Solving The Hardest Logic Puzzle Ever and its generalizations

TL;DR

This paper introduces a systematic bottom-up approach to the Hardest Logic Puzzle Ever and its generalizations, providing solvability conditions, an efficient solution, and an algorithmic implementation.

Contribution

It offers a new bottom-up method, proves solvability criteria for any number of gods, and presents an algorithm with bounds for the generalized puzzle.

Findings

Puzzle is solvable if and only if random gods are fewer than non-random gods.

An average of 4.15 questions suffices for the 5-god case with 2 random and 3 lying gods.

An algorithm and implementation for solving the generalized puzzle are provided.

Abstract

Raymond Smullyan came up with a puzzle that George Boolos called The Hardest Logic Puzzle Ever.[1] The puzzle has truthful, lying, and random gods who answer yes or no questions with words that we don't know the meaning of. The challenge is to figure out which type each god is. The puzzle has attracted some general attention -- for example, one popular presentation of the puzzle has been viewed 10 million times.[2] Various "top-down" solutions to the puzzle have been developed.[1,3] We present a systematic bottom-up approach to the puzzle and its generalization. We prove that an n gods puzzle is solvable if and only if the random gods are less than the non-random gods, for arbitrary cardinals. We develop a solution using 4.15 questions on average to the 5 gods variant with 2 random and 3 lying gods. Finally, we introduce an algorithm and an implementation for finding solutions to the generalized problem, together with upper bounds.