12 replies to this topic

[bonanova]

• At least one of the last two statements in this list is true.
• This is either the first true or the first false statement in the list.
• There are three consecutive false statements in this list.
• The difference between the numbers of the last true statement and first true statement is a factor of x.
• The sum of the numbers of the true statements equals x.
• This is not the last true statement.
• Each true statement's number is a factor of x.
• The percentage of true statements in this list equals x.
• The number of different factors of x [not counting 1 and x] is greater than the sum of the numbers of the true statements
• No three consecutive statements in this list are all true.

Find x.

[Yoruichi-san]

Nice puzzle. Hehe...
...I dare say my life has ten time more meaning now.
;P

[Molly Mae]

Spoiler for Checking now

But so far, I've got x=28 420

At least one of the last two statements in this list is true. FALSE
This is either the first true or the first false statement in the list. TRUE
There are three consecutive false statements in this list. TRUE
The difference between the numbers of the last true statement and first true statement is a factor of x. TRUE
The sum of the numbers of the true statements equals x. FALSE
This is not the last true statement. TRUE
Each true statement's number is a factor of x. TRUE
The percentage of true statements in this list equals x. FALSE
The number of different factors of x [not counting 1 and x] is greater than the sum of the numbers of the true statements FALSE
No three consecutive statements in this list are all true. FALSE

EDIT: Nope. 3 isn't a factor of 28...neither is 4 (from clue 4)

420

Edited by Molly Mae, 28 June 2012 - 07:42 PM.

[EventHorizon]

Spoiler for Answer and my method

If 2 is true, 1 is false. (statement 2)
If 2 is false, 1 cannot be true (statement 2)
So 1 is false. (previous two lines)
Since 1 is false, so are 9 and 10 (statement 1)
6 is true (assuming 6 is false says 6 is the last true statement = contradiction)
If 2 is false, then 3 must be true (if 1,2,and 3 are false, there are 3 consecutive false, so 3 is true =contradiction), and if 3 is false, 2 must be true. So how about we assume 3 is false and see what happens.
-----------
Assume 3 is false
2 is true since if 1,2,and 3 are false then 3 is true (statement 3)
8 is true since if 8,9,and 10 are false then 3 is true (statement 3)
5 is false since 8 is true and sum of numbers from 1 to 8 is 36, if 5 was true 8 would be false (percentage would need to be simultaneously greater than 40 and less than 36).
4 is true (statement 3)
x has a factor of 6. (statement 4)
7 is true (10 is false)
The true statements are 2,4,6,7,8. So x=50. (50% are true and statement 8)
But 50 does not have a factor of 6
Contradiction!
-------------
3 is true (previous block)
8 is false (the three consecutive false statements (from 3) must include 8 now)
7 is true (statement 6)
5 is false (since 7 is true and 6 and 7 are factors of x, but adding up 1 through 7 is less than 42)
2 and 4 are true. (There needs to be 3 consecutive true statements or 10 would need to be true)
5 is a factor of x (statement 4)
2,3,4,6 and 7 are factors of x (statement 7)
Since 2,3,4,5,6, and 7 are factors of x, x is a multiple of 420. (least common multiple)

The factors of 420 (besides 1 and 420), are:
2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, and 210.
There are 22 of them.
Adding the numbers of the true statements also gives 22.
If x was any multiple of 420 other than 420 itself, it would make statement 9 true by having too many factors.

x=420

Looks like I agree with Molly Mae

Edit: Fixed terminology.

Edited by EventHorizon, 29 June 2012 - 06:37 PM.

[psykomakia]

I got a different answer.

Spoiler for Where did I go wrong?

If S1 is true, then if S2 were true, it would be false, and if S2 were false, it would be true. A contradiction. S1 must be false. And so S1, S9, and S10 must be false.

Assume S2 is false. Then if S3 were false, it would be true, and so it must be true.

Assume all other statements are false. The statements are consistent, but ‘x’ is undefined.

Assume that ‘x’ is defined and determinable from the statements, so other statement(s) must be true.

S5 or S8 must be true to determine ‘x’.

If S8 were true, then S6 would be true. But then S3 would be false, and we know it to b true. S8 must be false. S5 must be true.

If S3 and S5 are the only true statements, then S4 must be true because 2 is a factor of 8. Another statement must be true.

If S6 were true, S7 must be true, and vice versa. But S7 claims, then, that 6 is a factor of 21 or 24, which is false. S6 and S7 must be false.

S4 must be true, which makes three consecutive true numbers, which satisfies S10 (making it false).

So: S1-F, S2-F, S3-T, S4-T, S5-T, S6-F, S7-F, S8-F, S9-F, S10-F

The value of ‘x’, according to S5, is 12.

EDIT to add:

Spoiler for Ah... my mistake.

I missed S6. In my answer, I have S6 as False, though it is not the last true statement because it isn't true, which means it is true that it isn't the last true statement, which means it is true. Bah.

Edited by psykomakia, 29 June 2012 - 03:55 AM.

[EventHorizon]

Spoiler for Answer and my method

If 2 is true, 1 is false. (statement 2)
If 2 is false, 1 cannot be true (statement 2)
So 1 is false. (previous two lines)
Since 1 is false, so are 9 and 10 (statement 1)
6 is true (assuming 6 is false says 6 is the last true statement = paradox)
If 2 is false, then 3 must be true (if 1,2,and 3 are false, there are 3 consecutive false, so 3 is true =paradox), and if 3 is false, 2 must be true. So how about we assume 3 is false and see what happens.
-----------
Assume 3 is false
2 is true since if 1,2,and 3 are false then 3 is true (statement 3)
8 is true since if 8,9,and 10 are false then 3 is true (statement 3)
5 is false since 8 is true and sum of numbers from 1 to 8 is 36, if 5 was true 8 would be false (percentage would need to be simultaneously greater than 40 and less than 36).
4 is true (statement 3)
x has a factor of 6. (statement 4)
7 is true (10 is false)
The true statements are 2,4,6,7,8. So x=50. (50% are true and statement 8)
But 50 does not have a factor of 6
PARADOX
-------------
3 is true (previous block)
8 is false (the three consecutive false statements (from 3) must include 8 now)
7 is true (statement 6)
5 is false (since 7 is true and 6 and 7 are factors of x, but adding up 1 through 7 is less than 42)
2 and 4 are true. (There needs to be 3 consecutive true statements or 10 would need to be true)
5 is a factor of x (statement 4)
2,3,4,6 and 7 are factors of x (statement 7)
Since 2,3,4,5,6, and 7 are factors of x, x is a multiple of 420. (least common multiple)

The factors of 420 (besides 1 and 420), are:
2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, and 210.
There are 22 of them.
Adding the numbers of the true statements also gives 22.
If x was any multiple of 420 other than 420 itself, it would make statement 9 true by having too many factors.

x=420

Looks like I agree with Molly Mae

So it looks like I had paradoxes on the mind when drafting that post. What I meant (in each of the three mentions of the word paradox) was contradiction. I hate when I read through one of my posts and find misused words. Too bad I'm no longer a VIP or I'd be able to fix them.

Edit: Hey, I'm a VIP again, thanks rookie1ja.

Edited by EventHorizon, 29 June 2012 - 06:36 PM.

[Yoruichi-san]

Well, apparently my hidden hitchhiker went unnoticed...but...

Spoiler for Straightforward method

A. 1 has to be false, per 2, so 9 and 10 have to be false
B. 6 has to be true
C. 7 and 8 cannot be simultaneously true (mathematically) as there are 10 total statements, so 8 would say X is a multiple of 10 b/w 0 and 100, but 7 would say X is a multiple of 56
D. Either 7 or 8 are true, per B and A
E. 5 cannot be true, mathematically, per D and B...the smallest common multiple of 5,6,7 is 210 and all the numbers are <10, so there is no way for 5 and 7 or 8 to be simultaneously true.
F. 2,3,4 must all 3 be true per A,C and E.
G. There must be a total of 5 true statements, per all the above.
H. 8 must be false, per G and F, as 4 is true.
I. 7 must be true per D and H.

So using 4 and 7, and satisfying the constraints of 9, there is only one number...ten times the meaning of life, everything, and the universe ;P

[bonanova]

Well, apparently my hidden hitchhiker went unnoticed...but...

Spoiler for Straightforward method

A. 1 has to be false, per 2, so 9 and 10 have to be false
B. 6 has to be true
C. 7 and 8 cannot be simultaneously true (mathematically) as there are 10 total statements, so 8 would say X is a multiple of 10 b/w 0 and 100, but 7 would say X is a multiple of 56
D. Either 7 or 8 are true, per B and A
E. 5 cannot be true, mathematically, per D and B...the smallest common multiple of 5,6,7 is 210 and all the numbers are <10, so there is no way for 5 and 7 or 8 to be simultaneously true.
F. 2,3,4 must all 3 be true per A,C and E.
G. There must be a total of 5 true statements, per all the above.
H. 8 must be false, per G and F, as 4 is true.
I. 7 must be true per D and H.

So using 4 and 7, and satisfying the constraints of 9, there is only one number...ten times the meaning of life, everything, and the universe ;P

And I'm so glad it is. 7 and 6 are two of my fav numbers.

[bonanova]

So it looks like I had paradoxes on the mind when drafting that post. What I meant (in each of the three mentions of the word paradox) was contradiction. I hate when I read through one of my posts and find misused words. Too bad I'm no longer a VIP or I'd be able to fix them.

Edit: Hey, I'm a VIP again, thanks rookie1ja.

Rookie would prob reinstate you. PM him. Eternal editing is cool.

[psykomakia]

And I'm so glad it is. 7 and 6 are two of my fav numbers.

Don't you mean 6 and 9?