My Favorite Mathematical

ParadoxesDan 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:

focus

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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

Rr

x

2 2 10R r 2 ( ) 10R r

101.59 feet

2R r x

Gabriel’s Horn

y = x1–

1

2Area = lim lim 2ln

1

lim 2 ln ln1

k

k k

k

kdx x

x

k

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

21

1Volume = lim

1lim

1

1lim 1

k

k

k

k

dxx

k

x

k

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.

3Volume = 0.01 0.000001

2Area = 0.01 0.0001

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?

64 192 1 seconds = 1.84467 10 seconds

19 19

15

14

11

1 hr1.8447 10 sec = 1.8447 10 sec

3600 sec1 day

5.1241 10 hrs 24 hrs

1 yr2.135 10 days

365.25 days

5.8454 10 years

584.54 billion years!

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:

Team Games Won Lost Percentage

Killz 7 5 2 .714

Settz 10 7 3 .700

Both teams struggled after the break:

Team Games Won Lost Percentage

Killz 12 2 10 .160

Settz 10 1 9 .100

Team Games Won Lost Percentage

Killz 7 5 2 .714

Settz 10 7 3 .700

Team Games Won Lost Percentage

Killz 12 2 10 .160

Settz 10 1 9 .100

Team Games Won Lost Percentage

Settz 20 8 12 .400

Killz 19 7 12 .368

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.

1 2 3

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:

364

365

This product is already less than 90%, and only ten people are in the room.

363

365

362 361 360 359 358 357 356

365 365 365 365 365 365 365

Last 40 Oscar-winning Best Actress Birthdays

Marlee Matlin Aug 24

Geraldine Page Nov 22

Sally Field Nov 6

Shirley MacLaine Apr 24

Meryl Streep May 27

Katharine Hepburn May 12

Sissy Spacek Dec 25

Jane Fonda Dec 21

Diane Keaton Jan 5

Faye Dunaway Jan 14

Louise Fletcher Jul 22

Ellen Burstyn Dec 7

Glenda Jackson May 9

Liza Minnelli Mar 12

Maggie Smith Dec 28

Barbra Streisand Apr 24

Elizabeth Taylor Feb 27

Sophia Loren Sep 20

Anne Bancroft Sep 17

Patricia Neal Jan 20

Kate Winslet Oct 5

Marion Cotillard Sep 30

Helen Mirren Jul 26

Reese Witherspoon Mar 22

Hilary Swank Jul 30

Charlize Theron Aug 7

Nicole Kidman Jun 20

Halle Berry Aug 14

Julia Roberts Oct 28

Gwyneth Paltrow Sep 27

Helen Hunt Jun 15

Frances McDormand Jun 23

Susan Sarandon Oct 4

Jessica Lange Apr 20

Holly Hunter Mar 20

Emma Thompson Apr 15

Jodie Foster Nov 19

Kathy Bates Jun 28

Jessica Tandy Jun 7

Cher May 20

Last 40 Oscar-winning Best Actor BirthdaysDaniel Day-Lewis Apr 29

Forest Whitaker Jul 15

Philip Seymour Hoffman Jul 23

Jamie Foxx Dec 13

Sean Penn Aug 17

Adrien Brody Apr 14

Denzel Washington Dec 28

Russell Crowe Apr 7

Kevin Spacey Jul 26

Roberto Benigni Oct 27

Jack Nicholson Apr 22

Geoffrey Rush Jul 6

Nicolas Cage Jan 7

Tom Hanks Jul 9

Al Pacino Apr 25

Anthony Hopkins Dec 31

Jeremy Irons Sep 19

Dustin Hoffman Aug 8

Michael Douglas Sep 25

Paul Newman Jan 26

William Hurt Apr 20

F. Murray Abraham Oct 24

Robert Duvall Jan 5

Ben Kingsley Dec 31

Henry Fonda May 16

Robert De Niro Aug 17

Jon Voight Dec 29

Richard Dreyfuss Oct 29

Peter Finch Sep 28

Art Carney Nov 4

Jack Lemmon Feb 8

Marlon Brando Apr 3

Gene Hackman Jan 30

George C. Scott Oct 18

John Wayne May 26

Cliff Robertson Sep 9

Rod Steiger Apr 14

Paul Scofield Jan 21

Lee Marvin Feb 19

Rex Harrison Mar 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 11th.

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.

7 7 6 4

3 1 6 4 3 3

2 5 3 5 2 4

4 1 7 2 3 3

7 7 6 4

3 1 6 4 3 43

dkennedy@baylorschool.org

www.baylorschool.org