A simple solution to the hardest logic puzzle ever

Abstract

We present the simplest solution ever to 'the hardest logic puzzle ever'. We then modify the puzzle to make it even harder and give a simple solution to the modified puzzle. The final sections investigate exploding god-heads and a two-question solution to the original puzzle. The simplest solution to the 'hard' puzzleThe puzzle. Three gods A, B, and C are called, in some order, 'True', 'False', and 'Random'. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for 'yes' and 'no' are 'da' and 'ja', in some order. You do not know which word means which. 1

A Framework for Riddles about Truth that do not involve Self-Reference

“…Left to the reader. [4] call the "Embedded Question Lemma". In natural language, Proposition 3 says that we can reveal the truth-value of σ by addressing the following question to a brother:…”

“…Whenever we ask a question Q to Random, he flips a coin and then, depending on the outcome of the coin-flip, he answers Q truly or falsely. Elaborating on suggestions in [4], we think of the coin-flip as determining the mental state in which Random answers Q. The mental state, on its turn, determines whether Random answers Q correctly or falsely.…”

“…The essential feature of the two-question solution of Rabern and Rabern for HLP E omn sem is given by their Tempered Liar Lemma ( [4], p. 110). The lemma can be stated as follows.…”

“…We only find Tim Roberts [5] coming up with an alternative solution for HLP E, criticizing Boolos' solution as being 'unnecessarily complicated'. Recently, Rabern and Rabern [4] interestingly observed an ambiguity in Boolos' instruction for HLP E concerning the behavior of Random. They observed that there is a genuine difference between the following two courses of action that may be followed by Random upon being addressed some question Q.…”

How to solve the hardest logic puzzle ever in two questions

“…Whether Random answers 'da' or 'ja' should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he answers 'ja'; if tails, 'da'. 1 Rabern and Rabern (2008) provide a clever two-question solution to the original puzzle, which exploits the fact that there are both self-referential questions that no truth-teller can answer and self-referential questions that no liar can answer. So, if we assume that all of A, B and C set out to speak the truth or lie, albeit randomly in one case, then we can glean more information from their inability to answer certain self-referential yes-no questions.…”