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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**All aboard…**… for Reasoning and Sense Making, with a smile. SCCTM**Martin Gardner**• Dedicated to Martin Gardner whose birthday was yesterday (October 21, 1914) and who passed away on May 22, 2010. • G4G Celebrations Worldwide SCCTM**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) SCCTM**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.” SCCTM**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) SCCTM**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. SCCTM**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. SCCTM**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 SCCTM**1. Magic Squares**SCCTM**1. Magic Squares**Age of Jesus at the time of the Passion? SCCTM**1. Magic Squares**• Applets for generating Magic Squares • http://www.allmath.com/magicsquare.php SCCTM**2. Mobius Strip and Klein Bottle**• Mobius Strip • August Ferdinand Möbius • (1790 –1868) SCCTM**2. Mobius Strip and Klein Bottle**• Recycling • Some properties of the Mobius Strip SCCTM**2. Mobius Strip and Klein Bottle**• Klein Bottle • Felix Christian Klein (1849 –1925) SCCTM**2. Mobius Strip and Klein Bottle**• A better view of the Klein Bottle • Buy one at the Acme Klein Bottle Company SCCTM**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. SCCTM**3. Monty’s Dilemma**• Marilyn vos Savant “Ask Marilyn” in Parade magazine 1990. • “World’s highest IQ” 228 • Mrs. Robert Jarvik SCCTM**3. Monty’s Dilemma**• As posed on the CBS Show NUMB3RS SCCTM**3. Monty’s Dilemma**• NCTM Illuminations Site Lesson • http://illuminations.nctm.org/LessonDetail.aspx?id=L377 SCCTM**3. Monty’s Dilemma**• Facebook? SCCTM**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 SCCTM**4. Buffon’s Needle Problem**• George-Louis Leclerc, Comte de Buffon (1707 – 1788) SCCTM**4. Buffon’s Needle Problem**• Java Applet Simulation • http://mste.illinois.edu/reese/buffon/bufjava.html • Video from Wolfram SCCTM**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? SCCTM**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. SCCTM**5. Curry’s or Hooper’s Paradox**• Applet to simulate • 13 x 5 • 8 x 3 • 5 x 2 SCCTM**5. Curry’s or Hooper’s Paradox**• Illusion! • Of a Linear Hypotenuse in the 2nd Triangle SCCTM**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. SCCTM**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. SCCTM**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 SCCTM**6. The Birthday Problem**• Get the pattern for n people? • And P(A) is • How about a picture: SCCTM**6. The Birthday Problem**• A Table? SCCTM**6. The Birthday Problem**SCCTM**6. The Birthday Problem**• NCTM Illuminations Birthday Paradox • http://illuminations.nctm.org/LessonDetail.aspx?id=L299 SCCTM**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) SCCTM**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% SCCTM**6. The Birthday Problem**• How about with this group? • Here are 15 birthdays of people mentioned this presentation. • Can we get a match? OPEN SCCTM**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. SCCTM**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 SCCTM**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 SCCTM**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. SCCTM**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.” SCCTM**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 SCCTM**8. Symmetry: M.C. Escher & Scott Kim**• Symmetry SCCTM**8. Symmetry: M.C. Escher & Scott Kim**• Infinity SCCTM**8. Symmetry: M.C. Escher & Scott Kim**• Tessellations SCCTM