A Mathematical Mystery Tour: 10 Mathematical Wonders and Oddities

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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

Ed Dickey

College of Education

Instruction &Teacher Education

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
• 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
• “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

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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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• 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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• 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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• Applet to simulate
• 13 x 5
• 8 x 3
• 5 x 2

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• 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

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6. The Birthday Problem
• 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
• 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
• Birthday Problem http://en.wikipedia.org/wiki/Birthday_problem

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References – Web
• 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