A logical treatment of noise: Solving The Hardest Logic Puzzle Ever and its generalizations
Pith reviewed 2026-05-24 12:33 UTC · model grok-4.3
The pith
An n-gods puzzle is solvable exactly when fewer gods answer at random than answer consistently, for any cardinal.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
An n gods puzzle is solvable if and only if the random gods are less than the non-random gods, for arbitrary cardinals. The proof proceeds by constructing explicit questioning strategies from below, checking consistency of answer sequences against the possible type assignments.
What carries the argument
The strict cardinality inequality between the set of random gods and the set of non-random gods, which decides whether a questioning strategy can distinguish all type assignments.
If this is right
- Solutions exist for every finite or infinite case obeying the random < non-random rule.
- The five-god instance with two random and three lying gods admits a strategy averaging 4.15 questions.
- An algorithm computes solutions and produces explicit upper bounds on the number of questions for any given n.
Where Pith is reading between the lines
- The same bottom-up enumeration of answer sequences may apply to variants in which random answers carry limited dependence on prior questions.
- The cardinality threshold supplies a clean dividing line that could be tested in related identification problems with mixed reliable and unreliable sources.
- Implementation of the algorithm permits systematic search for minimal-question solutions beyond the cases already solved by hand.
Load-bearing premise
Random gods answer yes or no completely at random, independently of the question and of all earlier answers.
What would settle it
An explicit counter-example: either a collection with strictly fewer random gods for which no finite questioning strategy works, or a collection with at least as many random gods for which a strategy does work.
read the original 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. We introduce an algorithm and an implementation for finding solutions to the generalized problem, together with upper bounds. Finally, we note that random gods act like noisy sources, which provides a connection to fault-tolerant computing.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript presents a bottom-up approach to the Hardest Logic Puzzle Ever (truthful, lying, and random gods with unknown yes/no lexicon) and its n-god generalizations. It claims to prove that such a puzzle is solvable if and only if the cardinality of random gods is strictly less than the cardinality of non-random gods, for arbitrary cardinals. It also reports an average of 4.15 questions for the 5-god case with 2 random and 3 lying gods, and introduces an algorithm together with an implementation and upper bounds.
Significance. If the if-and-only-if characterization holds, the result supplies a complete solvability criterion that extends prior top-down finite-case solutions to arbitrary cardinals, constituting a notable theoretical contribution. The bottom-up construction, the explicit average-question count for the 5-god instance, and the provision of a reproducible algorithm with implementation are additional strengths that support practical verification and extension.
minor comments (1)
- The abstract cites references [1], [2], and [3] but the manuscript should ensure the bibliography is complete and formatted consistently with journal style.
Simulated Author's Rebuttal
We thank the referee for their positive assessment, recognition of the bottom-up approach extending to arbitrary cardinals, and recommendation to accept. No major comments were raised requiring point-by-point rebuttal.
Circularity Check
No significant circularity
full rationale
The paper presents an independent bottom-up construction for the n-gods puzzle and proves an if-and-only-if solvability condition in terms of cardinal comparison |random| < |non-random|. The random-answerer model is the standard external one referenced in the abstract and prior literature; the central theorem is not justified by self-citation chains, fitted inputs renamed as predictions, or self-definitional reductions. The reported average-question counts for finite cases follow directly from the model without circular reduction to inputs. The derivation is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Gods answer yes/no questions using two unknown words whose meanings must also be discovered; random gods answer randomly.
discussion (0)
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