# 100 Wrong Statements puzzle

Puzzle: There are 100 statements.
1st one says : at least one is wrong.
2nd one says : at least two are wrong.
3rd one says : at least three are wrong.
4th one says : at least four are wrong.
and so on.
100th one says : at least 100 are wrong.

How many statements are actually wrong and how many actually right ?

Solution:

100th statement is definitely wrong because it says at least 100 are wrong.
But if that is correct, then 100 statement itself cannot be right.
=> 100th statement is wrong and
=> 1st statement is correct.

99th statement cannot be correct because if it were correct,
then two statements would become correct (1st and 99th itself.)

But 99th statement says that atleast 99 are wrong.
=> 99th is wrong and
=> 2nd is correct.

calculating so on…
50 statements are right (the first 50 ones)
remaining 50 statements are wrong.

### 5 Thoughts on “100 Wrong Statements puzzle”

1. errr…what is the statement?

2. Nitin Gupta on April 28, 2015 at 12:08 pm said:

Question not clear, Please share more details

3. harshitha on June 23, 2015 at 7:30 pm said:

Question not clear

4. Nice explanation…!!!

5. Question is that, in those 100 statements, not all are right. So they wanted to know how many are right and how many wrong.

Consider from 100th statement, which says at least 100 are wrong. That means, according to 100th statement, none of them are correct including itself, hence 100th statement cannot be correct at all. This conclusion clears the path for 1st statement, which tells us that, at least one is wrong, in this case we concluded that 100th one is definitely wrong.
Result: 100th is wrong, and 1st is correct, Total: 1 Correct, and 1 wrong

Moving to 99th statement, which says, at least 99 are wrong, means according to it 1 among 100 is correct, which we already concluded that 1st is correct based on above conclusion. If it is so, then 99th cannot be correct, because if we conclude 99th is correct, then we will get total of 2 correct so far, which is conflicting with 99th statement. Therefore, so far, 100 and 99th statements are not correct, which makes us conclude that 2nd is correct which says 2 are wrong (as we concluded 100 and 99).

If we continue like this, it will go on till 51st statement, where we will end up totalling 50 correct and 50 wrong statements.

Hope it clears.