Saturday, November 7, 2015

Solution to: Daring Thoughtless Thief Puzzle

A daring, rather thoughtless thief once stole a car of the police chief. The police immediately started an investigation and on the basis of witness depositions, four suspects were arrested that were seen near the car at the time of the crime. Because the chief of police took the case very seriously, he decided to examine the four suspects personally using a lie detector. Each suspect gave three statements during the examinations, as follows:
Suspect A:1. In high-school I was in the same class as suspect C.2. Suspect B has no driving license.3. The thief didn't know that it was the car of the chief of police.
Suspect B:1. Suspect C is the guilty one.2. Suspect A is not guilty.3. I never sat behind the wheel of a car.
Suspect C:1. I never met suspect A until today.2. Suspect B is innocent.3. Suspect D is the guilty one.
Suspect D:1. Suspect C is innocent.2. I didn't do it.3. Suspect A is the guilty one.
With so many contradicting statements, the chief of police lost track. To make things worse, it appeared that the lie-detector didn't quite work yet as it should, because the machine only reported that exactly four of the twelve statements were true, but not which ones.Now the big Question : Who is the thief??

Solution:
There are five statements in which nothing is said about the possible offender: A1, A2, A3, B3, and C1. The statements A1 and C1 seem to be completely contradictory, but that is not the case! Although at most one of these statements can be true, they can also be both false!For example, suspects A and C might only know each other from primary school. About the statements A2 and B3 not much can be said (although it seems unlikely that statement A2 would be false and at the same time statement B3 would be true). In addition, it follows from the introduction that statement A3 is true.On the basis of an assumption about which suspect is the offender, we can count how many of the remaining statements are true:
 Statement: A is the offender: B is the offender: C is the offender: D is the offender: None of the suspects is the offender: B1 false false true false false B2 false true true true true C2 true false true true true C3 false false false true false D1 true true false true true D2 true true true false true D3 true false false false false Total: 4 true, 3 false 3 true, 4 false 4 true, 3 false 4 true, 3 false 4 true, 3 false

Combined with the fact that statement A3 is true, this gives:
 A is the offender: B is the offender: C is the offender: D is the offender: None of the suspects is the offender: Total: 5 true, 3 false 4 true, 4 false 5 true, 3 false 5 true, 3 false 5 true, 3 false

Because it was given that exactly four statements were true, the statements A1, A2, B3, and C1 must be false, and suspect B must be the offender.

Wednesday, November 4, 2015

"You cannot prove this sentence is true."

Now, the question is - is that sentence true or false?

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The answer is that this is something of a trick question brainteaser, or a paradox. This can be seen by following through the logic that the sentence is either true or false, and seeing what happens in each case.
If you can prove the sentence is true, then the sentence is false. If you can prove the sentence is false, then it is true. In this way a paradox is created.

What is the solution to the puzzle? Well, there is no direct answer to that as it is a paradox and is self-referential in a way that seems to admit of no true or false answer. One way out is to say that the sentence appears meaningless or has no definite meaning, and get out of the problem by admitting of a potential third possible logical state to a question rather than simply having to have a true or false answer.