Solutions
Back to life. Here is the solution to the Christmas puzzles:
1. The king said that one day somebody will find a sound proof that he hasn't always said the truth. Now either this is true or it isn't. If it isn't, the king hasn't always said the truth. If it is, somebody will find such a proof, and since the conlusion of any sound proof is true, again the king hasn't always said the truth. So in any case, the king hasn't always said the truth.
2. The king had uttered only two sentences. By the above argument we know that one of them must be false. But we also know that the first one was true: Somebody really found the requested argument. So the second sentence must have been the false one. It said that the person who finds the argument will get the kingdom. Hence it was logically impossible to give the kingdom to the court jester.