## February 23, 2010

### Find the correct statement

A sheet of paper has statements numbered from 1 to 100. Statement
N says "Exactly N of the statements on this sheet are false."

How many statements are true?

Tiger Woods said...

Only one statement is true. Statement 99

Yaffar said...

100-N

Rahul said...
This comment has been removed by the author.
Rahul said...

Tiger Woods is very correct

Anonymous said...

why..? can explain..?

Rahul said...

In these 100 statements one and only one statement can be correct. That means 99 statements have to be incorrect and thus statement 99th is true.

Zac said...

Is only 1 true because it only tells you one of the statements?

Ali said...

so the way I see it is that if statement N is true you will have any number of combinations that add up to 100:

it can be statement 1, and any other one statement between 2-100 will be false.

it can be statement 2, and any other 2 statements between 1,3-100 will be false.
.
.
.
and it can be up to statement 99, which must be true for the other 99 statements to be false.

however if statement N is false then you have no way to verify how many is true or false.

Varun Rao said...

100-N

Anonymous said...

if statement- N is true.. no included in N-no of statements tat are false, result would be except 100- N no of statements.
if it is false then 100- statement-N

Ashwin Kumar said...

I think you're getting it wrong.
What tiger woods said is right.
Let me explain.

If statement one is right, then only one statement is false.
Thus, the remaining 98 have to be true.
Let's say the second one was the false one, then the third one is true. which means that 3 statements are false, contradicting the first statement.
Similarly, it can be seen that if statement 2 is true, 97 statements will be true, and then it will be contradicted by any of the remaining ones.

Now let us see the 98th one. If it is true, then 2 and only 2 statements are true.
Now, if the 99th statement is the second true statement, then It will mean that only one is true, which is again contradicting statement 98.

So, frankly, I think only the 99th statement can be true. It will imply that only one statement is true, and it will be the 99th one itself.

Hence Proved!