"The Hardest Logic Puzzle Ever"

“The 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”.

A simple solution to the hardest logic puzzle ever, © Brian Rabern & Landon Rabern, Analysis 68 (2), April 2008, pp. 105–112.

Also, Why the hardest logic puzzle ever cannot be solved in less than three questions, Gregory Wheeler & Pedro Barahona

There’s one other referenced paper, How to solve the hardest logic puzzle ever in two questions, Gabriel Uzquiano, Analysis, 70 (1), 39-44, that’s behind paywalls but is anonymously posted here http://mesosyn.com/mental1-5b3.pdf

(oh, The apparent paradox is resolved I think.)


T F ?
T ? F
F ? T

F F ?
T T ?

double false would be true and likewise for double true for there being a random or false

You should a link to the statistical definition of random to help people solve on there own before checking. Its required

Ref: New Oxford American Dictionary
"random - 1. made, done, happening, or chosen without method or conscious decision…• Statistics - governed by or involving equal chances for each item".

“whether Random speaks truly or falsely is a completely random matter”

Here “random” means exactly what is commonly understood about an experiment such as a coin toss: the coin is fair (will land heads or tails with equal chances) and the result of one toss has no effect on the result of subsequent tosses.

To be more precise and unnecessarily confusing, Random’s answers to questions are discrete, independent, identically (and uniformly) distributed random variables.

Chapter 2: Axioms of Probability, Néhémy Lim

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At least 2 different categories of randomness are considered in the paper, and the first ‘coin-toss’ model leads to trivialisation of the question by removing randomness. It’s done by a non-trivial trick around that particular randomness definition, i.e. a state of random machine, which has to be referred to in the question! Which is by no means natural or commonly understood notion of randomness (unless you are a programmer).

The whole considerations in paper lead to a simple (trivial?) observation, that the ‘Hardest Logic Puzzle Ever’ needs to be rephrased, to get the ‘spirit’ of the puzzle right and prevent the trick of removing randomness out of it.

Thanks for sharing the links! How did you ran across them?

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…almost at rundom. (gotcha :wink:)

I logged out of e-mail which dumps onto a MSN page with a news and nonsense slide show that had this story from Reader’s Digest https://www.msn.com/en-us/lifestyle/whats-hot/an-mit-professor-called-this-the-hardest-puzzle-ever-can-you-solve-it/ar-BBHoJ9. which includes a link to the Raberns’ paper at the end. mafiadoc helpfully suggested Wheeler & Barahona. Note: The RD story says “The Harvard Business Review may have published this logic puzzle back in 1996”. In fact, George Boolos, The hardest logic puzzle ever. The Harvard Review of Philosophy, 1996.

That’s cool.

One more thought I had - it’s tempting to think that the ‘coin-toss’ model is the only representation of randomness. It’s however just a specific representation of randomness, or it’s implementation that is practical and follows probability and consequently statistical principles. In complex enough, chaotic systems, randomness can be achieved without a notion of a ‘state’ as a side-effect or noise coming from the system. There is also one system to which it was proven that no underlying material state exists - photon spin and quantum effects - that is completely random and nicely follows probability and statistical principles.

The point is - the trick from original paper was rather cheap one :wink: