Here's an unsolved problem I found in my Algebra book.(adsbygoogle = window.adsbygoogle || []).push({});

See if you can spot where the error in reasoning lies:

A serial killer is sentenced to be executed.

He asks the judge not to let him know the day of the exectuion.

The judge says: "I sentence you to be executed at 10 A.M. some

day of this coming January, but I promise that you will not be aware

that you are being executed that day untill they come to get you

at 8 A.M." The killer goes to his cell and proceeds to prove, as follows,

that he can't be executed in January.

Let P(n) be the statement that I can't be executed on January (31-n).

I want to prove P(n) for [itex]0 \leq n \leq 30[/itex]. Now I can't be

executed on January 31, for since that is the last day of the month

and I am to be executed that month, I would know that was the day

before 8 A.M. (when they wouldn'y come to get me), contrary to the judge's sentence. Thus P(0) is true.

Suppose that P(m) is true for [itex]0\leq m \leq k[/itex], where [itex]k\leq 29[/itex]. That is, suppose I can't be executed on January (31-k) through

January 31. Then January (31-k-1) must be the last possible day for

execution, and I would be aware that was the day before 8 A.M. contrary

to the judge's sentence. Thus I can't be executed on January (31-(k+1)),

so P(k+1) is true. Therefore I can't be executed in January.

(Of course, the serial killer was executed on January 17.)

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# Mathematical Induction

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