1. One side of a card reads, "Statement on the other side is true". The other side reads, "Statement on the other side is false". Which one is saying the truth? How?

2. Both sides read, "Statement on this side is false or the statement on the other side is true". which side is saying the truth? How?

1. Both sides are false. If the first side is true then by the statement on the first side, the second side is true, and so by the statement on the second side the first side is false. Contradiction. Thus, the first side is false, and so the negation of the first side is true. The negation of the first side is that the second side is false. Therefore both sides are false.

2. Both sides are true. If either side is false, we have a contradiction to the statement on that side (assuming we are using the inclusive "or" as is the norm in math/logic). Conversely, if both sides are true, then there is no contradiction.

p.s. Have you ever read any of Ramond Smullyan's books? I'd recommend them if you like logic puzzles like this one.

1. From the first part of your answer I got that the first side is false. This is exactly what the second side is saying. So the second side must be true. Ain't it?

1. Oops! I should have said that there is no answer (or rather that no truth value can be consistently assigned to the cards), rather than both sides are false. The first part of my answer shows that the first side is false. But if the first side is false, then the second side is false (the negation of the statement on the first side), so the first side is true (negation of the statement on the second side). Contradiction.

2. Xor is exclusive or, I was assuming inclusive or (one or the other or both). However, looking at it again, I believe that it doesn't matter which "or" is intended. The answer is the same regardless.