True or False Statements (Gardner)

[MrJones]

This is the sort of teaser that appears in a book written by Martin Gardner, and I was hoping someone could help me see my way to the answer.

Of the following six statements, which are true and which are false?

1. There are no false statements in this card.
2. There is exactly one false statement in this card.
3. There are exactly two false statements in this card.
4. There are exactly three false statements in this card.
5. There are exactly four false statements in this card.
6. There are exactly five false statements in this card


I feel like I must be overanalyzing this one

 

[drizzle]

...Where is this card?

 

[D H]

...Where is this card?

I assume the OP meant this set. Don't overanalyze.

Each statement makes an exact claim that is contrary to the exact claims made by the other statements. Thusly, there cannot be two or more true statements, which in turn means at least five of the statements must be false.

1. Assume exactly five statements are false. This is statement six. Statement six is true and the others are false.

2. Assume all six statements are false. This is also consistent.


Bottom line: Statements 1-5 are false, the veracity of statement 6 cannot be determined.

 

[mugaliens]

This is the sort of teaser that appears in a book written by Martin Gardner, and I was hoping someone could help me see my way to the answer.

Of the following six statements, which are true and which are false?

1. There are no false statements in this card.
2. There is exactly one false statement in this card.
3. There are exactly two false statements in this card.
4. There are exactly three false statements in this card.
5. There are exactly four false statements in this card.
6. There are exactly five false statements in this card


I feel like I must be overanalyzing this one

6. The previous 5 statements are falese, which is true. It's the only true answer.

 

[D H]

The only thing that can be said with certainty is that statements 1-5 must be false. The last statement can be either true or false and still be consistent. It's truth value is indeterminate.

 

[Redbelly98]

6. The previous 5 statements are falese, which is true. It's the only true answer.

But #6 doesn't say there are at least 5 false statements, it says there are exactly 5 false statements.

D H is correct, #6 could be either true or false.

This reminds me of a question I heard on the TV show Whose Line Is It Anyway (American version):

Multiple choice: the correct answer is
A. B
B. C
C. A
D. D​

 

[DaveC426913]

Multiple choice: the correct answer is
A. B
B. C
C. A
D. D​

 

[mugaliens]

But #6 doesn't say there are at least 5 false statements, it says there are exactly 5 false statements.

D H is correct, #6 could be either true or false.

Let's review the initial problem:

"Of the following six statements, which are true and which are false?"

And the sixth question:

"6. There are exactly five false statements in this card"

The initial problem provides no option for indeterminancy. Therefore, it must either be true, or must either be false. "...could be either true or false" is not an option.

#6 is the only statement which can be true. Since it must be either true or false, the choice is up to the reader. I choose that it's ture.