Answer to #34: First she has to draw one of the papers and get rid of it (by eating it, throwing it, etc.). Then they will check the hat and see that the other paper says 'YES', so they know that she drew the paper that said 'NO'. In this case, she doesn't have to marry the king.

You stand at a fork in the road. Next to each of the two forks, there stands a guard. You know the following things: First that one path leads to paradise, the other leads to Death. You cannot distinguish between the two paths. You also know that one of the two guards always tells the truth and the other always lies. You have permission to ask one guard one question to discover which path leads to paradise. What one question would you ask to guarantee you take the path to paradise? [Slide for answer]