[Guest]

At midnight, John, as a cryptologist, was called to try to solve a riddle that the wealthy 2-hours-ago-killed Peter Hilton left.

John was told that the police found in Peter's house a fixed safe that, apparently, the killers couldn't open, or move.

Peter was well-known for his wealth and charity works, and had no descendants. The lock was tightly closed and it could be opened using a combination of unknown number of digits (more than one digit), each digit could be a number from 0-9. The words "Key Number" were inscribed above the keypad.

A note was on the safe in which the following statements were written.

1. At least one of statements 9 and 10 is true.

2. This either is the first true or the first false statement.

3. There are three consecutive statements, which are false.

4. The difference between the numbers of the last true and the first true statement divides the "Key Number".

5. The sum of the numbers of the true statements is the "Key Number".

6. This is not the last true statement.

7. The number of each true statement divides the "Key Number".

8. The "Key Number" is the percentage of true statements.

9. The number of divisors of the "Key Number", (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.

The key for this lock "Key Number", shall be the MINIMUM possible "Key Number" that could be deduced from the previous statements.

What is the "Key Number"??

Edited March 13, 2009 by HellFire

[Gmaster479]

30. This one is really tough

Hmmm, this one is tricky

[CaptainEd]

#6 is T: If it were F, then it would not be the last true statement, hence true.

#1 is F: Look at #2. It is either the first true (in which case 1 is F) or NOT the first false (in which case 1 is F).

#9 is F because #1 is F

#10 is F because #1 is F

#7,8 at least one true. However, #8 says that the Key number is 20, 30, 40, 50, 60, 70 (nothing higher or lower, as there are already 3 False statements and 2 True ones), and each one of those is missing either 7 or 8 as a divisor, so

#78 only one T

----

is 8 T?

8T => Key is 20, 30, 40, 50, 60.

--is 5 T?

--5T => 5,6,8 are true (sum is 19). remaining unknowns (2,3,4) don't add to 20 or 30 or higher.

--8T=>5F

Now (still assuming 8T) the truth list is:

123456789t

FxxxFTFTFF

Since 10F, there are 3 consecutive T. They can only be 234. But #3 says there are 3 consective F. There aren't. So, it's not true that 8T.

#8F.

---------------------------------

#7T (because one of 7 or 8 is T)

#3T (because 8, 9, 10 are F)

#5F: 7 says the number of each T divides the Key. So far, we know that 3, 6, 7 are T. The smallest such key is 42. Since 10, 9, 8, 1 are known false, the largest total of numbers remaining is 27, less than 42.

Since 10 is F, there have to be three consecutive T. They must be 234.

#2T

#4T

So the truth list is:

123456789t

FTTTFTTFFF

#4T says (7-2) divides the Key

#7T says the true statement numbers (2, 3, 4, 6, 7) divide the key

The smallest such key is 3 x 4 x 5 x 7 = 420. This has more divisors than the sum of the T statement numbers (sum = 22), so #9 becomes T!

So it appears to be a contradiction.

I can't verify GMaster's number, as I find a total contradiction. I presume I've made a blunder, but I can't see it.

Hats off to your intricate puzzle! Every line seems to deal with an additional attribute of the numbers. Well done!

Edited March 13, 2009 by CaptainEd

[Gmaster479]

I took a shot in the dark and based on your findings Captain Ed, I think there you have most of it but I lose you at the end for some reason. Maybe recheck that?

Edited March 13, 2009 by Gmaster479

[Guest]

I didn't get what u're saying CaptainEd at the end. but anyway, Great Job!!

I'm glad u liked it CaptainEd!!!

Edited March 14, 2009 by HellFire

[Guest]

ok, i got u now, but u seem having a misconception there!!

[Guest]

2 for the lowest key number. The true answers are 2,3, & 4.

I originally had 12 for the key number with the true answers as 3, 4, and 5.

It all started with if 2=T then 1=F, but if 1=T then 2 cannot be F cuz then it's T, so 1=F

If 1=F then 9&10 = F

If 10 = F then there must be 3 consecutive T so, (2,3,4)=T, (4,5,6)=T, (5,6,7)=T, or (6,7,8)=T

Then if 5=T then 8=F and if 8=T then 5=F

If 8=T then (2,3,4)= T or (6,7,8)=T or (2,3,4,6,7,8)=T

If (2,3,4,6,7,8)=T then

from 4: (8-2=6) KN/6=integer

from 7: KN/2, KN/3, KN/4, KN/6, KN/7, & KN/8 = integers and

from 8: KN=(6/10)*100=60

but 60/7 is not an integer.... not a solution

If (6,7,8) =T then

from 7: KN/6, KN/7, & KN/8 = integers and

from 8: KN=100*(3+x)/10

30, 40, & 50 all are not divisible by 7... not a solution

If (2,3,4)=T then

from 4: (8-2=6) KN/6=integer

from 8: KN= 100*(4+x)/10

only x=2 makes this true and if x=2 then 6,7 =T but we already saw that as false, so ... not a solution

Therefore, 8=F

So Far: 1=F, 8=F, 9=F, 10=F

Three consecutive false means 3=T

So Far: 1=F, 3=T, 8=F, 9=F, 10=F Still don't know: 2, 4, 5, 6, 7

If 6=T then 7=T

If 7=F then 6=F

If 6=F then 7=?

If 7=T then 6=?

If 6=T

7=T

from 10: (2,3,4)=T, (3,4,5)=T, (4,5,6)=T, or (5,6,7)=T

so we have the following options:

(2,3,4,6,7)=T

from 4:(7-2=5) KN/5=integer

from 7:KN/2, KN/3, KN/4, KN/6, KN/7 all must equal an integer

KN=420... POSSIBLY a solution

(3,4,5,6,7)=T

from 4:(7-3=4) KN/4=integer

from 5:KN=(3+4+5+6+7=25)

25/4 is not an integer so... NOT a solution

(3,5,6,7)=T

from 5:KN=(3+5+6+7=21)

from 7:KN/3, KN/5, KN/6, KN/7 must all equal an integer

21/5 is not an integer so... NOT a solution

(2,3,5,6,7)=T

from 5:KN=(2+3+5+6+7=23)

from 7:

23/2 is not an integer so... NOT a solution

(2,3,4,5,6,7)=T

from 4:(7-2=5) KN/5=integer

from 5:KN=(2+3+4+5+6+7=27)

27/5 is not an integer so... NOT a solution

Possible Solution #1: KN=420; 1=F, 2=T, 3=T, 4=T, 5=F, 6=T, 7=T, 8=F, 9=F, 10=F

But 420 has more divisors than the sum of the numbers, which would make 9=T so NOT a solution

So Far: 1=F, 3=T, 6=F, 8=F, 9=F, 10=F Still don't know: 2, 4, 5, 7

If 7=F

6=F

from 10: (2,3,4)=T, or (3,4,5)=T so 4=T

so we have the following options:

(2,3,4,5)=T

from 4:(5-2=3) KN/3=integer

from 5:KN=(2+3+4+5=14)

14/3 is not an integer so... NOT a solution

(2,3,4)=T

from 4:(4-2=2) KN/2=integer

KN/2=integer KN=Even so.... POSSIBLY a solution

(3,4,5)=T

from 4:(5-3=2) KN/2=integer

from 5:KN=(3+4+5=12)

12/2=6 so... POSSIBLY a solution

So Far: 1=F, 3=T, 4=T, 6=F, 7=F, 8=F, 9=F, 10=F Still don't know: 2, 5

Solution #1 KN=2, (2,3,4)=T all others F

Solution #2 KN=12, (3,4,5)=T all others F

Because it asks for the lowest, the Key Number is 2.

[Guest]

2 for the lowest key number. The true answers are 2,3, & 4.

I originally had 12 for the key number with the true answers as 3, 4, and 5.

It all started with if 2=T then 1=F, but if 1=T then 2 cannot be F cuz then it's T, so 1=F

If 1=F then 9&10 = F

If 10 = F then there must be 3 consecutive T so, (2,3,4)=T, (4,5,6)=T, (5,6,7)=T, or (6,7,8)=T

Then if 5=T then 8=F and if 8=T then 5=F

If 8=T then (2,3,4)= T or (6,7,8)=T or (2,3,4,6,7,8)=T

If (2,3,4,6,7,8)=T then

from 4: (8-2=6) KN/6=integer

from 7: KN/2, KN/3, KN/4, KN/6, KN/7, & KN/8 = integers and

from 8: KN=(6/10)*100=60

but 60/7 is not an integer.... not a solution

If (6,7,8) =T then

from 7: KN/6, KN/7, & KN/8 = integers and

from 8: KN=100*(3+x)/10

30, 40, & 50 all are not divisible by 7... not a solution

If (2,3,4)=T then

from 4: (8-2=6) KN/6=integer

from 8: KN= 100*(4+x)/10

only x=2 makes this true and if x=2 then 6,7 =T but we already saw that as false, so ... not a solution

Therefore, 8=F

So Far: 1=F, 8=F, 9=F, 10=F

Three consecutive false means 3=T

So Far: 1=F, 3=T, 8=F, 9=F, 10=F Still don't know: 2, 4, 5, 6, 7

If 6=T then 7=T

If 7=F then 6=F

If 6=F then 7=?

If 7=T then 6=?

If 6=T

7=T

from 10: (2,3,4)=T, (3,4,5)=T, (4,5,6)=T, or (5,6,7)=T

so we have the following options:

(2,3,4,6,7)=T

from 4:(7-2=5) KN/5=integer

from 7:KN/2, KN/3, KN/4, KN/6, KN/7 all must equal an integer

KN=420... POSSIBLY a solution

(3,4,5,6,7)=T

from 4:(7-3=4) KN/4=integer

from 5:KN=(3+4+5+6+7=25)

25/4 is not an integer so... NOT a solution

(3,5,6,7)=T

from 5:KN=(3+5+6+7=21)

from 7:KN/3, KN/5, KN/6, KN/7 must all equal an integer

21/5 is not an integer so... NOT a solution

(2,3,5,6,7)=T

from 5:KN=(2+3+5+6+7=23)

from 7:

23/2 is not an integer so... NOT a solution

(2,3,4,5,6,7)=T

from 4:(7-2=5) KN/5=integer

from 5:KN=(2+3+4+5+6+7=27)

27/5 is not an integer so... NOT a solution

Possible Solution #1: KN=420; 1=F, 2=T, 3=T, 4=T, 5=F, 6=T, 7=T, 8=F, 9=F, 10=F

But 420 has more divisors than the sum of the numbers, which would make 9=T so NOT a solution

So Far: 1=F, 3=T, 6=F, 8=F, 9=F, 10=F Still don't know: 2, 4, 5, 7

If 7=F

6=F

from 10: (2,3,4)=T, or (3,4,5)=T so 4=T

so we have the following options:

(2,3,4,5)=T

from 4:(5-2=3) KN/3=integer

from 5:KN=(2+3+4+5=14)

14/3 is not an integer so... NOT a solution

(2,3,4)=T

from 4:(4-2=2) KN/2=integer

KN/2=integer KN=Even so.... POSSIBLY a solution

(3,4,5)=T

from 4:(5-3=2) KN/2=integer

from 5:KN=(3+4+5=12)

12/2=6 so... POSSIBLY a solution

So Far: 1=F, 3=T, 4=T, 6=F, 7=F, 8=F, 9=F, 10=F Still don't know: 2, 5

Solution #1 KN=2, (2,3,4)=T all others F

Solution #2 KN=12, (3,4,5)=T all others F

Because it asks for the lowest, the Key Number is 2.

there's something wrong there.. anyway, the question said it's more than one digit, so it can't be 2!!

[CaptainEd]

2 for the lowest key number. The true answers are 2,3, & 4.

I originally had 12 for the key number with the true answers as 3, 4, and 5.

It all started with if 2=T then 1=F, but if 1=T then 2 cannot be F cuz then it's T, so 1=F

If 1=F then 9&10 = F

If 10 = F then there must be 3 consecutive T so, (2,3,4)=T, (4,5,6)=T, (5,6,7)=T, or (6,7,8)=T

Then if 5=T then 8=F and if 8=T then 5=F

If 8=T then (2,3,4)= T or (6,7,8)=T or (2,3,4,6,7,8)=T

If (2,3,4,6,7,8)=T then

from 4: (8-2=6) KN/6=integer

from 7: KN/2, KN/3, KN/4, KN/6, KN/7, & KN/8 = integers and

from 8: KN=(6/10)*100=60

but 60/7 is not an integer.... not a solution

If (6,7,8) =T then

from 7: KN/6, KN/7, & KN/8 = integers and

from 8: KN=100*(3+x)/10

30, 40, & 50 all are not divisible by 7... not a solution

If (2,3,4)=T then

from 4: (8-2=6) KN/6=integer

from 8: KN= 100*(4+x)/10

only x=2 makes this true and if x=2 then 6,7 =T but we already saw that as false, so ... not a solution

Therefore, 8=F

So Far: 1=F, 8=F, 9=F, 10=F

Three consecutive false means 3=T

So Far: 1=F, 3=T, 8=F, 9=F, 10=F Still don't know: 2, 4, 5, 6, 7

If 6=T then 7=T

If 7=F then 6=F

If 6=F then 7=?

If 7=T then 6=?

If 6=T

7=T

from 10: (2,3,4)=T, (3,4,5)=T, (4,5,6)=T, or (5,6,7)=T

so we have the following options:

(2,3,4,6,7)=T

from 4:(7-2=5) KN/5=integer

from 7:KN/2, KN/3, KN/4, KN/6, KN/7 all must equal an integer

KN=420... POSSIBLY a solution

(3,4,5,6,7)=T

from 4:(7-3=4) KN/4=integer

from 5:KN=(3+4+5+6+7=25)

25/4 is not an integer so... NOT a solution

(3,5,6,7)=T

from 5:KN=(3+5+6+7=21)

from 7:KN/3, KN/5, KN/6, KN/7 must all equal an integer

21/5 is not an integer so... NOT a solution

(2,3,5,6,7)=T

from 5:KN=(2+3+5+6+7=23)

from 7:

23/2 is not an integer so... NOT a solution

(2,3,4,5,6,7)=T

from 4:(7-2=5) KN/5=integer

from 5:KN=(2+3+4+5+6+7=27)

27/5 is not an integer so... NOT a solution

Possible Solution #1: KN=420; 1=F, 2=T, 3=T, 4=T, 5=F, 6=T, 7=T, 8=F, 9=F, 10=F

But 420 has more divisors than the sum of the numbers, which would make 9=T so NOT a solution

So Far: 1=F, 3=T, 6=F, 8=F, 9=F, 10=F Still don't know: 2, 4, 5, 7

If 7=F

6=F

from 10: (2,3,4)=T, or (3,4,5)=T so 4=T

so we have the following options:

(2,3,4,5)=T

from 4:(5-2=3) KN/3=integer

from 5:KN=(2+3+4+5=14)

14/3 is not an integer so... NOT a solution

(2,3,4)=T

from 4:(4-2=2) KN/2=integer

KN/2=integer KN=Even so.... POSSIBLY a solution

(3,4,5)=T

from 4:(5-3=2) KN/2=integer

from 5:KN=(3+4+5=12)

12/2=6 so... POSSIBLY a solution

So Far: 1=F, 3=T, 4=T, 6=F, 7=F, 8=F, 9=F, 10=F Still don't know: 2, 5

Solution #1 KN=2, (2,3,4)=T all others F

Solution #2 KN=12, (3,4,5)=T all others F

Because it asks for the lowest, the Key Number is 2.

The flaw that I see is that both your solutions involve #6 being F. If #6 is False, then it is definitely not the last true statement, which makes it true. So #6 has to be true, and I think that might change your key number.

[CaptainEd]

Hmmm, this one is tricky

30. This one is really tough

Yeah, I'm having a problem with that number, because it appears to lead to a contradiction as well:

if Key is 30, Let's look at #8

If #8 is True, then there are exactly 3 True statements

#6 must be true because if it were false, then it would definitely not be the last true statement. Which means it would be true.

#7 is false because 7 does not divide 30.

#3 is F, because if #3 is T, there is no stretch of 3 contiguous F statements

So, if #8T

123456789t

..F..TFT..

#9 is F, because the number of non-trivial divisors of 30 is 6 (2,3,5,6,10,15), and the sum of the number of the true statements is already 14.

123456789t

..F..TFTF.

is 5 T? If 5 T, then the sum is at least 19 (5,6,8).

123456789t

..F.TTFTF.

but to get to 30, we need the 10 and the 1, so for 5 to be true, the truth list must be

123456789t

TFFFTTFTFT

But the problem is that #2 is indeed the first false statement, which makes #2 True! So, 5 is false.

123456789t

..F.FTFTF.

At this point, there is no room for 3 consecutive T, so 10 is T

But that means #1 is T, so the truth list is now

123456789t

T.F.FTFTFT

But now look at 2. It is not the first T. If it is the first False, then it is True!

So all of this discussion has been false, which means

Start all over again with #8 False

123456789t

.......F..

Once again #7 is false, because 7 does not divide 30.

123456789t

......FF..

#9 is F, because if 9 were True, then the sum of the true statement numbers would be at least 9, which is greater than the number of divisors of 30 (6).

123456789t

......FFF.

This makes #3 True.

123456789t

..T...FFF.

if 5 were True, then the only way for the true statement numbers to add to 30 would be if they're all T but 1

(FTTTTTFFFT)But that would make #1 T, so 5 is not true

123456789t

..T.F.FFF.

is 1 True?

If so, 10 is True.

With 10 T, then 2 must be false (else 123T)

123456789t

TFT.F.FFFT

But this is a problem for #2, because 2 would be the first false statement, which would mean that a False statement is True.

So, 1 is False.

123456789t

F.T.F.FFF.

With 1 False, 9 and 10 are both F

123456789t

F.T.F.FFFF

But this causes a contradiction with #6:

If 6 is true, then there is a later True, but there isn't.

If 6 is false, then it is definitely not the last true statement, which makes it true.

So the entire discussion, from the beginning, results in contradiction. So how did you prove 30 to be the key number?

[CaptainEd]

Yeah, I'm having a problem with that number, because it appears to lead to a contradiction as well:

if Key is 30, Let's look at #8

If #8 is True, then there are exactly 3 True statements

#6 must be true because if it were false, then it would definitely not be the last true statement. Which means it would be true.

#7 is false because 7 does not divide 30.

#3 is F, because if #3 is T, there is no stretch of 3 contiguous F statements

So, if #8T

123456789t

..F..TFT..

#9 is F, because the number of non-trivial divisors of 30 is 6 (2,3,5,6,10,15), and the sum of the number of the true statements is already 14.

123456789t

..F..TFTF.

is 5 T? If 5 T, then the sum is at least 19 (5,6,8).

123456789t

..F.TTFTF.

but to get to 30, we need the 10 and the 1, so for 5 to be true, the truth list must be

123456789t

TFFFTTFTFT

But the problem is that #2 is indeed the first false statement, which makes #2 True! So, 5 is false.

123456789t

..F.FTFTF.

At this point, there is no room for 3 consecutive T, so 10 is T

But that means #1 is T, so the truth list is now

123456789t

T.F.FTFTFT

But now look at 2. It is not the first T. If it is the first False, then it is True!

So all of this discussion has been false, which means

Start all over again with #8 False

123456789t

.......F..

Once again #7 is false, because 7 does not divide 30.

123456789t

......FF..

#9 is F, because if 9 were True, then the sum of the true statement numbers would be at least 9, which is greater than the number of divisors of 30 (6).

123456789t

......FFF.

This makes #3 True.

123456789t

..T...FFF.

if 5 were True, then the only way for the true statement numbers to add to 30 would be if they're all T but 1

(FTTTTTFFFT)But that would make #1 T, so 5 is not true

123456789t

..T.F.FFF.

is 1 True?

If so, 10 is True.

With 10 T, then 2 must be false (else 123T)

123456789t

TFT.F.FFFT

But this is a problem for #2, because 2 would be the first false statement, which would mean that a False statement is True.

So, 1 is False.

123456789t

F.T.F.FFF.

With 1 False, 9 and 10 are both F

123456789t

F.T.F.FFFF

But this causes a contradiction with #6:

If 6 is true, then there is a later True, but there isn't.

If 6 is false, then it is definitely not the last true statement, which makes it true.

So the entire discussion, from the beginning, results in contradiction. So how did you prove 30 to be the key number?

OK, maybe my interpreation of #6 is not what OP intended. With a different interpretation, I can confirm Gmaster479's answer.

Then, sparing you the long analysis, the truth list

123456789t

FTTTFFFTFF

is consistent with the KeyNumber of 30.

[Guest]

I haven't done the work on this but just wondering if #6 is false could it be interpreted as 7,8,9,10 are false, (It being false cancels the "not" in the statement)?

[Guest]

1: 9 or 10 or both True T

2: 1st True or False F

3: 3 cons. statements False T

4: (Last True - First True) devides Key No F

5: sum of Nos. of True Statements = Key No. T

6. Not last True statement T

7. no of each true devides key number F

8. key no = % of True statements - F

9. no of divisors of Key no. > sum of numbers of True statements F

10. no 3 cons. True stat. T

so as statement 5 is true, my answer is 25

Edited March 16, 2009 by Archana Khedkar

[Guest]

Plz ignore my previous reply..... coz thats wrong

1: 9 or 10 or both True F

2: 1st True or False F

3: 3 cons. statements False T

4: (Last True - First True) devides Key No T

5: sum of Nos. of True Statements = Key No. T

6. Not last True statement F

7. no of each true devides key number F

8. key no = % of True statements - F

9. no of divisors of Key no. > sum of numbers of True statements F

10. no 3 cons. True stat. F

so as statement 5 is true, my actual answer is 12

[Guest]

The Key Number is 420.

6 has to be true because if it was false that it WOULD be the last true statement which is a paradox.

1 has to be false because of 2. If 1 was true, then 2 couldn't be either true or false.

Since 1 is false, 9 & 10 are false.

Since 6 is true and 9 & 10 are false, one of either 7 or 8 is true.

Since 10 is false, there ARE 3 consecutive truths in the list.

5 and 7 will not be true at the same time. This would imply a perfect number, which would mean that 1 would have to be true.

5 and 8 will not be true at the same time, because 8 is the same thing as 5 except counting 10 for each number instead of that number's value.

Since one of 7 or 8 is true, then 5 must be false.

So far we have: F X X X F T X X F F

7 and 8 will not be true at the same time, because the only percentage divisible by 7 would be 70, meaning 7 true statments, which cannot be true since we already know 4 are false.

This means the only place left for 3 consecutive true statements is 2, 3, 4. Thus all those are true.

Since 3 is true, we must have 3 consecutive false statements, so 8 must be false (and so 7 is true).

So now we have: F T T T F T T F F F

This means (from 4 and 7) that the Key Number has the following divisors: 2, 3, 4, 5 (7-2), 6, and 7.

The LCM of these is 420.

[Guest]

Hats off to your intricate puzzle! Every line seems to deal with an additional attribute of the numbers. Well done!

#6 is T: If it were F, then it would not be the last true statement, hence true.

#1 is F: Look at #2. It is either the first true (in which case 1 is F) or NOT the first false (in which case 1 is F).

#9 is F because #1 is F

#10 is F because #1 is F

#7,8 at least one true. However, #8 says that the Key number is 20, 30, 40, 50, 60, 70 (nothing higher or lower, as there are already 3 False statements and 2 True ones), and each one of those is missing either 7 or 8 as a divisor, so

#78 only one T

----

is 8 T?

8T => Key is 20, 30, 40, 50, 60.

--is 5 T?

--5T => 5,6,8 are true (sum is 19). remaining unknowns (2,3,4) don't add to 20 or 30 or higher.

--8T=>5F

Now (still assuming 8T) the truth list is:

123456789t

FxxxFTFTFF

Since 10F, there are 3 consecutive T. They can only be 234. But #3 says there are 3 consective F. There aren't. So, it's not true that 8T.

#8F.

---------------------------------

#7T (because one of 7 or 8 is T)

#3T (because 8, 9, 10 are F)

#5F: 7 says the number of each T divides the Key. So far, we know that 3, 6, 7 are T. The smallest such key is 42. Since 10, 9, 8, 1 are known false, the largest total of numbers remaining is 27, less than 42.

Since 10 is F, there have to be three consecutive T. They must be 234.

#2T

#4T

So the truth list is:

123456789t

FTTTFTTFFF

#4T says (7-2) divides the Key

#7T says the true statement numbers (2, 3, 4, 6, 7) divide the key

The smallest such key is 3 x 4 x 5 x 7 = 420. This has more divisors than the sum of the T statement numbers (sum = 22), so #9 becomes T!

So it appears to be a contradiction.

I can't verify GMaster's number, as I find a total contradiction. I presume I've made a blunder, but I can't see it.

Nice job CaptainEd.

Double check your factoring. I'm seeing 20 divisors of 420 (not counting 1 and 420, as the puzzle states).

[CaptainEd]

Nice job CaptainEd.

Double check your factoring. I'm seeing 20 divisors of 420 (not counting 1 and 420, as the puzzle states).

Yes, indeed, miscounted my divisors. What a relief! Thanks for restoring rationality. I really didn't like the possibility that #6 could be false.

Thank you JB, Thank you!

[Guest]

Using the KISS method (keep it simple stupid), I make the going-in assumption that all items are False unless they must be True. Using this method, I believe the Key Number is 10.

Evaluating each statement:

#1 assumed False, and therefore both #9 and #10 must also be False. If #10 is False, there must be 3 consecutive True statements.

#2 must be True.

#3 is likely True because of my going-in assumption.

#4 could be True to comply with #10 being False.

#5 assumed False.

#6 assumed False.

#7 assumed False.

#8 assumed False.

#9 assumed False because of #1.

#10 assumed False because of #1.

Therefore, 2, 3, and 4 are True. All others are False.

Item #4 indicates that the Key Number is a multiple of the difference between the highest True item (#4) and the lowest True item (#2). The difference is 2. The other requirements of the Key Number indicated are that it’s the MINIMUM number with more than one digit. The minimum multi-digit number that meets all the requirements is 10. (10 = 2 * 5)

Evaluating Key Number = 10 against each statement:

#1 is still False. Both 9 & 10 prove to be False.

#2 is still True. This is the first True statement.

#3 is still True. #5 – 10 prove to be False.

#4 is still True. The difference is 2, which evenly divides the Key Number.

#5 is still False. The sum is 9. The Key Number is 10.

#6 is still False. #6 is a paradoxical statement.

#7 is still False. Only 2 divides the Key Number, not 3 or 4.

#8 is still False. The % of True statements is 30%.

#9 is still False. The number of divisors is 2. The sum of the True statements is 9.

#10 is still False. There are 3 consecutive True statements.

[CaptainEd]

Using the KISS method (keep it simple stupid), I make the going-in assumption that all items are False unless they must be True. Using this method, I believe the Key Number is 10.

Evaluating each statement:

#1 assumed False, and therefore both #9 and #10 must also be False. If #10 is False, there must be 3 consecutive True statements.

#2 must be True.

#3 is likely True because of my going-in assumption.

#4 could be True to comply with #10 being False.

#5 assumed False.

#6 assumed False.

#7 assumed False.

#8 assumed False.

#9 assumed False because of #1.

#10 assumed False because of #1.

Therefore, 2, 3, and 4 are True. All others are False.

Item #4 indicates that the Key Number is a multiple of the difference between the highest True item (#4) and the lowest True item (#2). The difference is 2. The other requirements of the Key Number indicated are that it’s the MINIMUM number with more than one digit. The minimum multi-digit number that meets all the requirements is 10. (10 = 2 * 5)

Evaluating Key Number = 10 against each statement:

#1 is still False. Both 9 & 10 prove to be False.

#2 is still True. This is the first True statement.

#3 is still True. #5 – 10 prove to be False.

#4 is still True. The difference is 2, which evenly divides the Key Number.

#5 is still False. The sum is 9. The Key Number is 10.

#6 is still False. #6 is a paradoxical statement.

#7 is still False. Only 2 divides the Key Number, not 3 or 4.

#8 is still False. The % of True statements is 30%.

#9 is still False. The number of divisors is 2. The sum of the True statements is 9.

#10 is still False. There are 3 consecutive True statements.

Which statement is the last T? #4. So #6 is not the last true statement. So it doesn't make sense to say that #6 is False. #6 is not the last true statement.

It's only paradoxical if you call it False. It can be meaningful if you call it T (but then you have to have another true statement later)

[Guest]

Yes, indeed, miscounted my divisors. What a relief! Thanks for restoring rationality. I really didn't like the possibility that #6 could be false.

Thank you JB, Thank you!

Now I'm confused. I thought you were correct initially.

420 has 22 factors (excluding 1 and 420):

1. (2, 210)

2. (3, 140)

3. (4, 105)

4. (5, 84)

5. (6, 70)

6. (7, 60)

7. (10, 42)

8. (12, 35)

9. (14, 30)

10. (15, 28)

11. (20, 21)

[Guest]

Whoops! I should have listed them out before I jumped to that conclusion. Your list looks right, vinays. Well, CaptainEd wins again!

[CaptainEd]

Now I'm confused. I thought you were correct initially.

420 has 22 factors (excluding 1 and 420):

1. (2, 210)

2. (3, 140)

3. (4, 105)

4. (5, 84)

5. (6, 70)

6. (7, 60)

7. (10, 42)

8. (12, 35)

9. (14, 30)

10. (15, 28)

11. (20, 21)

It's good that somebody can count around here (I sure can't, and I guess JB is only slightly better than I am).

Still, that doesn't change the correct conclusion, which is

(first) the puzzle statement is totally consistent and has a correct answer and

(second)

yes, 420 is the key number.

I was concerned because I thought that 420 had too many divisors (I miscounted to something like 28!). Then jb_riddler corrected the count to 20, showing that 420 is still valid. Now you have accurately corrected the number to 22. But there's no change:

Statement 9 is true if the number of divisors is greater than the sum of the true statements. Since 22=22, the number is NOT greater, hence 9 can still be False.

So, it takes a village to solve a puzzle. jb_riddler, vinays84, and CaptainEd arrive before you, riddled corpse of a puzzle in hand.

We deposit it at the feet of HellFire, whose puzzles are clearly and accurately stated, and hard.

Edited March 18, 2009 by CaptainEd