Multiple Choice Question (Logic)

BigBeachBanana

BigBeachBanana

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Which answer in this list is the correct answer to this question?
  1. All of the below
  2. None of the below
  3. All of the above
  4. One of the above
  5. None of the above
  6. None of the above
Note: "below or "above" means below or above that statement. For example, in statement 5 ", none of the above" refers to statements 1,2,3,4. Whereas statement 6 refers to statements 1,2,3,4, and 5.
 
I'll give this a try.
Suppose 6 is correct, that is none of the statements above 6 is true. But then 5 contradicts that. So 6 is out.

Suppose 5 is correct, that is none of the statements above 5 is correct. 1 contradicts that.

Suppose 4 is correct, that is one of the above is correct. Now 2 contradicts 4, so 2 is out. 1 and 2 together contradicts 3, so 3 is out. 2 and 3 together contradicts 1, so 1 is out. In the end 4 is not correct.

Suppose 3 is correct, that is above 3 is correct. 2 contradicts that.

Suppose 2 is correct, that is all the statements below 2 are false. 6 contradicts that.

Suppose 1 is not correct.
 
If 1 is correct. then both 2 and 5 are correct. If 5 is correct, then 2 is not correct, which makes 1 incorrect.1 is correct leads to contradiction. 1 is not correct.

If 2 is correct, then 3 is not correct (because of 1) and 4 is correct, which makes 2 incorrect. 2 is correct leads to contradiction. 2 is not correct.

Given 1 and 2 are not correct, 3 is not correct.

Given 1, 2, and 3 are incorrect, 4 is not correct.

Given 1, 2, 3, and 4 are incorrect 5 is correct.

Given 5 is correct, 6 is incorrect.

5 is the correct answer.
 
JeffM said:
If 1 is correct. then both 2 and 5 are correct. If 5 is correct, then 2 is not correct, which makes 1 incorrect.1 is correct leads to contradiction. 1 is not correct.

If 2 is correct, then 3 is not correct (because of 1) and 4 is correct, which makes 2 incorrect. 2 is correct leads to contradiction. 2 is not correct.

Given 1 and 2 are not correct, 3 is not correct.

Given 1, 2, and 3 are incorrect, 4 is not correct.

Given 1, 2, 3, and 4 are incorrect 5 is correct.

Given 5 is correct, 6 is incorrect.

5 is the correct answer.
Click to expand...
You got it.
 
This is the solution by the author:

  • If 1 is true, that implies 5 is true. But if 5 is true, then none of the above can be true, meaning 1 has to be false. This means 1 is a self-contradicting statement and cannot be true.
  • If 3 is true, that implies 1 is true. But we already concluded 1 is self-contradictory and cannot be true. 3 cannot be true.
  • If 2 is true, that implies 4 has to be false.
  • If 4 is false, since 1 and 3 are false, then we must have that 2 is false. Thus, 2 is a self-contradictory statement and cannot be true.
  • If 4 is true, then 1, 2, or 3 is true. But we already concluded 1, 2, and 3 cannot be true, so 4 has to be false.
  • If 6 is true, then 5 has to be false, implying some statement 1, 2, 3, or 4 has to be true. But we know 1, 2, 3 and 4 are false, so we cannot have 6 as a true statement.
  • If 5 is true, we need 1, 2, 3, and 4 to be false, which we already worked out.
Therefore the answer is 5.
 
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