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!