Logical confusion

Question

This time, there are 20 questions, no bold or strike or anything. Just the puzzle. In #12, another way of saying it is that 10 statements are true.

If you don't want to do that, provide a concrete true or false answer for every problem in the bonus.

I am also wondering if cream cheese is cheese. Please tell.

Also, I like strawberries. But that doesn't count for the problems.

Bonus

A. Statement B is true.

B. Statement A is false.

The actual problem

Level: easy-medium

1. 7 and 20 are true, and 2 and 13 are false.

2. I like blueberries.

3. 17 and 15 are true.

4. 1 is false, and 2 is true.

5. If 2 is false, 3 is false. Also, 7 is true.

6. I like strawberries.

7. 13 and 5 are false.

8. 1 is true, and 2 is false.

9. 19 and 4 are not true.

10. If 12 is true, 20 is true, and 9 is false.

11. If 20 is true, 7 is false.

12. Half of the statements are true.

13. Either 7 is false, 8 is false, or 9 is false.

14. Statement 14 is true.

15. Of statements 1,2,3,4,5,6, and 7, four of them are false.

16. 4 is false, and 14 is true.

17. 3 is true, and 13 is false.

18. The first battery was invented in 1800

19. Somehow, 14,3,9,7,18, and 4 are all true.

20. 12 is true. 5+3 is 8.

Answer 1

True

1,3,7,8,9,12,15,16,17,18,20

False

2,4,5,6,9,10,11,13,14,16,19

Starting with the premise that

1 is true

We can assign

7 & 20 to true, 2 & 13 to False

Next because

20 is true, 12 is true, and we're looking for 10 true and ten false. Because 7 is also true, we confirm that 13 is false, and assign 5 to false. Because 2 is false, 8 must be true. We'll skip 13 for now.

With our new info

That 5 is false, we determine 3 must be true. Because 3 is true, so are 17 and 15. Fifteen tells us that 4 and 6 are false, because only three of the first seven can be true.

A quick Google search tells us that 18

is true

So far we've got

True: 1,3,7,8,12,15,17,18,20 (9/10) and False: 2,4,5,6,13 (5/10)

Because only one more can be true, let's scan through for some easy eliminations.

11 is pretty clearly false at this point, as is 19. Leaving us with 9, 10, 14, and 16.

16

Can't be true, because if it is true, then 14 is also true, which brings us to 11/10. 16 is false.

9

Oh wait, look at this, it's true! This must be our last one. Hurrah!

Answer 2

I am only gonna answer the bonus part because the actual problem is unclear. Here goes :

We have two statements :
A. Statement B is true.
B. Statement A is false.

We only have 4 possibilities :

Possibility #1

Both statements are correct
If statement A is correct, then B should be true --> If Statement B is true, then A should be false.
Hence, this will just be an endless cycle and so we move on to #2.

Possibility #2

Statement A is correct while Statement B is not.
This will make statement B true while statement A will become also correct --> These 2 statements then contradict one another and so, we move on to possibility #3

Possibility #3

Statement B is correct while A is not.
This will make statement A false while B is also false. --> These 2 statements also contradict one another and so, we move on to possibility #4

Possibility #4

Both statements are false.
This will make statement B false while A is true --> Both statements contradict each other again.

Conclusion :

There is no answer (impossible) to the bonus part as this is called a liar's dilemma