Number of True Statements

Which of the following statements are true?

  1. The number of true statements is 0 or 1 or 3.
  2. The number of true statements is 1 or 2 or 3.
  3. The number of true statements (excluding this one) is 0 or 1 or 3.
  4. The number of true statements (excluding this one) is 1 or 2 or 3.

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Statements #2 and #4 are true.

The “0” part of #1 can never be true, because it would contradict the statement itself being true. Temporarily rewrite the problem as:

  1. The number of true statements is 1 or 3.
  2. The number of true statements is 1 or 2 or 3.
  3. The number of true statements (excluding this one) is 0 or 1 or 3.
  4. The number of true statements (excluding this one) is 1 or 2 or 3.

But now #1 is just a subset of #2, so if #1 is true, #2 must also be true. That means the “1” part of both statements can never be true. Temporarily rewrite the problem:

  1. The number of true statements is 3.
  2. The number of true statements is 2 or 3.
  3. The number of true statements (excluding this one) is 0 or 1 or 3.
  4. The number of true statements (excluding this one) is 1 or 2 or 3.

The “3” part of #3 and #4 can never be true, because it would mean all 4 statements are true, which contradicts #1 and #2.

  1. The number of true statements is 3.
  2. The number of true statements is 2 or 3.
  3. The number of true statements (excluding this one) is 0 or 1.
  4. The number of true statements (excluding this one) is 1 or 2.

The “0” part of #3 can’t be true, because if #3 were the only true statement, #4 would also be true, which contradicts this part of #3.

Similarly, the “1” part of #3 can’t be true, because then both #2 would be true (2 total statements true) and #4 would be true (1 statement other #4 is true), which contradicts this part of #3.

We have ruled out every part of #3, so #3 is false.

Now, the remaining 3 statements don’t seem to contradict each other. However, if all 3 were true, then the original #3 (“The number of true statements (excluding this one) is 0 or 1 or 3.”) would also be true, which we already ruled out. This means there can’t be 3 true statements, which means #1 is false.

#2 and #4 don’t contradict each other, and the two of them being true does not make #1 or #3 true, so this is a valid solution.

However, to double-check that this solution is unique, we need to rule out the other possibilities. At the beginning, we already ruled out the possibility of 0 true statements (#1 would contradict itself). And if one of #2 or #4 were true, the other one would have to be true (since #1 and #3 are confirmed to be false). This means this is indeed the unique solution.

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