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atgdax
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 THE MIRRORED MAZE   « on: Dec 8th, 2006, 2:33am » Quote Modify

Below are 10 statements, all either true or false:

1-At least one of statements 9 and 10 is true.
2-This is either the first true statement or the first false stament.
3-There are three consecutive false statements.
4-The difference between the number of the last true statement and the first true statement divides the number which is to be found.
5-The sum of the numbers of the true statements is the number which is to be found.
6-This is not the last true statement.
7-The number of each true statement divides the number which is to be found.
7-The number that is to be found is the percentage of true statements.
8-The number of divisors of the number that is to be found (apart from 1 and itself) is greater than 9-the sum of the numbers of the true statements.
10-There are no three consecutive true statements.

Find the minimum admissible number (which is to be found).

statment 2 must be false, because if all are true or false, statment 2 must be the same as s.1 thus it cannot be true...

im confused, any one know the answer...
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atgdax
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 Re: THE MIRRORED MAZE   « Reply #1 on: Dec 8th, 2006, 2:35am » Quote Modify

looking at statment 10. .... i don't get this!
if its false, it means they are all right - but they can't be can they ?!!!
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towr
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 Re: THE MIRRORED MAZE   « Reply #2 on: Dec 8th, 2006, 2:41am » Quote Modify

on Dec 8th, 2006, 2:35am, atgdax wrote:
 looking at statment 10. .... i don't get this! if its false, it means they are all right - but they can't be can they ?!!!
If statement 10 is false that would mean there are at least three consecutive true statements, not that all statements are true.
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towr
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 Re: THE MIRRORED MAZE   « Reply #3 on: Dec 8th, 2006, 2:44am » Quote Modify

Why are there two statements 7, doesn't 9 start on it's own line, and is 8 left unfinished?
 « Last Edit: Dec 8th, 2006, 2:45am by towr » IP Logged

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atgdax
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 Re: THE MIRRORED MAZE   « Reply #4 on: Dec 8th, 2006, 2:55am » Quote Modify

oh... yea yea, i just missclicked the numbers.
they are in corret order just wrong numbers at the end.

the key thing - is that they should ALL be true or ALL false.
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towr
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 Re: THE MIRRORED MAZE   « Reply #5 on: Dec 8th, 2006, 3:29am » Quote Modify

on Dec 8th, 2006, 2:55am, atgdax wrote:
 the key thing - is that they should ALL be true or ALL false.
Really? I thought they should all be either true or false. (The reason to mention that they're all either true or false is that otherwise some statements might be neither)

"all either true or false" is not the same as "either all true or all false". And generally these puzzles use the former codnition, as it's more interesting.
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atgdax
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 Re: THE MIRRORED MAZE   « Reply #6 on: Dec 8th, 2006, 4:13am » Quote Modify

OH.... tnx !
now i need to think of it again....

but do you know the answer ?
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ThudnBlunder
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 Re: THE MIRRORED MAZE   « Reply #7 on: Dec 8th, 2006, 6:26am » Quote Modify

on Dec 8th, 2006, 4:13am, atgdax wrote:
 but do you know the answer ?

420
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SMQ
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 Re: THE MIRRORED MAZE   « Reply #8 on: Dec 8th, 2006, 7:19am » Quote Modify

Assuming you meant this:

#1 - At least one of statements 9 and 10 is true.
#2 - This is either the first true statement or the first false stament.
#3 - There are three consecutive false statements.
#4 - The difference between the number of the last true statement and the first true statement divides the number which is to be found.
#5 - The sum of the numbers of the true statements is the number which is to be found.
#6 - This is not the last true statement.
#7 - The number of each true statement divides the number which is to be found.
#8 - The number that is to be found is the percentage of true statements.
#9 - The number of divisors of the number that is to be found (apart from 1 and itself) is greater than the sum of the numbers of the true statements.
#10 - There are no three consecutive true statements.

I agree with T&B:

 hidden: - If #2 is true then it is the first true statement and #1 must be false.  If #2 is false then it is not the first false statement and #1 must be false.  Either way, #1 must be false. - Since #1 is false, #9 and #10 must both be false. - If #6 were false, it would be the last true statement, but this would be a contradiction; therefore #6 is true. - Since #6 is true, at least one of #7 and #8 is true. - If #3 is true, #8 must be false to create three consecutive false statements.  If #3 is false, both #2 and #8 must be true to prevent three consecutive false statements. - Mathematically, #5 and #7 cannot both be true. - Putting these together, we find that there must be from three to six true statements. - Since there are at least three true statements, #5 and #8 cannot both be true. - Since there are from three to six true statements, #7 and #8 cannot both be true. - Therefore at most one of #5, #7, and #8 is true. - If #8 is true, #3 must be false (because if #3 were true #8 would have to be false), #5 must be false, and #7 must be false, but this would contradict #10 being false; therefore #8 is false. - Since #6 is true and #8 is false, #7 must be true. - Since both #5 and #7 cannot both be true, #5 must be false. - Since #10 is false, #2, #3, and #4 must all be true to give three consecutive true statements. - Therefore, #2, #3, #4, #6 and #7 are true, and #1, #5, #8, #9, and #10 are false. - Since #4 is true, the number which is to be found must be divisible by 7 - 2 = 5. - Since #7 is true, the number which is to be found must be divisible by 2, 3, 4, 6 and 7. - The least number satisfying the properties is 420. - 420 has 22 factors not including 1 and itself, and 2+3+4+6+7 = 22 so #9 false is satisfied.

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azalia
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 Re: THE MIRRORED MAZE   « Reply #9 on: Dec 8th, 2006, 9:47am » Quote Modify

I came up with a different answer, which I now know is wrong, maybe. I didn't realize that that "9-" was a misplaced statement number. I thought it said '9 minus the sum..." So I just changed the second 7 to 8, and the 8 to 9.
I concluded that the only way to have 3 consecutive true statements and #9 to be false is if the only true statements are 2,3 and 4. That would give you a final answer of 2 (with 0 divisors, which is not greater than 9-9), and everything would work except #6. But it's possible for 6 to be false if you parse the English differently.
Instead of interpreting it to mean This is ~(the last true statement), read it as This is ~(the last) true statement, or This is a non-last true statement. Then it can be false. I have more of a background in language interpretation and translation than in either math or logic. If I hold something up and say, "This is not the last donut," normal rules of English would assume the most likely interpretation as "This is a donut, but not the last one," rather than that I'm holding a cumquat. I could be holding a cumquat, and the statement would be true, but it's not the normal sense of that statement.
Seeing now that the sum of the true statements can be more than 9, the other answer is obviously the intended one. But would formal logic allow for the language interpretation that allows 6 to be false? Because 2 is definitely less than 420.
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SMQ
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 Re: THE MIRRORED MAZE   « Reply #10 on: Dec 8th, 2006, 10:18am » Quote Modify

on Dec 8th, 2006, 9:47am, azalia wrote:
 Instead of interpreting it to mean This is ~(the last true statement), read it as This is ~(the last) true statement, or This is a non-last true statement. Then it can be false.

But to me the negation of any of those statements is still "this is the last true statement."  Just as, whether the statement "This is not the last donut" is referring to a non-last donut or to a kumquat, it's negation: "this is the last donut" is unambiguously referring to a donut which is last.

In the case of a self-referencing statement, this leads to a contradiction.  Call the statement "This is not the last true statement" S.  It's negation, ~S is "this is the last true statement.", but now ~S implies S -- a contradiction.  Since we are told that each statement is either true or false -- implicitly excluding "ambiguous" and "contradictory" as possible values -- then the only consistent interpretation is that S is true.

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Ulkesh
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 Re: THE MIRRORED MAZE   « Reply #11 on: Dec 8th, 2006, 10:55am » Quote Modify

I was going to link to the version of this that T&B posted a few years ago, but after reading through the replies I don't think it'd add anything new!

Ahh, those were the days...
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azalia
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 Re: THE MIRRORED MAZE   « Reply #12 on: Dec 8th, 2006, 11:10am » Quote Modify

Bearing in mind that, because of the typo, I was forced to find a way for 6 to be false and would have taken your interpretation if there was any way to solve the riddle that way, and that I've never studied logic.
"It's negation, ~S is "this is the last true statement.", but now ~S implies S -- a contradiction."
One way to negate a sentence with the word "not" in it is to take out the "not." Another way is to add another "not" modifying a different word.
Suppose S is comprised of several substatements A AND B AND C, then ~S could imply either ~A OR ~B OR ~C.
This is the last true statement could be parsed as
This is a statement, and it is true, and it is the last.
And it is false if either This is not a statement or it is not true or it is not the last.
So, This is not the last true statement could be understood as This is ~(the last) true statement. This is a statement and it is true and it is not the last. If we were talking about donuts, and I couldn't see what you had in your hand, that's the way I would expect the sentence to be understood. And if any one of those three is false, then the statement as a whole is false.
If S=A AND B AND ~C then ~S=~A OR ~B OR C.
I suppose logic requires that every possible interpretation of a statement to be false for the statement itself to be false, and "This is not the last true statement" has to be parsed as "This is not a statement, or this is not true, or this is not last." But to me, a lie is when a statement reasonably interpreted communicates a falsehood, and not just when a statement has no possible true interpretation. That's why I'm not a logician, or a lawyer, or a politician, or a used car salesmen...
 « Last Edit: Dec 8th, 2006, 11:22am by azalia » IP Logged
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 Re: THE MIRRORED MAZE   « Reply #13 on: Dec 8th, 2006, 12:20pm » Quote Modify

Even linguistically, I guess I don't see how a reasonable interpretation of "this is not the last true statement" could communicate a falsehood without becoming contradictory.  If we take A = "this is a statement", B = "this is true", and C = "this is the last", we agree that ~S = ~A OR ~B OR C.  You seem to agree that asserting C leads to a contradiction overall -- I don't see how asserting ~A or ~B is any better: ~A would be "this is not a statement", and if that has any meaning at all it is clearly false.  ~B would be "this is not true", which is in-and-of itself contradictory, and so conveys neither truth nor falsehood.

How about this: regardless of how you apply the negation, the statement S = "this is not the last true statement" is fully equivalent to the statement T = "another of the statements is the last true statement OR there is no true statement." right?  But now the negation of T is clearly ~T = "no other of the statements is the last true statement AND there is a true statement."  Which leads back to the impossible situation where ~T, by implying T (if there is at least one true statement and no other statement is the last true statement then T must be the last true statement), implies its own negation ~(~T).  Thus ~T can not convey a clear meaning, either true or false, and T (and so its equivalent, S) must be true to be meaningful.

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wahsplayer101
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 Re: THE MIRRORED MAZE   « Reply #14 on: Dec 13th, 2006, 10:44am » Quote Modify

Maybe I am really off with this, but from what I can tell, the smallest number is 25, with 2, 3, 5, 6, and 10 being true. any comments?
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SMQ
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 Re: THE MIRRORED MAZE   « Reply #15 on: Dec 13th, 2006, 11:12am » Quote Modify

If #1 is false, #10 can't be true (b/c if it were it would make #1 true.)

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ifurgf
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 Re: THE MIRRORED MAZE   « Reply #16 on: Nov 28th, 2011, 2:24pm » Quote Modify

the all true and all false requirement can't be correct:

If all are false, 3 is true
If all are true, 3 is false

if 6 is false then it must be true which is a paradox so 6 must be true

for numbers 1 and 2 there are 4 combinations
a. false false
b. false true
c. true false
d. true true
c and d both lead to paradoxes so #1 must be false

1 is false so 9 and 10 are both false

by trial and error there is no combination that allows 8 to be true and thus 8 is false

since #'s 8,9, and 10 are false, #3 is true

since #6 is true and #'s 8,9, and 10 are false, 7 must be true

7 and 5 can not both be true, so since 7 is true, 5 is false

since 10 is false , 2 and 4 must both be true

thus 2,3,4,6 and 7 are true

based on 4 and 7, the number is divisible by 2,3,4,5,6, and 7

210 is the smallest number that satisfies this

thus:

if you ignore that requirement then #s 2,3,4,6, and 7 are true and 210 is the minimum number
 « Last Edit: Nov 28th, 2011, 2:34pm by ifurgf » IP Logged
SMQ
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 Re: THE MIRRORED MAZE   « Reply #17 on: Nov 29th, 2011, 4:46am » Quote Modify

I interpreted the requirement to be "each statement is either true or false" i.e. not contradictory or ambiguous, but with a well-defined truth value.

As to your conclusion, close, but not quite. You solved the questions but made a small algebra error: 210 isn't divisible by 4.

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