Math Paradoxes

My Favorite Mathematical ParadoxesDan Kennedy Baylor School Chattanooga, TN

Mathematics and Mirrors: The Mirage

The reflective property of a parabola:

The Mirage Illusion Explained.

The Marvelous Mbius Strip

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

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?The Band Around the Earth Paradox

The Band Around the Earth (not to scale):

A little geometryRrx

The best part about this paradox is that you have to trust the mathematics. You cant perform the experiment!

Gabriels Horn

The area of this region is infinite. Heres a proof:

The volume of this solid is finite. Heres a proof:

So Gabriels Horn is a mathematical figure which has a finite volume (), 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. Gabriels Horn is just an infinite extension of this less paradoxical phenomenon.

The Tower of Hanoi PuzzleRules: 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 = 2^1 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. Sohow much time do we have left?

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

Simpsons Paradox

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

TeamGames WonLostPercentageKillz752.714Settz1073.700


Despite having a poorer winning percentage than the Killz before and after spring break, the Settz won the trophy!.368.400



TeamGames WonLostPercentageKillz19712Settz20812

Lets 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.

123Before 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!

123When you pick door 2, the chance that the car is behind one of the other doors is 2/3.Remember: Monty knows where the goats are.When he opens door 1 to show you a goat, he is shifting that 2/3 probability to door 3 alone!The door 2 probability is still 1/3, but the door 3 probability is now 2/3. Switch doors!

The Monty Hall Paradox got some recent notoriety when it appeared in Mark Haddons novel The Curious Incident of the Dog in the Night-time.However, it had been notorious well before that.

In 1990, Marilyn Vos Savant published the question (and her correct answer) in her Ask Marilyn column in Parade magazine. She later ran two more columns with letters from Ph. D. mathematicians (unwisely signed) calling her wrong. Since then, several journal articles have appeared with variations on the problem.

The Birthday ParadoxIf 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.

By the way, Marilyn Vos Savant also wrote about the Birthday Paradox:It is a well-established fact that in any randomly chosen group of 50 people, it is virtually certain that two will have birthdays on the same day. Since there are 365 days in a year, I find it almost impossible to understand why this is the case. Can you provide an explanation of this phenomenon? -- Robert Shearn, Loleta, Calif.

Here was Marilyns reply:This is a persistent, erroneous extrapolation of the fact that if 23 people are chosen at random, the probability is just a bit greater than 50/50 that at least two of them will share the same birthday.people are taking the correct number of 23, doubling it to about 50 and incorrectly reasoning thatthere must be a 100% chance that at least two out of 50 will! Thats just plain wrong.

In fact, for 50 people the probability of a birthday match is 97%!This is not 100%, but it certainly conforms to the letter-writers claim of virtually certain. It is certainly not, as Marilyn said, just plain wrong.OOPS.

Last 40 Oscar-winning Best Actress Birthdays

CherMay 20Marlee MatlinAug 24Geraldine PageNov 22Sally FieldNov 6Shirley MacLaineApr 24Meryl StreepMay 27Katharine HepburnMay 12Sissy SpacekDec 25Jane FondaDec 21Diane KeatonJan 5Faye DunawayJan 14Louise FletcherJul 22Ellen BurstynDec 7Glenda JacksonMay 9Liza MinnelliMar 12Maggie SmithDec 28Barbra StreisandApr 24Elizabeth TaylorFeb 27Sophia LorenSep 20Anne BancroftSep 17

Sandra BullockJul 26Kate WinsletOct 5Marion CotillardSep 30Helen MirrenJul 26Reese WitherspoonMar 22Hilary SwankJul 30Charlize TheronAug 7Nicole KidmanJun 20Halle BerryAug 14Julia RobertsOct 28Gwyneth PaltrowSep 27Helen HuntJun 15Frances McDormandJun 23Susan SarandonOct 4Jessica LangeApr 20Holly HunterMar 20Emma ThompsonApr 15Jodie FosterNov 19Kathy BatesJun 28Jessica TandyJun 7

Last 40 Oscar-winning Best Actor Birthdays

Jeff BridgesDec 4Daniel Day-LewisApr 29Forest WhitakerJul 15Philip Seymour HoffmanJul 23Jamie FoxxDec 13Sean PennAug 17Adrien BrodyApr 14Denzel WashingtonDec 28Russell CroweApr 7Kevin SpaceyJul 26Roberto BenigniOct 27Jack NicholsonApr 22Geoffrey RushJul 6Nicolas CageJan 7Tom HanksJul 9Al PacinoApr 25Anthony HopkinsDec 31Jeremy IronsSep 19Dustin HoffmanAug 8Michael DouglasSep 25

Paul NewmanJan 26William HurtApr 20F. Murray AbrahamOct 24Robert DuvallJan 5Ben KingsleyDec 31Henry FondaMay 16Robert De NiroAug 17Jon VoightDec 29Richard DreyfussOct 29Peter FinchSep 28Art CarneyNov 4Jack LemmonFeb 8Marlon BrandoApr 3Gene HackmanJan 30George C. ScottOct 18John WayneMay 26Cliff RobertsonSep 9Rod SteigerApr 14Paul ScofieldJan 21Lee MarvinFeb 19

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 2nd.

The Paradox of the Kruskal CountorThe 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.

7 7 6 43 1 6 4 3 32 5 3 5 2 44 1 7 2 3 37 7 6 43 1 6 4 3 43

Imagine the paradoxical implications

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Math Paradoxes

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