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". The key to this solution is that, for any yes/no question Q, asking either True or False to the question: :If I asked you Q, would you say
ja? the answer
ja means the truthful answer to Q is
yes, and the answer
da means the truthful answer to Q is
no (Rabern and Rabern (2008) call this result the embedded question lemma). It also works with any other expression for yes and no - if the given expression is repeated, the Answer to Q is yes, and otherwise it is no. This can be seen by reasoning through all possible cases (the meaning of
ja and
da, which god we ask, and what their answer is): • Assume that
ja means
yes and
da means
no. • True is asked and responds with
ja. Since he is telling the truth, the truthful answer to Q is
ja, which means
yes. • True is asked and responds with
da. Since he is telling the truth, the truthful answer to Q is
da, which means
no. • 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. • 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. • True is asked and responds with
ja. Since he is telling the truth, the truthful answer to Q is
da, which means
yes. • True is asked and responds with
da. Since he is telling the truth, the truthful answer to Q is
ja, which means
no. • 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. • 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. The reason this works can also be seen by studying the
logical form of the expected answer to the question. This logical form (
Boolean expression) is developed below (
Q is true if the answer to Q is 'yes',
God is true if the god to whom the question is asked is acting as a truth-teller and
Ja is true if the meaning of
ja is 'yes'): • How a god would choose to answer Q is given by the negation of the
exclusive disjunction between
Q and
God (if the answer to Q and the nature of the god are opposite, the answer given by the god is bound to be 'no', while if they are the same, it is bound to be 'yes'): • ¬ ( Q ⊕ God) • Whether the answer given by the god would be
ja or not is given again by the negation of the exclusive disjunction between the previous result and
Ja • ¬ ( ( ¬ ( Q ⊕ God) ) ⊕ Ja ) • The result of step two gives the truthful answer to the question: '''If I ask you Q, would you say ja'? ''What would be the answer the God will give can be ascertained by using reasoning similar to that used in step 1 • ¬ ( ( ¬ ( ( ¬ ( Q ⊕ God) ) ⊕ Ja ) ) ⊕ God ) • Finally, to find out if this answer will be
ja or
da, (yet another) negation of the exclusive disjunction of
Ja with the result of step 3 will be required • ¬ ( ( ¬ ( ( ¬ ( ( ¬ ( Q ⊕ God) ) ⊕ Ja ) ) ⊕ God ) ) ⊕ Ja ) This final expression evaluates to true if the answer is
ja, and false otherwise. The eight cases are worked out below (1 represents true, and 0 false). Comparing the first and last columns makes it plain that the answer is
ja only when the answer to the question is 'yes'. The same results apply if the question asked were instead: '''If I asked you Q, would you say da'?
because the evaluation of the counterfactual does not depend superficially on meanings of ja
and da. '' 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. The solution below constructs its three questions using the lemma described above. This effectively extracts the truth-teller and liar personalities from Random and forces him to be only one of them. By doing so the puzzle becomes completely trivial: that is, truthful answers can be easily obtained. However, it assumes that Random has decided to lie or tell the truth prior to determining the correct answer to the question – something not stated by the puzzle or the clarifying remark. :Ask god A, "If I asked you 'Are you Random?' in your current mental state, would you say
ja?" :# If A answers
ja, A is Random: Ask god B, "If I asked you 'Are you True?', would you say
ja?" :#* If B answers
ja, B is True and C is False. :#* If B answers
da, B is False and C is True. In both cases, the puzzle is solved. :# If A answers
da, A is not Random: Ask god A, "If I asked you 'Are you True?', would you say
ja?" :#* If A answers
ja, A is True. :#* If A answers
da, A is False. :# Ask god A, "If I asked you 'Is B Random?', would you say
ja?" :#* If A answers
ja, B is Random, and C is the opposite of A. :#* If A answers
da, C is Random, and B is the opposite of A. One can elegantly obtain truthful answers in the course of solving the original problem as clarified by Boolos ("if the coin comes down heads, he speaks truly; if tails, falsely") without relying on any purportedly unstated assumptions, by making a further change to the question: : If I asked you Q,
and if you were answering as truthfully as you are answering this question, would you say
ja? Here, the only assumption is that Random,
in answering the question, is either answering truthfully ("speaks truthfully") OR is answering falsely ("speaks falsely") which are explicitly part of the clarifications of Boolos. The original unmodified problem (with Boolos' clarifications) in this way can be seen to be the "Hardest Logical Puzzle Ever" with the most elegant and uncomplicated looking solution. Rabern and Rabern (2008) suggest making an amendment to Boolos' original puzzle so that Random is actually random. The modification is to replace Boolos' third clarifying remark with the following: : Whether Random says
ja or
da should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he says
ja; if tails, he says
da With this modification, the puzzle's solution demands the more careful god-interrogation given at the top of
The Solution section. ==Unanswerable questions and exploding god-heads==