My Favorite Mathematical Paradoxes Dan Kennedy Baylor School Chattanooga, TN T^3 International Conference – Seattle, WA February 28, 2009

Mathematics and Mirrors: The Mirage ®

The reflective property of a parabola:

The Mirage Illusion Explained.

The Marvelous Möbius Strip

The Klein Bottle This region of apparent intersection is actually not there. This requires a fourth dimension for actual assembly!

The Band Around the Earth (not to scale):

Imagine a flexible steel band wrapped tightly around the equator of the Earth. Imagine that we have 10 feet left over. We cut the band, add the 10 feet, and then space the band evenly above the ground all around the Earth to pick up the extra slack. Could I crawl under the band?

A little geometry… R r x

R r x

Gabriel’s Horn

The area of this region is infinite. Here’s a proof:

The volume of this solid is finite. Here’s a proof:

So Gabriel’s Horn is a mathematical figure which has a finite volume (π), but which casts an infinite shadow!

If you find that this paradox challenges your faith in mathematics, remember that a cube with sides of length 0.01 casts a shadow that is 100 times as big as its volume. Gabriel’s Horn is just an infinite extension of this less paradoxical phenomenon.

The Tower of Hanoi Puzzle Rules: Entire tower of washers must be moved to the other outside peg. Only one washer may be moved at a time. A larger washer can never be placed on top of a smaller washer.

The minimum number of moves required to move a tower of n washers is 2^n – 1. The proof is a classic example of mathematical induction. Clearly, 1 washer requires 1 = 2^n – 1 move. Assume that a tower of k washers requires a minimum of 2^k – 1 moves. Then what about a tower of k + 1 washers?

First, you must uncover the bottom washer. By hypothesis, this requires 2^k – 1 moves. Then you must move the bottom washer. Finally, you must move the tower of k washers back on top of the bottom washer. By hypothesis, this requires 2^k – 1 moves. Altogether, it requires 2*(2^k – 1) + 1 = 2^(k +1) – 1 moves to move k + 1 washers. We are done by mathematical induction!

The typical Tower of Hanoi games comes with a tower of 7 washers. At one move per second, this can be solved in a minimum time of 2^7 – 1 = 127 seconds (or about 2 minutes). Now comes the paradox. Legend has it that God put one of these puzzles with 64 golden washers in Hanoi at the beginning of time. Monks have been moving the washers ever since, at one move per second.

When the tower is finally moved, that will signal the End of the World. So…how much time do we have left?

The age of the universe is currently estimated at just under 14 billion years. So relax.

Simpson’s Paradox

Bali High has an intramural volleyball league. Going into spring break last year, two teams were well ahead of the rest: TeamGamesWonLostPercentage Killz Settz Both teams struggled after the break: TeamGamesWonLostPercentage Killz Settz

TeamGamesWonLostPercentage Killz Settz TeamGamesWonLostPercentage Killz Settz TeamGamesWonLostPercentage Settz Killz Despite having a poorer winning percentage than the Killz before and after spring break, the Settz won the trophy!

Let’s Make a Deal! Monty Hall offers you a choice of three closed doors. Behind one door is a brand new car. Behind the other two doors are goats. You choose door 2.

123 Before he opens door 2, just to taunt you, Monty opens door 1. Behind it is a goat. He then offers you a chance to switch from door 2 to door 3. What should you do? Switch doors !

The Birthday Paradox If there are 40 people in a room, would you bet that some pair of them share the same birthday? You should. The chance of a match is a hefty 89%!

The key to this wonderful paradox is that the probability of NO match gets small faster than you would expect: This product is already less than 90%, and only ten people are in the room.

Last 40 Oscar-winning Best Actress Birthdays Marlee MatlinAug 24 Geraldine PageNov 22 Sally FieldNov 6 Shirley MacLaineApr 24 Meryl StreepMay 27 Katharine HepburnMay 12 Sissy SpacekDec 25 Jane FondaDec 21 Diane KeatonJan 5 Faye DunawayJan 14 Louise FletcherJul 22 Ellen BurstynDec 7 Glenda JacksonMay 9 Liza MinnelliMar 12 Maggie SmithDec 28 Barbra StreisandApr 24 Elizabeth TaylorFeb 27 Sophia LorenSep 20 Anne BancroftSep 17 Patricia NealJan 20 Kate WinsletOct 5 Marion CotillardSep 30 Helen MirrenJul 26 Reese WitherspoonMar 22 Hilary SwankJul 30 Charlize TheronAug 7 Nicole KidmanJun 20 Halle BerryAug 14 Julia RobertsOct 28 Gwyneth PaltrowSep 27 Helen HuntJun 15 Frances McDormandJun 23 Susan SarandonOct 4 Jessica LangeApr 20 Holly HunterMar 20 Emma ThompsonApr 15 Jodie FosterNov 19 Kathy BatesJun 28 Jessica TandyJun 7 CherMay 20

Last 40 Oscar-winning Best Actor Birthdays Daniel Day-LewisApr 29 Forest WhitakerJul 15 Philip Seymour HoffmanJul 23 Jamie FoxxDec 13 Sean PennAug 17 Adrien BrodyApr 14 Denzel WashingtonDec 28 Russell CroweApr 7 Kevin SpaceyJul 26 Roberto BenigniOct 27 Jack NicholsonApr 22 Geoffrey RushJul 6 Nicolas CageJan 7 Tom HanksJul 9 Al PacinoApr 25 Anthony HopkinsDec 31 Jeremy IronsSep 19 Dustin HoffmanAug 8 Michael DouglasSep 25 Paul NewmanJan 26 William HurtApr 20 F. Murray AbrahamOct 24 Robert DuvallJan 5 Ben KingsleyDec 31 Henry FondaMay 16 Robert De NiroAug 17 Jon VoightDec 29 Richard DreyfussOct 29 Peter FinchSep 28 Art CarneyNov 4 Jack LemmonFeb 8 Marlon BrandoApr 3 Gene HackmanJan 30 George C. ScottOct 18 John WayneMay 26 Cliff RobertsonSep 9 Rod SteigerApr 14 Paul ScofieldJan 21 Lee MarvinFeb 19 Rex HarrisonMar 5

The 44 U.S. Presidents are surprisingly well spread-out. From Washington to Obama, there has only been one birthday match: James Polk (#11) and Warren Harding (#29) were both born on November 11 th.

The Paradox of the Kruskal Count or The Amazing Secret of Twinkle Twinkle Little Star

One of the neatest math articles I ever read was a piece by Martin Gardner in the September 1998 issue of Math Horizons. He called it “Ten Amazing Mathematical Tricks.”

Twinkle, Twinkle, little star; How I wonder what you are, Up above the world so high, Like a diamond in the sky; Twinkle, twinkle, little star; How I wonder what you are.