The Hardest Logic Puzzle Ever - The Solution

The Solution

Boolos provided his solution in the same article in which he introduced the puzzle. Boolos states that the "first move is to find a god that you can be certain is not Random, and hence is either True or False". There are many different questions that will achieve this result. One strategy is to use complicated logical connectives in your questions (either biconditionals or some equivalent construction).

Boolos' question was to ask A:

Does da mean yes iff you are True iff B is Random?

Equivalently:

Are an odd number of the following statements true: you are False, da means yes, B is Random?

It was observed by Roberts (2001) and independently by Rabern and Rabern (2008) that the puzzle's solution can be simplified by using certain counterfactuals. The key to this solution is that, for any yes/no question Q, asking either True or False the question

If I asked you Q, would you say ja?

results in the answer ja if the truthful answer to Q is yes, and the answer da if the truthful answer to Q is no (Rabern and Rabern (2008) call this result the embedded question lemma). The reason it works can be seen by looking at the eight possible cases.

• Assume that ja means yes and da means no.

1. True is asked and responds with ja. Since he is telling the truth, the truthful answer to Q is ja, which means yes.
2. True is asked and responds with da. Since he is telling the truth, the truthful answer to Q is da, which means no.
3. False is asked and responds with ja. Since he is lying, it follows that if you asked him Q, he would instead answer da. He would be lying, so the truthful answer to Q is ja, which means yes.
4. False is asked and responds with da. Since he is lying, it follows that if you asked him Q, he would in fact answer ja. He would be lying, so the truthful answer to Q is da, which means no.

• Assume ja means no and da means yes.

1. True is asked and responds with ja. Since he is telling the truth, the truthful answer to Q is da, which means yes.
2. True is asked and responds with da. Since he is telling the truth, the truthful answer to Q is ja, which means no.
3. False is asked and responds with ja. Since he is lying, it follows that if you asked him Q, he would in fact answer ja. He would be lying, so the truthful answer to Q is da, which means yes.
4. False is asked and responds with da. Since he is lying, it follows that if you asked him Q, he would instead answer da. He would be lying, so the truthful answer to Q is ja, which means no.

Regardless of whether the asked god is lying or not and regardless of which word means yes and which no, you can determine if the truthful answer to Q is yes or no. If, however, the god is answering randomly, the answer to the Question is still without meaning.

The solution below constructs its three questions using the lemma described above.

Q1: Ask god B, "If I asked you 'Is A Random?', would you say ja?". If B answers ja, either B is Random (and is answering randomly), or B is not Random and the answer indicates that A is indeed Random. Either way, C is not Random. If B answers da, either B is Random (and is answering randomly), or B is not Random and the answer indicates that A is not Random. Either way, you know the identity of a god who is not Random.

Q2: Go to the god who was identified as not being Random by the previous question (either A or C), and ask him: "If I asked you 'Are you False?', would you say ja?". Since he is not Random, an answer of da indicates that he is True and an answer of ja indicates that he is False.

Q3: Ask the same god the question: "If I asked you 'Is B Random?', would you say ja?". If the answer is ja, B is Random; if the answer is da, the god you have not yet spoken to is Random. The remaining god can be identified by elimination.