Illustrations of
Mathematical Induction
An Illustration of Mathematical Induction
by John Allen Poulos
Robert Louis Stevenson wrote "The Imp in the Bottle," the story of a genie
in a bottle who will satisfy your every wish for love, money, and power.
You can buy this amazing bottle for any amount that you care to offer.
The only constraint is that when you are finished with the bottle,
you must sell it for less than what you paid for it.
If you don't sell it to someone for a lower price, you will lose everything
and suffer everlasting torment in hell. What would you pay for such a bottle?
Certainly you won't pay 1 cent for it because then you won't be able to sell
it for a lower price. You won't pay 2 cents for it either, since no one will
buy it from you for 1 cent for the same reason. Neither will you pay 3 cents
for it; the person to whom you would have to sell it for 2 cents wouldn't be
able to sell it for 1 cent. A similar argument applies to a price of 4 cents,
5 cents, 6 cents, and so on.
Mathematical induction can be used to formalize this argument, which proves
conclusively that you shouldn't buy this magic bottle for any amount of money.
Yet you would almost certainly buy it for $1,000.
I know I would. At what point does the argument against buying the bottle become
practically convincing?
Another Illustration of Mathematical Induction
by Marilyn vos Savant
A logic teacher announces that a surprise test will be given the next week.
The students reason that the quiz cannot be given on Friday -- since, if it
were, the students would know at the end of class on Thursday that the quiz
would be on Friday, so it would not be a surprise. Likewise, the quiz cannot
be given on Thursday -- since, it if were, they would know at the end of
class on Wednesday that the quiz would be on Thursday (because it cannot be
given on Friday), also making it no surprise. Continuing with this logic,
the students eliminate every day of the week, conclude that the test
cannot be given at all and are shocked when it is given on Tuesday.
What, if anything, is wrong with their thinking? -- Eric Thurschwell, Langhorne, PA.
Marilyn vos Savant replies:
Talk about a foregone conclusion! The students draw a conclusion about Friday
based on their premise that the quiz has not already been given. So, when
they use the same conclusion to form additional conclusions about earlier
days, they continue to rely on their premise that the quiz has not already been
given. In effect, they say that if the quiz is not given by next Friday, it
won't be given by next Friday. Well, hello? Their logic guarantees a surprise!