- #1
- 1,995
- 7
Here's an unsolved problem I found in my Algebra book.
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 until 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.)
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 until 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.)