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# test 1 review: The Puzzle Instinct

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University of Toronto St. George

Victoria College Courses

VIC101H1

Marcel Danesi

Winter

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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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