The Hardest Logic Puzzle Ever - Unanswerable Questions and Exploding God-heads

Unanswerable Questions and Exploding God-heads

In A simple solution to the hardest logic puzzle ever, B. Rabern and L. Rabern develop the puzzle further by pointing out that it is not the case that ja and da are the only possible answers a god can give. It is also possible for a god to be unable to answer at all. For example, if the question "Are you going to answer this question with the word that means no in your language?" is put to True, he cannot answer truthfully. (The paper represents this as his head exploding, "...they are infallible gods! They have but one recourse – their heads explode.") Allowing the "exploding head" case gives yet another solution of the puzzle and introduces the possibility of solving the puzzle (modified and original) in just two questions rather than three. In support of a two-question solution to the puzzle, the authors solve a similar simpler puzzle using just two questions.

Three gods A, B, and C are called, in some order, Zephyr, Eurus, and Aeolus. The gods always speak truly. Your task is to determine the identities of A, B, and C by asking yes-no questions; each question must be put to exactly one god. The gods understand English and will answer in English.

Note that this puzzle is trivially solved with three questions. Furthermore, to solve the puzzle in two questions, the following lemma is proved.

Tempered Liar Lemma. If we ask A "Is it the case that { OR (B is Eurus)}?", a response of 'yes' indicates that B is Eurus, a response of 'no' indicates that B is Aeolus, and an exploding head indicates that B is Zephyr. Hence we can determine the identity of B in one question.

Using this lemma it is simple to solve the puzzle in two questions. Rabern and Rabern (2008) use a similar trick (tempering the liar's paradox) to solve the original puzzle in just two questions. In "How to solve the hardest logic puzzle ever in two questions" G. Uzquiano uses these techniques to provide a two question solution to the amended puzzle. Two question solutions to both the original and amended puzzle take advantage of the fact that some gods have an inability to answer certain questions. Neither True nor False can provide an answer to the following question.

Would you answer the same as Random would to the question 'Is Dushanbe in Kirghizia?'?

Since the amended Random answers in a truly random manner, neither True nor False can predict whether Random would answer ja or da to the question of whether Dushanbe is in Kirghizia. Given this ignorance they will be unable to tell the truth or lie – they will therefore remain silent. Random, however, who spouts random nonsense, will have no problem spouting off either ja or da. Uzquiano (2010) exploits this asymmetry to provide a two question solution to the modified puzzle. Yet, one might assume that the gods have an "oracular ability to predict Random's answers even before the coin flip in Random’s brain?" In this case, a two question solution is still available by using self‐referential questions of the style employed in Rabern and Rabern (2008).

Would you answer ja to the question of whether you would answer da to this question?

Here again neither True nor False are able to answer this question given their commitments of truth-telling and lying, respectively. They are forced to answer ja just in case the answer they are committed to give is da and this they cannot do. Just as before they will suffer a head explosion. In contrast, Random will mindlessly spout his nonsense and randomly answer ja or da. Uzquiano (2010) also uses this asymmetry to provide a two question solution to the modified puzzle. However, Uzquiano's own modification to the puzzle, which eliminates this asymmetry by allowing Random to either answer "ja", "da", or remain silent, cannot be solved in less than three questions.