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A Mathematical Mystery Tour: 10 Mathematical Wonders and Oddities

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### A Mathematical Mystery Tour: 10 Mathematical Wonders and Oddities

Ed Dickey

College of Education

Instruction &Teacher Education

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

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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

- NCTM Illuminations Site Lesson
- http://illuminations.nctm.org/LessonDetail.aspx?id=L377

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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

- 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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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

- 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

- 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

- 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

- 11111 x 11 =
- 122221
- 11111 x 111 =
- 1233321
- 222 x 111 =
- 24642
- 333 x 111 =
- 36963
- 444 x 111=
- 49284

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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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