Logic Problem

[DougOzzzz]

Posted this in OOT a few days ago without any interest (did get 1 possible (I think incorrect) answer, but no explanation from DBowling).

Anyways, here's the problem:

Below are ten statements concerning X, a whole number between 1 and 10
(inclusive). Each statement may or may not be true. What number is X?

1. X equals the sum of the statement numbers of the false statements in
this list.

2. X is less than the number of false statements in this list, and
statement 10 is true.

3. There are exactly three true statements in this list, or statement 1
is false, but not both.

4. The previous three statements are all false, or statement 9 is true,
or both.

5. Either X is odd, or statement 7 is true, but not both.

6. Exactly two of the odd-numbered statements are false.

7. X is the number of a true statement.

8. The even-numbered statements are either all true or all false.

9. X equals three times the statement number of the first true statement
in this list, or statement 4 is false, or both.

10. X is even, or statement 6 is true, or both.

[DougOzzzz]

I have a solution to the problem... but will let everyone else have a shot at it.

The original wording was "Not all of the statements are true, but not all of them are false either." When I looked at the statements it did not take me long to prove that they logically could not all be true and logically could not all be false, either.

That threw me off for a while, because for some reason I thought that was a key piece of information and that it was somehow a trick question. That's why I reworded it here...

[Jazza]

[ QUOTE ]
Each statement may or may not be true

[/ QUOTE ]

intuitivly i think you have to say each statement is either true or false

i'm guessing this whole thing revolves around statements like:

this statement is false

are these kind of statements allowed? your original wording seems to imply these statements are allowed, where as my wording i reckon says these statements are not allowed

....what i am saying may or may not be true /images/graemlins/tongue.gif

[DougOzzzz]

Yes, that's why the original wording threw me off so much.

I thought that the fact that NOT all the statements are true, and NOT all the statements are false, would eliminate 9 of the 10 possible numbers (i.e. they would all have to be true or all have to be false for all incorrect answers).

However - a better way to look at the problem is not to guess the actual number, but to try to evaluate each statement. You'll see alot of contradictions arise; if this is true then this has to be false; if A and B are true, then C has to be false, but then D has to be true, and E has to be false, but that makes A false - thus this combination of T/F's does not make sense. Just giving some examples... from my analysis if you look at it carefully enough, only one combination of true/false assignments makes sense logically - and the number is derived from those statements.

[Jazza]

[ QUOTE ]
thus this combination of T/F's does not make sense

[/ QUOTE ]

ah ok, i'm on the same wave length now, so each statement has to be either strictly true or strictly false yeah?

[DougOzzzz]

[ QUOTE ]
[ QUOTE ]
thus this combination of T/F's does not make sense

[/ QUOTE ]

ah ok, i'm on the same wave length now, so each statement has to be either strictly true or strictly false yeah?

[/ QUOTE ]

Yeah, something like that. You're not allowed to have something be true and false at the same time, if that's what you're saying...

Certain combinations don't work logically. For example, statement 1 and 2 both can't be true. Statement 8 clearly indicates that at least one of statements 2, 4, 6, or 10 are true (if they are all false, then if 8 is false, 8 would be true, but then 8 would be false, but then 8 would be true.... etc.)

[txag007]

Results below:

























X=9

Statements:
1. F
2. F
3. T
4. T
5. F
6. T
7. T
8. F
9. T
10. T

[DougOzzzz]

that's the answer I got. Care to explain? Your explanation is probably better than mine.

[DougOzzzz]

By the way, nice job....

My solution is based on the fact that 4 must be true (because of 9). For instance, if 4 is false, then 9 is true. But then 4 must be true also.

After that the rest follows.

[txag007]

Several of the statements contradict. The two most obvious (at least to me) were statements 9 and 4. They either both must be true or both false. Assume one or the other and see where it takes you. I assumed true. From there work through each statement logically until something fits. Once you evaluate one and two as being false and three as true, statement nine gives the value of X. From there, work through each statement and make sure nothing eliminates the number 9 as a possible answer.

That's how I did it. It actually takes a bit of trial and error. For example, if you assume statement one is true, then X=3 according to statement nine. But this will be eliminated as a possible answer when you work through the rest of the statements.

[DougOzzzz]

[ QUOTE ]
Several of the statements contradict. The two most obvious (at least to me) were statements 9 and 4. They either both must be true or both false.


[/ QUOTE ]
Actually, that's not correct. 4 MUST be true. 9 could be either true or false. They can't both be false, because if 4 is false, 9 is true. However, if you assume 4 is true, and 9 is false, then 1, 2, and 3 must be false.

It follows that statement 8 is false (since 2 is false and 4 is true). It follows that 9 is false (1, 2, 3 are false, 4 is true, and X cannot be 12). 8 also must be false (since 2 is false, and 4 is true). 6 then must be false (since 1, 3 and 9 are all false - more than 2 odd #'s).

Now, we have 1 true statement and 6 false statements. The status of statements 5, 7 and 10 are undetermined. However, given #3 is false, and #1 is false, there must be exactly 3 true statements. That means that 2 out of the 3 undetermined statements must be true.

There are 3 possibilities here - 5 = T, 7 = T, 10 = F; 5 = T, 7 = F, 10 = T; 5 = F, 7 = T, 10 = T .

If 5 is true, and 7 is true, then X cannot be odd. But that makes 10 true - which is not allowed, since that makes 3 true. If 5 is true, and 7 is false, then X must be odd. But if X is odd, then 10 must be false - it has to be true to meet the requirements of 3 being false. If 7 is true, and 10 is true, then X is even. But that makes 5 true - once again failing to meet the requirements of 3 being false.

Therefore, there is no possible combination with 1, 2, and 3 all being false that meets the logical requirements of all the other statements.

[ QUOTE ]

Assume one or the other and see where it takes you. I assumed true. From there work through each statement logically until something fits. Once you evaluate one and two as being false and three as true, statement nine gives the value of X. From there, work through each statement and make sure nothing eliminates the number 9 as a possible answer.

That's how I did it. It actually takes a bit of trial and error. For example, if you assume statement one is true, then X=3 according to statement nine. But this will be eliminated as a possible answer when you work through the rest of the statements.

[/ QUOTE ]

All in all, pretty much how I did it, even though your initial assumption was incorrect (that 4 and 9 must both be true or both be false). Clearly, everything starts with these 2 statements though. You can eventually determine that they must both be true, thus at least 1 of statements 1, 2, or 3 are true and X is either 3, 6, or 9. The rest is just a process of elimination.

[txag007]

You're correct about the statements 4 and 9. My initial reaction to your original post was that 4 and 9 could not be opposites. You're right, though. They cannot both be false. So the only logical solution is that they both are true which therefore defines the equation for which to determine X.

We worked the problem the same way. There was just one less assumption needed on my part.