# Thread: At most how many of the following statements are true?

1. ## At most how many of the following statements are true?

At most how many of the following statements are true?

1: Exactly one of statements 2 or 3 is true.
2: Exactly one of statements 3 or 4 is true.
.
.
.
998: Exactly one of statements 999 or 1000 is true.
999: Exactly one of statements 1000 or 1 is true.
1000: Exactly one of statements 1 or 2 is true.

I already know the solution to the problem; just giving this out as a fun little exercise.

2. ## Re: At most how many of the following statements are true?

None.

Each statement is the subject of the previous two statements and asserts the truth of exactly one of the two subsequent statements.

Let's assign a true value to a statement (let's say #1), then one of the subsequent statements must also be true.

If 2 is true, then 3 must be false (or 1 would be false); if 3 is false, then 4 is true (or 2 would be false); if 4 is true, then 5 is true (or 3 would be true); etc. IF there were only 999 statements this pattern would be okay (giving us 666 true statements):

$\frac{1}{T} \frac{2}{T} \frac{3}{F} \frac{4}{T} \frac{5}{T} ... \frac{997}{T} \frac{998}{T} \frac{999}{F} \frac{1000}{?}$

Unfortuantely, that 1000th statement (which would be true or 998 would be false) would require that 1 or 2 could be true (but not both).

If 2 is false, then 3 would have to be true (or 1 would be false). In order for 3 to be true and 2 to be false, 4 also has to be true - this is the same pattern as before, which doesn't work.

[Sorry about the multiple edits, I'm having issues submitting and keep losing my responses]

Yep

4. ## Re: At most how many of the following statements are true?

Excellent question ... I like this forum!