0 symbol probably originated around 600 B.C. in India  Al-Khwarizmi introduced symbol 0, as “number empltiness”  Philosophers argued that there is no use for it if it stood of nothing  Fibonacci claimed that 0 was a digital place-holder for separating columns of figures  547 = five hundred plus forty plus seven  506, 0 is used to show that there is no tens  his puzzles ften concealed traps of twists, requires large dose of insight thinking o e.g. a snake is at the bottom of a 30-foot well. Each day it crawls up 3 feet and slips back 2 feet. At that rate, when will the snake be able to reach the top of the well? o 28 days, start on 27 foot, goes up 3 feet to the top and can crawl out before sliding down  this book help establish Hindu-Arabic numeral system in Europe fibonacci‟s Rabbit Puzzle  a certain man put a pair of rabbits, male and female, in a very large cage. How many pairs of rabbits can be produced in that cage in a year if every month each pair produces a new pair which, from the second month of its existence on, also is productive?  Sequence of 0-1, 1-1-, 2-2, 3-3, 4-5, 5-8, 6-13, 7-21, 8-34, 9-55, 10-89, 11-144, 12-233  The number is the sum of the previous two- Fibonacci sequence  Contain various mathemiatic patterns. E.g. if the nth number in the sequence s x, then every nth number after x turns out to be a multiple of x  Series surfaced in Nature: petals of daisies, wild roses in Fibonacci numbers, major chords in octave, third and fifth tones of scale  „reification” – serendipitous actual manifestations of something that was originally conceived as an abstraction or as a figment of mind  don‟t know reason why, but raised much interest, including a Fibonacci society in 1962  Fibonacci‟s book of the Abacus was used as a textbook throughout Europe Ibn Kallikan  Arab mathematician  Devised a famous puzzle to illustrate the nature of geometric sequences – each term as a power of 2  How many grains of wheat are needed on the last square of a 64- square chessboard if 1 grain is to be put on the first square of the board, 2 on the second, 4 on the third, 8 on the fourth, and so on in this fashion?  All sorts of geometric sequences manifest themselves as patterns in Nature o Trigonometric sequence in the structure of waves Fifteenth and sixttenth centuries  Puzzles were thought to possess occult powers and secret aesthetic qualtities  Magic squares were carried on amulets and talisman to ward off evil o Magic square – square array of number in which row, columns and major diagonals all have the same sum  Increasing employment of the puzzle format to illustrate mathematical concepts  By 17 thcentury, widely accepted as material for delectation and tool for illustrating and probing mathematical ideas Classification  1612, French Jesuit poet and scholar Claude-Gaspar Bachet de Mezirac  published amusing and Delightful Number Problems st  1 ever classificiations of puzzles – rive-crossing puzzles, weighting puzzles, number tricks, etc  system adopted grosso modo by puzzlists Euler‟s Konigsberg‟s Bridge puzzle  In the town of Konigsberg, is it possible to cross each of its seven bridges over the Pregel river, which connect two islands and the mainland, without crossing over any bridge twice?(land as verticles and lines and arcs as path/bridges  Even point is one at which an even number of path converge  It is not possible to traverse a network with more than two odd points in it without having to double back over some of its path  An odd number of path meet can only be a starting point  This led to the branch of mathematics called topology, -properties of networks 19 century  growing interest incombinatorial mathematics  Kirkman‟s school girl puzzle by Thomas Penyngton Kirkman- had implications for matrix theory o How can 15 girls walk in 5 rows of 3 each for 7 days so that no girl walks with any other girl in the same triplet more than once?  The Ladies‟ Diary or Woman‟s Almanac found in 1704 became the first puzzle magazine in history Lewis Carroll  Nom de plume of Charles Lutwidge Dodgson o Raised puzzle genre to high literary art o Created puzzle storybooks of history e.t. pillow problems an A Tnagled Tale o Best known for Alice‟s advantues in wonderland and throught the looking-glass  Contains puzzles of ingenious mind play and double entendre o Finding solutions to puzzles provide reassurance and as ense of order th 19 century  increasing popular in print Tower of Hanoi  Frenchman Francois
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