A Mathematical Mystery Tour: 10 Mathematical Wonders and Oddities

1. A Mathematical Mystery Tour: 10 Mathematical Wonders and Oddities Ed Dickey College of Education Instruction &Teacher Education

2. All aboard… … for Reasoning and Sense Making, with a smile. SCCTM

3. 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

4. 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

5. 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

6. 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

7. 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

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9. 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

10. 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

11. 1. Magic Squares SCCTM

12. 1. Magic Squares Age of Jesus at the time of the Passion? SCCTM

13. 1. Magic Squares • Applets for generating Magic Squares • http://www.allmath.com/magicsquare.php SCCTM

14. 2. Mobius Strip and Klein Bottle • Mobius Strip • August Ferdinand Möbius • (1790 –1868) SCCTM

15. 2. Mobius Strip and Klein Bottle • Recycling • Some properties of the Mobius Strip SCCTM

16. 2. Mobius Strip and Klein Bottle • Klein Bottle • Felix Christian Klein (1849 –1925) SCCTM

17. 2. Mobius Strip and Klein Bottle • A better view of the Klein Bottle • Buy one at the Acme Klein Bottle Company SCCTM

18. 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

19. 3. Monty’s Dilemma • Marilyn vos Savant “Ask Marilyn” in Parade magazine 1990. • “World’s highest IQ” 228 • Mrs. Robert Jarvik SCCTM

20. 3. Monty’s Dilemma • As posed on the CBS Show NUMB3RS SCCTM

21. 3. Monty’s Dilemma • NCTM Illuminations Site Lesson • http://illuminations.nctm.org/LessonDetail.aspx?id=L377 SCCTM

22. 3. Monty’s Dilemma • Facebook? SCCTM

23. 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

24. 4. Buffon’s Needle Problem • George-Louis Leclerc, Comte de Buffon (1707 – 1788) SCCTM

25. 4. Buffon’s Needle Problem SCCTM

26. 4. Buffon’s Needle Problem • Java Applet Simulation • http://mste.illinois.edu/reese/buffon/bufjava.html • Video from Wolfram SCCTM

27. 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

28. 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

29. 5. Curry’s or Hooper’s Paradox • Applet to simulate • 13 x 5 • 8 x 3 • 5 x 2 SCCTM

30. 5. Curry’s or Hooper’s Paradox • Illusion! • Of a Linear Hypotenuse in the 2nd Triangle SCCTM

31. 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

32. 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

33. 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

34. 6. The Birthday Problem • Get the pattern for n people? • And P(A) is • How about a picture: SCCTM

35. 6. The Birthday Problem • A Table? SCCTM

36. 6. The Birthday Problem SCCTM

37. 6. The Birthday Problem • NCTM Illuminations Birthday Paradox • http://illuminations.nctm.org/LessonDetail.aspx?id=L299 SCCTM

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 7. Kissing Numbers and Packing Spheres SCCTM

46. 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

47. 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

48. 8. Symmetry: M.C. Escher & Scott Kim • Symmetry SCCTM

49. 8. Symmetry: M.C. Escher & Scott Kim • Infinity SCCTM

50. 8. Symmetry: M.C. Escher & Scott Kim • Tessellations SCCTM