Statements!!

Statements!!

Posers and Puzzles

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s
Standard membersloppyb

Joined
31 Jan 09
Moves
4200
23 Mar 10

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

T
Standard memberThomaster

ALG

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16 Dec 07
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6190
23 Mar 10

5 are true, 5 are false

Standard memberAThousandYoung
Insanity at Masada

tinyurl.com/mw7txe34

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23 Mar 10

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.

s
Standard membersloppyb

Joined
31 Jan 09
Moves
4200
23 Mar 10

it is tough - if you know answer explain

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

T
Standard memberThomaster

ALG

Joined
16 Dec 07
Moves
6190
23 Mar 10

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.

Standard memberglandmaster flash

Joined
29 Nov 08
Moves
2316
23 Mar 10

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.

l
Standard memberluskin

Joined
14 Dec 05
Moves
5694
24 Mar 10

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.

l
Standard memberluskin

Joined
14 Dec 05
Moves
5694
24 Mar 10

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

P
Standard memberPBE6
Bananarama

False berry

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28719
24 Mar 10

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.

T
Standard memberThomaster

ALG

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16 Dec 07
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24 Mar 10

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.

f
Standard memberforkedknight
Defend the Universe

127.0.0.1

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24 Mar 10

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

P
Standard memberPBE6
Bananarama

False berry

Joined
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Moves
28719
24 Mar 10

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! 😉

L
Standard memberLemonJello

Joined
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Moves
3061
24 Mar 10

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.

Standard memberiamatiger

Joined
26 Apr 03
Moves
26771
24 Mar 10

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.

T
Standard memberThomaster

ALG

Joined
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Moves
6190
24 Mar 10
1 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.