1. Joined
    31 Jan '09
    Moves
    4200
    23 Mar '10 19:41
    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
  2. ALG
    Joined
    16 Dec '07
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    6190
    23 Mar '10 21:00
    5 are true, 5 are false
  3. SubscriberAThousandYoung
    West Coast Rioter
    tinyurl.com/y7loem9q
    Joined
    23 Aug '04
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    23 Mar '10 21:08
    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.
  4. Joined
    31 Jan '09
    Moves
    4200
    23 Mar '10 21:28
    it is tough - if you know answer explain

    i know some easier riddles - might post all in one thread good idea?
  5. ALG
    Joined
    16 Dec '07
    Moves
    6190
    23 Mar '10 22:11
    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.
  6. Joined
    29 Nov '08
    Moves
    2251
    23 Mar '10 22:37
    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.
  7. Joined
    14 Dec '05
    Moves
    5692
    24 Mar '10 04:35
    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.
  8. Joined
    14 Dec '05
    Moves
    5692
    24 Mar '10 09:21
    On second thought I agree with Thomaster. The first 5 are true and last 5 are false. Any other "solution" leads to contradiction.
  9. Standard memberPBE6
    Bananarama
    False berry
    Joined
    14 Feb '04
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    28719
    24 Mar '10 15:19
    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.
  10. ALG
    Joined
    16 Dec '07
    Moves
    6190
    24 Mar '10 18:40
    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.
  11. Standard memberforkedknight
    Defend the Universe
    127.0.0.1
    Joined
    18 Dec '03
    Moves
    15977
    24 Mar '10 18:59
    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
  12. Standard memberPBE6
    Bananarama
    False berry
    Joined
    14 Feb '04
    Moves
    28719
    24 Mar '10 19:13
    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! 😉
  13. Joined
    24 Apr '05
    Moves
    3061
    24 Mar '10 21:42
    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.
  14. Joined
    26 Apr '03
    Moves
    25316
    24 Mar '10 23:09
    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.
  15. ALG
    Joined
    16 Dec '07
    Moves
    6190
    24 Mar '10 23:351 edit
    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.
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