Logic Problem

[DougOzzzz]

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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.


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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.

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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.

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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.