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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 l.jpg

A Mathematical Mystery Tour: 10 Mathematical Wonders and Oddities

Ed Dickey

College of Education

Instruction &Teacher Education


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All aboard…

… for Reasoning and Sense Making, with a smile.

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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

  • 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

  • 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 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 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 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 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 Squares

Age of Jesus at the time of the Passion?

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1. Magic Squares

  • Applets for generating Magic Squares

  • http://www.allmath.com/magicsquare.php

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2. Mobius Strip and Klein Bottle

  • Mobius Strip

  • August Ferdinand Möbius

  • (1790 –1868)

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2. Mobius Strip and Klein Bottle

  • Recycling

  • Some properties of the Mobius Strip

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2. Mobius Strip and Klein Bottle

  • Klein Bottle

  • Felix Christian Klein (1849 –1925)

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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

  • 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 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 Dilemma

  • As posed on the CBS Show NUMB3RS

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3. Monty’s Dilemma

  • NCTM Illuminations Site Lesson

  • http://illuminations.nctm.org/LessonDetail.aspx?id=L377

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3. Monty’s Dilemma

  • Facebook?

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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 Problem

  • George-Louis Leclerc, Comte de Buffon (1707 – 1788)

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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

  • 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 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 Paradox

  • Applet to simulate

  • 13 x 5

  • 8 x 3

  • 5 x 2

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5. Curry’s or Hooper’s Paradox

  • Illusion!

  • Of a Linear Hypotenuse in the 2nd Triangle

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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 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 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 Problem

  • Get the pattern for n people?

  • And P(A) is

  • How about a picture:

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6. The Birthday Problem

  • A Table?

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6. The Birthday Problem

  • NCTM Illuminations Birthday Paradox

  • http://illuminations.nctm.org/LessonDetail.aspx?id=L299

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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 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 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

  • 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 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 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 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 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

  • 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 Kim

  • Scott Kim

  • Puzzlemaster

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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 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 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 Kim

  • Shuffle Brain:

  • Smart games for a connected world at http://www.shufflebrain.com/

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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 Hanoi

Tower of Hanoi Applet

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

VIDEO Solution

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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

  • At least since 79 AD

  • Sator Arepo Tenet Opera Rotas (Sower Arepo sows the seeds)

  • Biological Structures (genomes)

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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. 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. Palindromes

  • Number Palindromes

  • 11 x 11 =

  • 121

  • 111 x 111 =

  • 12321

  • 1111111 x1111111 =

  • 1234567654321

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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. Palindromes

  • Lost Generation

    Palindrome Paragraph

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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

  • 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 – 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

  • 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

  • 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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