My Favorite
Mathematical Paradoxes
Dan Kennedy
Baylor School
Chattanooga, TN
Mathematics and Mirrors:
The Mirage ®
The reflective property of a parabola:
focus
The Mirage Illusion Explained.
The Marvelous Möbius Strip
This region
of apparent
intersection
is actually
not there.
This
requires a
fourth
dimension
for actual
assembly!
The Klein Bottle
The Band Around the Earth Paradox
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 (not to scale):
A little geometry…
r
R
x
r
R x
2 R  2 r  10
2 ( R  r )  10
10
Rr  x 
 1.59 feet
2
In fact, if you multiply this
number by  , you’ll find
that a fellow with a 60-inch
belt size could heave his
way under the band, just
barely scraping.
The best part about this paradox is that you
have to trust the mathematics. You can’t
perform the experiment!
Gabriel’s Horn
1
y = –x
The area of this region is
infinite. Here’s a proof:

2

Area = lim  dx  lim  2 ln x
k   x
k 
1

k
 lim 2  ln k  ln1  
k 
k

1
The volume of this solid is
finite. Here’s a proof:
k
1

Volume = lim   2 dx
k  
x
1
 1 k
 lim   

k 
 x 1
 1 
 lim     1  
k 
 k 
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.
Volume = 0.013  0.000001
Area = 0.012  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 = 2^1 – 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.
So…how much time do we have left?
264  1 seconds = 1.84467 1019 seconds
1 hr
1.8447 10 sec = 1.8447 10 sec 
3600 sec
1 day
15
 5.124110 hrs 
24 hrs
1 yr
14
 2.135 10 days 
365.25 days
19
19
 5.8454 1011 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
.167
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
.167
Settz
10
1
9
.100
Team
Games
Won
Lost
Killz
19
7
12
Settz
20
8
12
Percentage
.368
.400
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!
1
2
3
When 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 Haddon’s
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 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:
364 363 362 361 360 359 358 357 356








365 365 365 365 365 365 365 365 365
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 Marilyn’s 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 that…there
must be a 100% chance that at least two out
of 50 will! That’s 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-writer’s claim of
“virtually certain.” It is
certainly not, as Marilyn
said, “just plain wrong.”
OOPS
.
Last 40 Oscar-winning Best Actress Birthdays
Sandra Bullock
Jul 26
Cher
May 20
Kate Winslet
Oct 5
Marlee Matlin
Aug 24
Marion Cotillard
Sep 30
Geraldine Page
Nov 22
Helen Mirren
Jul 26
Sally Field
Nov 6
Reese Witherspoon
Mar 22
Shirley MacLaine
Apr 24
Hilary Swank
Jul 30
Meryl Streep
May 27
Charlize Theron
Aug 7
Katharine Hepburn
May 12
Nicole Kidman
Jun 20
Sissy Spacek
Dec 25
Halle Berry
Aug 14
Jane Fonda
Dec 21
Julia Roberts
Oct 28
Diane Keaton
Jan 5
Gwyneth Paltrow
Sep 27
Faye Dunaway
Jan 14
Helen Hunt
Jun 15
Louise Fletcher
Jul 22
Frances McDormand
Jun 23
Ellen Burstyn
Dec 7
Susan Sarandon
Oct 4
Glenda Jackson
May 9
Jessica Lange
Apr 20
Liza Minnelli
Mar 12
Holly Hunter
Mar 20
Maggie Smith
Dec 28
Emma Thompson
Apr 15
Barbra Streisand
Apr 24
Jodie Foster
Nov 19
Elizabeth Taylor
Feb 27
Kathy Bates
Jun 28
Sophia Loren
Sep 20
Jessica Tandy
Jun 7
Anne Bancroft
Sep 17
Last 40 Oscar-winning Best Actor Birthdays
Jeff Bridges
Dec 4
Paul Newman
Jan 26
Daniel Day-Lewis
Apr 29
William Hurt
Apr 20
Forest Whitaker
Jul 15
F. Murray Abraham
Oct 24
Philip Seymour Hoffman
Jul 23
Robert Duvall
Jan 5
Jamie Foxx
Dec 13
Ben Kingsley
Dec 31
Sean Penn
Aug 17
Henry Fonda
May 16
Adrien Brody
Apr 14
Robert De Niro
Aug 17
Denzel Washington
Dec 28
Jon Voight
Dec 29
Russell Crowe
Apr 7
Richard Dreyfuss
Oct 29
Kevin Spacey
Jul 26
Peter Finch
Sep 28
Roberto Benigni
Oct 27
Art Carney
Nov 4
Jack Nicholson
Apr 22
Jack Lemmon
Feb 8
Geoffrey Rush
Jul 6
Marlon Brando
Apr 3
Nicolas Cage
Jan 7
Gene Hackman
Jan 30
Tom Hanks
Jul 9
George C. Scott
Oct 18
Al Pacino
Apr 25
John Wayne
May 26
Anthony Hopkins
Dec 31
Cliff Robertson
Sep 9
Jeremy Irons
Sep 19
Rod Steiger
Apr 14
Dustin Hoffman
Aug 8
Paul Scofield
Jan 21
Michael Douglas
Sep 25
Lee Marvin
Feb 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 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
4
Imagine the paradoxical implications…
dkennedy@baylorschool.org
www.baylorschool.org