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My Favorite Mathematical Paradoxes
Dan Kennedy Baylor School Chattanooga, TN
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Mathematics and Mirrors The Mirage
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The reflective property of a parabola
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The Mirage Illusion Explained.
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The Marvelous Möbius Strip
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This region of apparent intersection is actually
not there. This requires a fourth dimension for
actual assembly!
The Klein Bottle
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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?
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The Band Around the Earth (not to scale)
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A little geometry
r
R
x
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The best part about this paradox is that you have
to trust the mathematics. You cant perform the
experiment!
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Gabriels Horn
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The area of this region is infinite. Heres a
proof
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The volume of this solid is finite. Heres a
proof
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So Gabriels Horn is a mathematical figure which
has a finite volume (p), but which casts an
infinite shadow!
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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.
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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.
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The minimum number of moves required to move a
tower of n washers is 2n 1. The proof is a
classic example of mathematical
induction. Clearly, 1 washer requires 1 21 1
move. Assume that a tower of k washers requires
a minimum of 2k 1 moves. Then what about a
tower of k 1 washers?
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First, you must uncover the bottom washer. By
hypothesis, this requires 2k 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
2k 1 moves. Altogether, it requires 2(2k
1) 1 2(k 1) 1 moves to move k 1
washers. We are done by mathematical induction!
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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 27 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.
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When the tower is finally moved, that will signal
the End of the World.
Sohow much time do we have left?
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The age of the universe is currently estimated at
just under 14 billion years.
So relax.
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Simpsons Paradox
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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
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.368
.400
Despite having a poorer winning percentage than
the Killz before and after spring break, the
Settz won the trophy!
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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.
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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!
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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!
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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.
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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.
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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!
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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.
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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.
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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.
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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.
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Last 40 Oscar-winning Best Actress Birthdays
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Last 40 Oscar-winning Best Actor Birthdays
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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.
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The Paradox of the Kruskal Count or The Amazing
Secret of Twinkle Twinkle Little Star
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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.
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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.
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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
3
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Imagine the paradoxical implications
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dkennedy_at_baylorschool.org www.baylorschool.org