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A Mathematical Mystery Tour: 10 Mathematical Wonders and Oddities. Ed Dickey. All aboard… … for Reasoning and Sense Making, with a smile. Martin Gardner. Dedicated to Martin Gardner whose birthday was yesterday (October 21, 1914) and who passed away on May 22, 2010.

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a mathematical mystery tour 10 mathematical wonders and oddities

A Mathematical Mystery Tour: 10 Mathematical Wonders and Oddities

Ed Dickey

College of Education

Instruction &Teacher Education

slide2
All aboard…

… for Reasoning and Sense Making, with a smile.

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martin gardner
Martin Gardner
  • Dedicated to Martin Gardner whose birthday was yesterday (October 21, 1914) and who passed away on May 22, 2010.
  • G4G Celebrations Worldwide

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10 wonders and oddities
10 Wonders and Oddities
  • Magic Squares (MG inspired)
  • Mobius Strip & Klein Bottle
  • Monty’s Dilemma
  • Buffon’s Needle Problem
  • Curry’s Paradox (MG inspired)
  • The Birthday Problem
  • Kissing Numbers & Packing Spheres
  • Symmetry: Escher & Scott Kim (MG inspired)
  • Tower of Hanoi
  • Palindromes (MG inspired)

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1 magic squares
1. Magic Squares
  • What is it?
  • “set of integers in serial order, beginning with 1, arranged in square formation so that the total of each row, column, and main diagonal are the same.”

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1 magic squares6
1. Magic Squares
  • The “order” of a magic square is the number of cells on one its sides
  • Order 2? (none)
  • Order 3? (one, counting symmetry only once)
  • Order 4? (880)

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1 magic squares7
1. Magic Squares
  • In 1514, Albrecht Dürer created an engraving named Melancholia that included a magic square.
  • In the bottom row of his 4 X 4 magic square you can see that he placed the numbers "15" and "14" side by side to reveal the date of his engraving.

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1 magic squares9
1. Magic Squares
  • Diabolical Magic Square
  • “… a magic square that remains magic if a row is shifted from top to bottom or bottom to top, and if a column is moved from one side to the other.

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1 magic squares10
1. Magic Squares
  • Temple Expiatori de la Sagrada Familia created by Antoni Gaudi (1852-1926) in Barcelona, Spain
  • Open to public but expected to be complete in 2026

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1 magic squares12
1. Magic Squares

Age of Jesus at the time of the Passion?

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1 magic squares13
1. Magic Squares
  • Applets for generating Magic Squares
  • http://www.allmath.com/magicsquare.php

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2 mobius strip and klein bottle
2. Mobius Strip and Klein Bottle
  • Mobius Strip
  • August Ferdinand Möbius
  • (1790 –1868)

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2 mobius strip and klein bottle15
2. Mobius Strip and Klein Bottle
  • Recycling
  • Some properties of the Mobius Strip

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2 mobius strip and klein bottle16
2. Mobius Strip and Klein Bottle
  • Klein Bottle
  • Felix Christian Klein (1849 –1925)

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2 mobius strip and klein bottle17
2. Mobius Strip and Klein Bottle
  • A better view of the Klein Bottle
  • Buy one at the Acme Klein Bottle Company

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3 monty s dilemma
3. Monty’s Dilemma
  • In search of a new car, the player picks a door, say 1.
  • The game host then opens one of the other doors, say 3, to reveal a goat and offers to let the player pick door 2 instead of door 1.

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3 monty s dilemma19
3. Monty’s Dilemma
  • Marilyn vos Savant “Ask Marilyn” in Parade magazine 1990.
  • “World’s highest IQ” 228
  • Mrs. Robert Jarvik

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3 monty s dilemma20
3. Monty’s Dilemma
  • As posed on the CBS Show NUMB3RS

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3 monty s dilemma21
3. Monty’s Dilemma
  • NCTM Illuminations Site Lesson
  • http://illuminations.nctm.org/LessonDetail.aspx?id=L377

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3 monty s dilemma22
3. Monty’s Dilemma
  • Facebook?

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4 buffon s needle problem
4. Buffon’s Needle Problem
  • Drop a need on a lined sheet of paper
  • What is the probability of the needle crossing one of the lines?
  • Probability related to p
  • Simulation of the probability lets you approximate p

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4 buffon s needle problem24
4. Buffon’s Needle Problem
  • George-Louis Leclerc, Comte de Buffon (1707 – 1788)

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4 buffon s needle problem26
4. Buffon’s Needle Problem
  • Java Applet Simulation
  • http://mste.illinois.edu/reese/buffon/bufjava.html
  • Video from Wolfram

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5 curry s or hooper s paradox
5. Curry’s or Hooper’s Paradox
  • In one case as two triangles, but with a 5×3 rectangle of area 15.
  • In the other case, same two triangles, but with an 8×2 rectangle of area 16.
  • How?

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5 curry s or hooper s paradox28
5. Curry’s or Hooper’s Paradox
  • A right triangle with legs 13 and 5 can be cut into two triangles (legs 8, 3 and 5, 2, respectively).
  • The small triangles could be fitted into the angles of the given triangle in two different ways.

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5 curry s or hooper s paradox29
5. Curry’s or Hooper’s Paradox
  • Applet to simulate
  • 13 x 5
  • 8 x 3
  • 5 x 2

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5 curry s or hooper s paradox30
5. Curry’s or Hooper’s Paradox
  • Illusion!
  • Of a Linear Hypotenuse in the 2nd Triangle

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6 the birthday problem
6. The Birthday Problem
  • What is the probability that in a group of people, some pair have the SAME BIRTHDAY?
  • If there are 367 people (or more), the probability is 100%
  • COUNTERINTUITIVE!
  • With a group of 57 people the probability is 99%
  • It’s “50-50” with just 23 people.

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6 the birthday problem32
6. The Birthday Problem
  • Let P(A) be the probability of at least two people in a group having the same birthday and A’, the complement of A.
  • P(A) = 1 – P(A’)
  • What is P(A’)?
  • Probability of NO two people in a group having the same birthday.

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6 the birthday problem33
6. The Birthday Problem
  • In a group of 2, 3 more more, what it probability that the birthdays will be different?
  • (Let’s ignore Feb 29 for now.)
  • Person #2 has 364 possible birthdays so
  • The probability is 365/365 x 364/365
  • Person #3 has 363 possible birthdays, so as not to match person #1 and #2
  • The probability is 365/365 x 364/365 x 363/365

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6 the birthday problem34
6. The Birthday Problem
  • Get the pattern for n people?
  • And P(A) is
  • How about a picture:

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6 the birthday problem37
6. The Birthday Problem
  • NCTM Illuminations Birthday Paradox
  • http://illuminations.nctm.org/LessonDetail.aspx?id=L299

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6 the birthday problem38
6. The Birthday Problem
  • Random: people and equally distributed birthdays
  • 2 US Presidents have the name birthday
  • Polk (11th) and Harding (29th) November 2
  • 67 actresses won a Best Actress Oscar
  • Only 3 pairs share the same birthdays
  • Jane Wyman and Diane Keaton (January 5), Joanne Woodward and Elizabeth Taylor (February 27) and Barbra Streisand and Shirley MacLaine (April 24)

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6 the birthday problem39
6. The Birthday Problem
  • Birthdays are NOT evenly distributed.
  • In Northern Hemisphere summer sees more births.
  • In the US, more children conceived around the holidays of Christmas and New Years.
  • In Sweden 9.3% of the population is born in March and 7.3% in November when a uniform distribution would give 8.3%

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6 the birthday problem40
6. The Birthday Problem
  • How about with this group?
  • Here are 15 birthdays of people mentioned this presentation.
  • Can we get a match? OPEN

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7 kissing numbers and packing spheres
7. Kissing Numbers and Packing Spheres
  • What is the largest number identical spheres that can be packed into a fixed space?
  • In two-dimensions, the sphere packing problem involves packing circles. This problem can be modeled with coins or plastic disks and is solvable by high school students.

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7 kissing numbers and packing spheres42
7. Kissing Numbers and Packing Spheres
  • In 1694, Isaac Newton and David Gregory argued about the 3D kissing number.
  • 12 or 13?
  • Proof that 12 is the maximum (“all physicists know and most mathematicians believe…”) was not accepted until 1953

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7 kissing numbers and packing spheres43
7. Kissing Numbers and Packing Spheres
  • Kissing Number problem from Martin Gardner
  • Rearrange the triangle of six coins into a hexagon,
  • By moving one coin at a time, so that each coin moved is always touching at least two others

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7 kissing numbers and packing spheres44
7. Kissing Numbers and Packing Spheres
  • Problems of 4, 5, and n-dimension sphere packing have application in radio transmissions (cell phone signals) across different frequency spectrum.
  • Kenneth Stephenson tells the Mathematical Tale in the Notices of the AMS.

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7 kissing numbers and packing spheres46
7. Kissing Numbers and Packing Spheres
  • “It is an article of mathematical faith that every topic will find connections to the wider world—eventually.
  • For some, that isn’t enough. For some it is real-time exchange between the mathematics and the applications that is the measure of a topic.”

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8 symmetry m c escher scott kim
8. Symmetry: M.C. Escher & Scott Kim
  • M.C. Escher (1898-1972)
  • Produced mathematically inspired woodcuts and lithographs
  • Many including concepts of symmetry, infinity, and tessellations

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8 symmetry m c escher scott kim51
8. Symmetry: M.C. Escher & Scott Kim
  • Scott Kim
  • Puzzlemaster

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8 symmetry m c escher scott kim52
8. Symmetry: M.C. Escher & Scott Kim
  • Ambigram:

a word or words that can be read in more than one way or from more than a single vantage point, such as both right side up and upside down.

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8 symmetry m c escher scott kim53
8. Symmetry: M.C. Escher & Scott Kim
  • Game Related to Math
  • Figure Ground
  • Rush Hour
  • Shoe Repair (for Isabel)
  • Kokontsu-Super Sudoku

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8 symmetry m c escher scott kim54
8. Symmetry: M.C. Escher & Scott Kim
  • Smart Games for Social Media
  • Scott Kim and wife Amy
  • “Not so smart games
  • “Shoot ‘em up,” “Bejeweled, or Farmville
  • Games that are good for you
  • To “stave off Alzheimer”
  • As part of social media like Facebook
  • Healthy + Stylish = On Trend

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8 symmetry m c escher scott kim55
8. Symmetry: M.C. Escher & Scott Kim
  • Shuffle Brain:
  • Smart games for a connected world at http://www.shufflebrain.com/

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9 tower of hanoi
9. Tower of Hanoi
  • There is a legend about a Vietnamese temple which contains a large room with three time-worn posts in it surrounded by 64 golden disks.
  • The priests of Hanoi, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the rules of the puzzle, since that time.
  • The puzzle is therefore also known as the Tower of Brahma puzzle.
  • According to the legend, when the last move of the puzzle is completed, the world will end.

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9 tower of hanoi57
9. Tower of Hanoi

Tower of Hanoi Applet

http://www.mazeworks.com/hanoi/index.htm

VIDEO Solution

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9 tower of hanoi58
9. Tower of Hanoi
  • If the legend were true, and if the priests were able to move disks at a rate of one per second, using the smallest number of moves, it would take them 264−1 seconds or roughly 585 billion years; it would take 18,446,744,073,709,551,615 turns to finish.

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10 palindromes
10. Palindromes
  • At least since 79 AD
  • Sator Arepo Tenet Opera Rotas (Sower Arepo sows the seeds)
  • Biological Structures (genomes)

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10 palindromes60
10. Palindromes
  • Able was I ere I saw Elba
  • A man, a plan, a canal, Panama
  • Madam I’m Adam
  • Girl, bathing on Bikini, eyeing boy, sees boy eyeing bikini on bathing girl
  • January 2, 2010 (01/02/2010),
  • and the next one will be on November 2, 2011 (11/02/2011)

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10 palindromes61
10. Palindromes
  • Rotation
  • Reflection
  • Translation
  • Scaling
  • Dissection
  • Regrouping
  • STRIPE – RIPEST
  • STRESSED – DESSERT
  • Filer a l’anglaise – To take French leave
  • LASER – Light Amplification by Stimulated Emission of Radiation
  • ESCHER – CHEERS
  • now here – no where

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10 palindromes62
10. Palindromes
  • Number Palindromes
  • 11 x 11 =
  • 121
  • 111 x 111 =
  • 12321
  • 1111111 x1111111 =
  • 1234567654321

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10 palindromes63
10. Palindromes
  • 11111 x 11 =
  • 122221
  • 11111 x 111 =
  • 1233321
  • 222 x 111 =
  • 24642
  • 333 x 111 =
  • 36963
  • 444 x 111=
  • 49284

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10 palindromes64
10. Palindromes
  • Lost Generation

Palindrome Paragraph

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references print
References – Print
  • Gardner, Martin. Mathematical Puzzles & Diversions. Simon & Schuster, 1961.
  • Stephenson, Kenneth (2003). Circle Packing: A Mathematical Tale. Notices of the American Mathematical Society, 50, 11, pp . 1376-1388.
  • Kim, Scott. Inversions. Key Curriculum Press, 1996.

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references web
References – Web
  • Gathering for Gardner http://www.g4g-com.org/
  • Magic Squares. Suzanne Alejandre http://mathforum.org/alejandre/magic.square.html
  • Magic Square Applet http://www.allmath.com/magicsquare.php
  • Acme Klein Bottle Company http://www.kleinbottle.com/
  • Monty’s Dilemma Applet http://mste.illinois.edu/reese/monty/MontyGame5.html
  • Stick or Switch Lesson http://illuminations.nctm.org/LessonDetail.aspx?id=L377
  • Buffon’s Needle Problem http://mste.illinois.edu/reese/buffon/buffon.html
  • Curry’s Paradox http://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml
  • Birthday Problem http://en.wikipedia.org/wiki/Birthday_problem

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references web67
References – Web
  • Illuminations: Birthday Paradox http://illuminations.nctm.org/LessonDetail.aspx?id=L299
  • Newton and the Kissing Numberhttp://plus.maths.org/issue23/features/kissing/index.html
  • Kissing Numberhttp://mathworld.wolfram.com/KissingNumber.html
  • Sphere Packinghttp://mathworld.wolfram.com/SpherePacking.html
  • The Official M.C. Escher Websitehttp://www.mcescher.com/Gallery/gallery.htm
  • Scott Kim-Puzzles, Ambigrams, Brain Games, Math Educationhttp://www.scottkim.com/
  • Meet the Artist: Scott Kim. Ambigram Magazine (2009) http://www.ambigram.com/scott-kim
  • Shuffle Brain | Smart games for a Connected Worldhttp://www.shufflebrain.com/
  • Tower of Hanoihttp://www.mazeworks.com/hanoi/index.htm

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web puzzles and games
Web Puzzles and Games
  • Figure Ground Game http://clockworkgoldfish.com/figureground/play.php
  • Rush Hour Game http://www.puzzles.com/products/rushhour.htm
  • Shoe Repair http://www.puzzles.com/Projects/LogicProblems/ShoeRepair.htm
  • Kokonotsu Super Sodoku http://www.kokonotsu.info/9/kokoplay3.htm

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references video
References – Video
  • Mobius Strip Video http://www.youtube.com/watch?v=4bcm-kPIuHE
  • Klein Bottle Video http://www.youtube.com/watch?v=E8rifKlq5hc
  • Monty on NUMB3RS http://www.youtube.com/watch?v=Aw3r1QMu82M
  • Buffon Simulation Video http://www.youtube.com/watch?v=l2VOPdQHi-s
  • Scott Kim takes apart the art of puzzles http://www.ted.com/talks/scott_kim_takes_apart_the_art_of_puzzles.html
  • Rush Hour Game http://www.youtube.com/watch?v=4gcW6JIL10o
  • Lost Generation Palindrome http://www.youtube.com/watch_popup?v=42E2fAWM6rA

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