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This is the line of thought I followed:

Statement #3

is impossible because of #1 and #2 contradicting each other (let's consider only the last three statements, for simplicity). So, #3 must be false.

As a consequence,

#1 must be false.

If #4 were true, then #2 must be true (by exclusion), but this would imply that #4 itself is false. Then,

#4 is false.

If #5 were true,

then #2 must be false. So far, this holds.
If #5 were false, then #2, by exclusion, must be true. But this implies that #3 is true too, which is a contradiction, as seen above.
Then #5 is true, and #2 is false.

Accordingly,

#6 is false because it being true would imply that #5 is false.

In conclusion,

there is only one true statement, as said in the title, and is #5.

True statement is

5th statement

Reason

1 is false(only 5 is true)

2 is false(5 is true)

3 is false(both above are false)

4 is false(all are false)

6 is false(5is true)

Another nice way to approach this puzzle is by constructing chains of implications. We know there's only one true statement, so if one statement implies another one, then it's false.

• 
  

  Firstly, $3Rightarrow1Rightarrow6Rightarrow5$, so $3$ and $1$ and $6$ are false.

  
  




• 
  

  Since $3$ and $1$ are not true, $4Leftrightarrow2$, so they're both false.

  
  




• 
  

  The only option left is $5$, so this is the answer.

  
  

The correct one is

5

Explanation:

1 is not possible, as only one is true.

2 is not possible, as it makes 4 true.

3 is not possible for similar reasons.

4 is not true as it makes one of 1, 2 or 3 true as well.

6 is self-contradictory.

Same answer as everyone else, slightly different reasoning

1 must be false (if true then 6 would be true and contradict it).

=> 3 is false.

As a result, if 2 were true then 4 would also be true, contradicting "which one statement", so 2 is false.

=> 4 is false

Trivially 5 is true, 6 is false.

My Answer

Statement 5 is the true statement.

Explanation

If Statement 1 is true then Statement 6 is true; however, statements 1 and 6 cannot both be true as they are mutually contradictory; therefore, Statement 1 is false.

If Statement 1 is false then Statement 3 is also false.

If 2 is true then 4 would be false; However, if statement 4 is false then statement 2 is negated. It is logically impossible for statement 2 to be true.

If 1, 2, and 3 are false then 4 is also false.

If 1, 2, 3, and 4 are false then 5 is true.

Finally, if 5 is true then 6 is false.

Hence, statement 5 is the only true statement.

edit: you can also eliminate statements 1 and 3 immediately because they imply that more than one statement is true while the questions states that there is only one true statement.

I have used substitution to determine the only true statement.

Indeed, I started by writing the logic equivalents of each statement.

$$1 leftarrow 2 land 3 land 4 land 5 land 6$$
$$2 leftarrow lnot 3 land lnot 4 land lnot 5 land lnot 6$$
$$3 leftarrow 1 land 2$$
$$4 leftarrow (1 land lnot 2 land lnot 3) lor ( lnot 1 land 2 land lnot 3) lor ( lnot 1 land lnot 2 land 3)$$
$$5 leftarrow lnot 1 land lnot 2 land lnot 3 land lnot 4$$
$$6 leftarrow lnot 1 land lnot 2 land lnot 3 land lnot 4 land lnot 5$$

From this, a simple replacement in $6$ gives us

$$6 leftarrow 5 land lnot 5$$

Which really is just,

$$6 leftarrow F$$

From there, you simply substitute the result in the other equations

$$1 leftarrow 2 land 3 land 4 land 5 land F $$
$$2 leftarrow lnot 3 land lnot 4 land lnot 5 land lnot F $$

Which gives us

$$ 1 leftarrow F $$

Again, substitution...

$$3 leftarrow F land 2$$
$$4 leftarrow (F land lnot 2 land lnot 3) lor ( lnot F land 2 land lnot 3) lor ( lnot F land lnot 2 land 3)$$
$$5 leftarrow lnot F land lnot 2 land lnot 3 land lnot 4 $$

Simplifying to

$$1 leftarrow F $$
$$2 leftarrow lnot 4 land lnot 5 $$
$$3 leftarrow F $$
$$4 leftarrow (F) lor (2 land T) lor (F) $$
$$5 leftarrow T land lnot 2 land T land lnot 4 $$
$$6 leftarrow F $$

At this point, we can simply rewrite as:

$$1 leftarrow F $$
$$2 leftarrow lnot 4 land lnot 5 $$
$$3 leftarrow F $$
$$4 leftarrow 2 $$
$$5 leftarrow lnot 2 $$
$$6 leftarrow F $$

This gives us the satisfaction that, in fact,

$$ 2 leftarrow lnot 2 land lnot lnot 2 $$

Which is a contradiction, thence

$$ 2 leftarrow F $$

Giving us the solution

$$5 leftarrow T $$

Alright, here's my try. I think I have a fairly straightforward explanation.

1. All five statements below are true.
2. None of the four statements below are true.
3. Both of the statements above are true.
4. Exactly one of the three statements above is true.
5. None of the four statements above are true.
6. None of the five statements above are true.

We can instantly eliminate

Statements 1, 3, and 4.

Why?

Well, they all say that there is another true answer. The design of the question precludes this from being true - "Which one is the true statement?" (emphasis mine).

This leaves

Statements 2, 5, and 6. We need a way to make two of them false. Statement 6 cannot be the true statement - it would make statement 5 true, which would make statement 6 false. Statement 2 and 5 can be true, if the other is false. Ignoring previously eliminated statements, statement 2 says statement 5 is false. Statement 5 says statement 2 is false. However, if statement 2 were true, statement 3 is also true.

Thus, as a final answer,

Statement 5 would work.