Here are a couple of problems that some might have heard already.
1) There are two doors with one guard in front of each. Behind one of the doors is the right path, while the other one leads to certain death. There is no way for you to know which one leads to what path except by asking each of the two guards. However, you can only ask one question to one of the guards, and one of them says only truths and the other only lies. You also don't know who is the liar and who tells the truth. What would be the question to ask?
2) There are two cars owned respectively by two drivers. Both are in a race where the car reaching last place is actually the winner. The problem is that at the start of such a race, both drivers logically do not move their cars and the race cannot even start. Without changing the rules, what would be done in order for the race to get underway?
Feel free to post any interesting question type things. Ethical dilemmas count too. Here is one that comes to mind:
3) If you were in a situation where you have to chose between two people you love from falling from a cliff, but you can only save one, which one would you choose? This is one really tough.