Gdddk

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.

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)

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.

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.

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.