*Originally posted by Thomaster*

**How many statements are true?
**

1. At least one of these statements is false.

2. At least two of these statements are false.

3. At least three of these statements are false.

...

999 998. At least 999 998 of these statements are false.

999 999. At least 999 999 of these statements are false.

1 000 000. At least 1 000 000 of these statements are false.

Using the same logic as above:

n = 1,000,000 - n

2n = 1,000,000

n = 500,000

Therefore, there are 500,000 true statements, and the first 500,000 statements are the true ones. I think a more interesting question is the following:

**How many statements are true?**
1. At least 1 of these statements is false.

2. At least 2 of these statements are false.

3. At least 3 of these statements are false.

Let's step through this case by case. First, let's assume there is 1 true statement. That means there are 2 false statements. Assigning a truth value to each statement based on this assumption, we have:

1. At least 1 of these statements is false. T

2. At least 2 of these statements are false. T

3. At least 3 of these statements are false. F

Summing up the T's and F's, we find that there is only 1 false statement and 2 true statements. This contradicts our original assumption of 1 true statement, and thus this cannot be the solution. If we assume there are 2 true statements and 1 false statement, we can repeat the process:

1. At least 1 of these statements is false. T

2. At least 2 of these statements are false. F

3. At least 3 of these statements are false. F

Summing up the T's and F's, we find that there are 2 false statements and only 1 true statement. This contradicts our original assumption of 2 true statements, and thus this cannot be the solution. If we assume all statements are true, then we can repeat the process:

1. At least 1 of these statements is false. F

2. At least 2 of these statements are false. F

3. At least 3 of these statements are false. F

Summing up the T's and F's, we find that there are 3 false statements and no true statements. This contradicts our original assumption of 3 true statements, and thus this cannot be the solution. So far none of our assumptions could possibly be right! What if we assume all statements are false?

1. At least 1 of these statements is false. T

2. At least 2 of these statements are false. T

3. At least 3 of these statements are false. T

Summing up the T's and F's, we find that there are 3 true statements and no false statements. This contradicts our original assumption of 3 false statements, and thus this cannot be the solution. So far, nothing is working. What does our previous logic have to say about this?

n = 3 - n

2n = 3

n = 1.5

And now it becomes clear. The first statement is true, the last statement is false, and the second statement is both true and false. Nonsense you say? You're just looking at it with the wrong frame of reference...this question is obviously political! 😉