Which of the following statements is true?

Question

i. One of these statements is true.
ii. Two of these statements are true.

iii. Three of these statements are true.
iv. Four of these statements are true.
v. Five of these statements are true.
vi. Six of these statements are true

Answer 1

Since these statements don't say "AT LEAST x of these statements are true", then you interpret these statements to say "EXACTLY x of these statements are true". In that light, the correct answer is (i).

Answer 2

If "exactly" is implied, all six statements contradict each other, so no more than one of them is true.
We have two possibilities, both of which are consistent:
a) All of the statements are false.
b) One of them is true, namely (i), and all the others are false.

If "at least" is implied, then the problem has seven solutions, namely {}, {i}, {i,ii}, {i,ii,iii}, {i,ii,iii,iv}, {i,ii,iii,iv,v} and {i,ii,iii,iv,v,vi} can all be the set of true statements. For example, you could say that only the first three statements are true and the last three are false and this would be consistent (it would indeed be true that one, two, and three statements are true, and that the statements saying that there are more than three true statements are false).

All this assumes that we have a non-contradictory set of statements, and that every statement is true or false. This isn't necessarily the case. For example, suppose you found three boxes on a beach. On each box is a message:

Box 1 says: "One of these three boxes contains 1000 gold pieces."

Box 2 says: "All three boxes are empty, and exactly two of the boxes have a true statement written on them."

Box 3 says: "All the boxes are lying."

Let's solve this problem. Box 3 must be lying because if it said the truth, it would says that it would be lying itself. Therefore Box 3 is lying, and Box 1 or Box 2 is telling the truth.

Is Box 2 telling the truth? If so, Box 1 is also telling the truth (says Box 2, 2 out of 3 are true) - not possible because Boxes 1 and 2 contradict each other.

So you arrive at the only logical conclusion that Box 1 says the truth, and the other two are lying. You open the boxes to get the 1000 gold pieces, and... they're all empty. You made the error of assuming that the set of statements was consistent, that there were no internal contradictions in the system.

Answer 3

Either none or 6. If you consider the first statement as false then necessarily they are all false. If you consider the first statement as true then they are all true (if 6 of these statement are true, then 1,2,3,4,5 of these statement is/are true). Every other statement is nothing else than a repetition of the first statement. Thus if 1 is true, then so is 2 and so is 3 and so on.

Answer 4

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Answer 5

one of the statements is true

Answer 6

all of the statements are true. each is a subset of the other.

a statement that one is true does not exclude the others or infer that the others are not true,just states that at least one is true. the statement that five are true does not infer that at least one is not true but rather that at least five are true.

following this logic all are true...

Answer 7

True

all statements

- - - - - s-

Answer 8

None...
means there is no true statement. so all are not true.

Answer 9

Any of the following are possible:
0. All false
1. i true, the rest false
2. i, ii true, the rest false
3. i, ii, iii true, the rest false
4. i, ii, iii, iv true, the rest false
5. i, ii, iii, iv, v true, vi false
6. All true

Answer 10

(1)i'm a boy,so true
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