College of Education

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A Mathematical Mystery Tour 10 Mathematical
Wonders and Oddities
Ed Dickey
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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

1. Magic Squares (MG inspired)
2. Mobius Strip Klein Bottle
3. Montys Dilemma
4. Buffons Needle Problem
5. Currys Paradox (MG inspired)
6. The Birthday Problem
7. Kissing Numbers Packing Spheres
8. Symmetry Escher Scott Kim (MG inspired)
9. Tower of Hanoi
10. 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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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. Montys 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. Montys Dilemma

• Marilyn vos Savant Ask Marilyn in Parade
  magazine 1990.
• Worlds highest IQ 228
• Mrs. Robert Jarvik

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3. Montys Dilemma

• As posed on the CBS Show NUMB3RS

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3. Montys Dilemma

• NCTM Illuminations Site Lesson
• http//illuminations.nctm.org/LessonDetail.aspx?id
  L377

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3. Montys Dilemma

• Facebook?

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4. Buffons 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. Buffons Needle Problem

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

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4. Buffons Needle Problem
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4. Buffons Needle Problem

• Java Applet Simulation
• http//mste.illinois.edu/reese/buffon/bufjava.html
  
• Video from Wolfram

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5. Currys or Hoopers Paradox

• In one case as two triangles, but with a 53
  rectangle of area 15.
• In the other case, same two triangles, but with
  an 82 rectangle of area 16.
• How?

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5. Currys or Hoopers 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. Currys or Hoopers Paradox

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

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5. Currys or Hoopers 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
• Its 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?
• (Lets 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
n probability that 2 of n have same birthday
10 11.70
20 41.10
23 50.70
30 70.60
50 97.00
57 99.00
100 99.999970
200 99.9999999999999999999999999998
300 (100 - (610-80))
350 (100 - (610-80))
360 100
note that it is possible to have 366 people in a room with different birthdays if one was born on February 29th on a leap year.
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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
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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
  worldeventually.
• For some, that isnt 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

• Symmetry

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8. Symmetry M.C. Escher Scott Kim

• Infinity

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8. Symmetry M.C. Escher Scott Kim

• 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/h
anoi/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 Im 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

• STRIPE RIPEST
• STRESSED DESSERT
• Filer a langlaise To take French leave
• LASER Light Amplification by Stimulated
  Emission of Radiation
• ESCHER CHEERS
• now here no where

• Rotation
• Reflection
• Translation
• Scaling
• Dissection
• Regrouping

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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/magicsq
  uare.php
• Acme Klein Bottle Company http//www.kleinbottle.c
  om/
• Montys Dilemma Applet http//mste.illinois.edu/re
  ese/monty/MontyGame5.html
• Stick or Switch Lesson http//illuminations.nctm.o
  rg/LessonDetail.aspx?idL377
• Buffons Needle Problem http//mste.illinois.edu/r
  eese/buffon/buffon.html
• Currys Paradox http//www.cut-the-knot.org/Curric
  ulum/Fallacies/CurryParadox.shtml
• Birthday Problem http//en.wikipedia.org/wiki/Birt
  hday_problem

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References Web

• Illuminations Birthday Paradox
  http//illuminations.nctm.org/LessonDetail.aspx?id
  L299
• Newton and the Kissing Number http//plus.maths.or
  g/issue23/features/kissing/index.html
• Kissing Number http//mathworld.wolfram.com/Kissi
  ngNumber.html
• Sphere Packing http//mathworld.wolfram.com/Sphere
  Packing.html
• The Official M.C. Escher Website
  http//www.mcescher.com/Gallery/gallery.htm
• Scott Kim-Puzzles, Ambigrams, Brain Games, Math
  Education http//www.scottkim.com/
• Meet the Artist Scott Kim. Ambigram Magazine
  (2009) http//www.ambigram.com/scott-kim
• Shuffle Brain Smart games for a Connected World
  http//www.shufflebrain.com/
• Tower of Hanoi http//www.mazeworks.com/hanoi/inde
  x.htm

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Web Puzzles and Games

• Figure Ground Game http//clockworkgoldfish.com/fi
  gureground/play.php
• Rush Hour Game http//www.puzzles.com/products/rus
  hhour.htm
• Shoe Repair http//www.puzzles.com/Projects/LogicP
  roblems/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?vAw
  3r1QMu82M
• Buffon Simulation Video http//www.youtube.com/wat
  ch?vl2VOPdQHi-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?v4gcW
  6JIL10o
• Lost Generation Palindrome http//www.youtube.com/
  watch_popup?v42E2fAWM6rA