Statements!!

Which of the following statements are true?

1. At least one of these ten statements is false.
2. At least two of these ten statements are false.
3. At least three of these ten statements are false.
4. At least four of these ten statements are false.
5. At least five of these ten statements are false.
6. At least six of these ten statements are false.
7. At least seven of these ten statements are false.
8. At least eight of these ten statements are false.
9. At least nine of these ten statements are false.
10. At least ten of these ten statements are false.

Key Word: AT LEAST

5 are true, 5 are false

Originally posted by sloppyb
Which of the following statements are true?

1. At least one of these ten statements is false.
2. At least two of these ten statements are false.
3. At least three of these ten statements are false.
4. At least four of these ten statements are false.
5. At least five of these ten statements are false.
6. At least six of these ten statements are false. ...[text shortened]... statements are false.
10. At least ten of these ten statements are false.

Key Word: AT LEAST

Tough one...I'll get back to it.

it is tough - if you know answer explain

i know some easier riddles - might post all in one thread good idea?

Originally posted by sloppyb
Which of the following statements are true?

1. At least one of these ten statements is false.
2. At least two of these ten statements are false.
3. At least three of these ten statements are false.
4. At least four of these ten statements are false.
5. At least five of these ten statements are false.
6. At least six of these ten statements are false. ...[text shortened]... statements are false.
10. At least ten of these ten statements are false.

Key Word: AT LEAST

Statements 6-10 are false, and therefore statements 1-5 are true.

Originally posted by sloppyb
Which of the following statements are true?

1. At least one of these ten statements is false.
2. At least two of these ten statements are false.
3. At least three of these ten statements are false.
4. At least four of these ten statements are false.
5. At least five of these ten statements are false.
6. At least six of these ten statements are false. ...[text shortened]... statements are false.
10. At least ten of these ten statements are false.

Key Word: AT LEAST

No.1
Even if the 1st 5 were true as posted earlier that would mean AT LEAST 1 of the remaining statements would be false.

Originally posted by sloppyb
Which of the following statements are true?

1. At least one of these ten statements is false.
2. At least two of these ten statements are false.
3. At least three of these ten statements are false.
4. At least four of these ten statements are false.
5. At least five of these ten statements are false.
6. At least six of these ten statements are false. ...[text shortened]... statements are false.
10. At least ten of these ten statements are false.

Key Word: AT LEAST

It's a good one.
I think any of the first 1,2,3,4 or 5 statements can be true, and statements 6-10 must be false.

On second thought I agree with Thomaster. The first 5 are true and last 5 are false. Any other "solution" leads to contradiction.

Yep, Thomaster has the right solution. I think the easiest way to solve this problem is to assume a certain number of true statements and then work through the logical implications.

Let's assume there are n true statements in the list. For any n true statements, there are (10-n) false statements in the list. Now, if the statement "there are at least (10-n) false statements in the list" is true, the the statement "there are at least (10-n)-1 false statements in the list" is also true (e.g. if there are at least 7 false statements in the list, then there must be at least 6 false statements in the list too). This logic applies all they way down to "there is at least 1 false statement in the list". Therefore, if we assume there are n true statements in the list, then the first (10-n) statements in the list must be true. Our original assumption was that there are n true statements in the list, so equating the two we get:

n = (10-n)

2n = 10

n = 5

Therefore, there are 5 true statements in total in the list, and the first 5 statements in the list are true.

How many statements are true?

1. At least one of these statements is false.
2. At least two of these statements are false.
3. At least three of these statements are false.
...
999 998. At least 999 998 of these statements are false.
999 999. At least 999 999 of these statements are false.
1 000 000. At least 1 000 000 of these statements are false.

Originally posted by Thomaster
How many statements are true?

1. At least one of these statements is false.
2. At least two of these statements are false.
3. At least three of these statements are false.
...
999 998. At least 999 998 of these statements are false.
999 999. At least 999 999 of these statements are false.
1 000 000. At least 1 000 000 of these statements are false.

half

Originally posted by Thomaster
How many statements are true?

1. At least one of these statements is false.
2. At least two of these statements are false.
3. At least three of these statements are false.
...
999 998. At least 999 998 of these statements are false.
999 999. At least 999 999 of these statements are false.
1 000 000. At least 1 000 000 of these statements are false.

Using the same logic as above:

n = 1,000,000 - n
2n = 1,000,000
n = 500,000

Therefore, there are 500,000 true statements, and the first 500,000 statements are the true ones. I think a more interesting question is the following:

How many statements are true?

1. At least 1 of these statements is false.
2. At least 2 of these statements are false.
3. At least 3 of these statements are false.

Let's step through this case by case. First, let's assume there is 1 true statement. That means there are 2 false statements. Assigning a truth value to each statement based on this assumption, we have:

1. At least 1 of these statements is false. T
2. At least 2 of these statements are false. T
3. At least 3 of these statements are false. F

Summing up the T's and F's, we find that there is only 1 false statement and 2 true statements. This contradicts our original assumption of 1 true statement, and thus this cannot be the solution. If we assume there are 2 true statements and 1 false statement, we can repeat the process:

1. At least 1 of these statements is false. T
2. At least 2 of these statements are false. F
3. At least 3 of these statements are false. F

Summing up the T's and F's, we find that there are 2 false statements and only 1 true statement. This contradicts our original assumption of 2 true statements, and thus this cannot be the solution. If we assume all statements are true, then we can repeat the process:

1. At least 1 of these statements is false. F
2. At least 2 of these statements are false. F
3. At least 3 of these statements are false. F

Summing up the T's and F's, we find that there are 3 false statements and no true statements. This contradicts our original assumption of 3 true statements, and thus this cannot be the solution. So far none of our assumptions could possibly be right! What if we assume all statements are false?

1. At least 1 of these statements is false. T
2. At least 2 of these statements are false. T
3. At least 3 of these statements are false. T

Summing up the T's and F's, we find that there are 3 true statements and no false statements. This contradicts our original assumption of 3 false statements, and thus this cannot be the solution. So far, nothing is working. What does our previous logic have to say about this?

n = 3 - n
2n = 3
n = 1.5

And now it becomes clear. The first statement is true, the last statement is false, and the second statement is both true and false. Nonsense you say? You're just looking at it with the wrong frame of reference...this question is obviously political! 😉

Originally posted by sloppyb
Which of the following statements are true?

1. At least one of these ten statements is false.
2. At least two of these ten statements are false.
3. At least three of these ten statements are false.
4. At least four of these ten statements are false.
5. At least five of these ten statements are false.
6. At least six of these ten statements are false. ...[text shortened]... statements are false.
10. At least ten of these ten statements are false.

Key Word: AT LEAST

Why should I think any one of these statements has a truth value? These kinds of self-referential (or at least quasi-self-referential) statements; I always find them somewhat perplexing trying to figure out how to treat them.

How about:

1. At least 1 of these statements is false.
2. At most 1 of these statements is true.
3. At least 2 of these statements are false.
4. At most 2 of these statements are true.
5. At least 3 of these statements are false.
6. At most 3 of these statements are true.
7. At least 4 of these statements are false.
8. At most 4 of these statements are true.

Originally posted by iamatiger
How about:

1. At least 1 of these statements is false.
2. At most 1 of these statements is true.
3. At least 2 of these statements are false.
4. At most 2 of these statements are true.
5. At least 3 of these statements are false.
6. At most 3 of these statements are true.
7. At least 4 of these statements are false.
8. At most 4 of these statements are true.

1. At least 1 of these statements is false.
3. At least 2 of these statements are false.
5. At least 3 of these statements are false.
7. At least 4 of these statements are false.

2. At most 1 of these statements is true.
4. At most 2 of these statements are true.
6. At most 3 of these statements are true.
8. At most 4 of these statements are true.

2, 4, and 6 are false. 7 is neither true nor false, since it is paradoxal.