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The most difficult puzzle in the world

The most difficult puzzle in the world

Three gods: Apollo, Bacchus, and Calliope are called True, False, and Random although we don't know who is who. The True God always tells the truth, the False God always lies and the response of the Random God may be true or false. We must determine the identities of Apollo, Bacchus and Calliope, that is, find out which is the True God, which is the False and which is the Random, asking three questions whose only possible answers are Yes or Do not. Each question must be asked to a single god and although the gods understand our language, they answer in their own language in which the words for Yes Y Do not They are Gives Y Ha, but we don't know which one corresponds to each one.

We can ask the same god more than one question and the addressee of the second and third questions may depend on the answer received to the first and second questions respectively.

You can get more information about this riddle on Wikipedia. Watch out!, the link also includes the solution.

Solution

The first question to ask is to find a god who is certain that it is not Random and therefore is either True or False. There are several possible questions. For example, we can ask Apollo the following:

Is that Gives it means Yes yes and only if you are True and yes and only if Bacchus is random?

This amounts to asking if an odd number of the following statements are true: you are False, Gives it means Yes, Bacchus is Random.

In general, for any question Q with possible answer Yes รณ Do not, by asking the True God or the False God the following:

If I ask you Q, is that you would answer Ha? God would answer us Ha if the false answer to the question Q it is Yes and answer us Gives yes the real answer to the question Q it is Do not.

We can analyze the eight possible cases to verify it:

If we assume that Ha it means Yes and that Gives it means Do not:

  • If we ask the True God and answer with Ha, since it always tells the truth, the true answer to Q would Hawhat would it mean Yes.
  • If we ask the True God and answer with Gives, since it always tells the truth, the true answer to Q would Giveswhat would it mean Do not.
  • If we ask the god False and answer with Ha, since he always lies, then we deduce that if we asked Q and answer Gives, I would be lying, so the true answer to 'Q' would be Hawhat would it mean Yes.
  • If we ask the god False and answer with Gives, since he always lies, then we deduce that if we asked Q he would answer Ha I would be lying, so the true answer to Q would be Giveswhat would it mean Do not.

If we assume instead that Ha it means Do not and that Gives it means Yes:

  • If we ask the True God and answer with Ha, since it always tells the truth, the true answer to Q would Giveswhat would it mean Yes.
  • If we ask the True God and answer with Gives, since it always tells the truth, the true answer to Q would Hawhat would it mean Do not.
  • If we ask the god False and answer with Ha, since he always lies then we deduce that if we asked Q and answer Ha, I would be lying, so the true answer to 'Q' would be given, which would mean Yes.
  • If we ask the god False and answer with Gives, since he always lies then we deduce that if we asked Q and answer Gives, I would be lying, so the true answer to 'Q' would be Hawhat would it mean Do not.

Based on this analysis we ask the following questions:

We ask the god Bacchus the following: "If I asked you, is it Apollo Random? Would you answer Ha?โ€.

If Bacchus responds Ha, then Either Bacchus is the Random God or Apollo is the Random God. In either case, Calliope would not be Random.

If Bacchus responds Gives, then Either Bacchus is the Random God or Apollo is not the Random God. In either case, Apollo would not be Random.

Now, we turn to the god we have identified as not Random from the answer to the previous question and ask him the following: "If I asked you, are you the True God? Would you answer Ha?โ€.

Since we know that he is not the Random God, if he answered Ha it would mean that it is the true god and if he answered Gives it would be the false god.

We would ask that same god now the following: "If I ask you, is it Random Bacchus? Your answer would be Ha?โ€.

If the answer is Ha then Bacchus is Random, if the answer is Gives So the god we haven't asked yet is Random. The god that remains can be identified by elimination.