1) THREE GODS.  In a country long ago and far away there was a temple to which the people came to request the favor of the gods.  In the temple there were three identical statues, and each statue could speak to worshippers on behalf of the god it represented.  Though the three statues were identical in appearance, each represented a different god.  One statue spoke for the God of Truth, and it said was absolutely true.  Another spoke for the God of Falsehood, and what it said was always false.  The third spoke for the God of Diplomacy; it sometimes spoke the truth, and sometimes did not.

The statues would answer any questions the people asked them, but of course different statues would give different answers to the same questions!  Since no one knew which statue represented which god, interpreting the answers which the statues gave was pretty tricky, and there were a good many "religious experts" who claimed to be able to interpret the statues' answers and who charged high fees for their services.  It seemed to most people, however, that these experts usually disagreed with each other.

One day a logic student appeared at the temple and announced that she knew how to discover which statue represented which god; it could be done by simply asking each statue one question.  The answers would, she said, reveal which one answered truly, which one answered falsely, and which one was diplomatic.

So she entered the temple and stood before the three statues and asked the one on the left, "What god is (represented by the statue) standing next to you?"  The answer was: "The God of Truth."  Then to the statue in the middle he said: "Which god are you?"  The answer was, "The God of Diplomacy." And finally he said to the statue on the right, "What god is standing next to you?"  And the reply was, "The God of Falsehood."

"Aha!" said the logic student, "That makes it perfectly clear!"  She then went into business as a religious expert and soon made a fortune interpreting the answers which the statues gave to people's questions, because it soon became obvious that her interpretations were always correct.

Which statue represented which god?  Prove that you are right!
 

2) THE HATS.  Once upon a time there were three prisoners in a certain jail, all in the same cell. One had normal vision, one had one eye, and one was blind. The jailor, to amuse himself, tells the prisoners this:

In my office I have three red hats and two white hats. I'm going to blindfold each of you, go to my office and pick three of those five hats, and put one on each of your heads. The hats are small, so you won't be able to see what color hat is on your own head even without a blindfold. Then I'll remove the blindfolds and let each of you look at the hats on the heads of the other two. Any of you who can, by looking at the hats on the other's heads, definitely prove (not just guess!) what color hat is on your own head, can go free.

So the jailor did as he said. He blindfolded all three and put a hat from his office on each’s  head. Of course there was no way for the prisoners to know the colors of the hats remaining in the office. Then the jailor removed the blindfold from the prisoner with normal eyesight and asked if he could prove what color hat was on his own head. This prisoner, after thinking a bit, said:

 “No, from the hats I see, I would have to just take a guess at the color of mine."

Then the blindfold was removed from the prisoner with only one eye. He thought for a while, and then said exactly the same thing. The jailor, thinking that his little game was over, was about to leave when the blind prisoner said:

    "Wait a minute! I don't need to have my sight!
      From what my friends with eyes have said,
      I clearly see my hat is ________!"

So the blind prisoner thought that he was able to figure out for sure what color his hat was.  But could he really, or was he only bluffing?
 

3) THE ODD BALL.  Suppose you are given a dozen billiard balls, numbered 1 to 12.  They are all the same weight except for one - the odd ball.  It is a little different in weight, either a little lighter OR a little heavier than the others.  You are also given a simple balance.  Each side of the balance can hold up to six billiard balls.  All that the balance can tell you is which side is heavier.

You are to use the balance to find out which billiard ball is the odd ball, and  you are also to find out whether it is heavier or lighter than the rest.  What is the minimum number of times you need to use the balance in order to get the answer every time?  That is, the minimum number of weighings that is guaranteed to identify the odd ball and whether it is light or heavy, no matter which ball it happens to be?