Looking for answers on 3 questions regarding the hardest logic puzzle ever

Question

Greetings! I’m an assistant of a mathematical scientific researcher, and my research programme evolves around finding/developing all the (possible) solutions regarding all unsolved mathematical, logic, exact, and IQ puzzles and problems ever created. If you search on the internet for: “The hardest unsolved logic math/iq puzzle/problem ever possible”, or any query which looks alike for that matter. You would find the well-known "The Hardest Logic Puzzle Ever" (https://en.wikipedia.org/wiki/The_Hardest_Logic_Puzzle_Ever). I would like to gather some of your thoughts around this puzzle.

Quote:

This puzzle involves three gods, A, B, and C, who are named True, False, and Random. True always speaks truly, False always speaks falsely, and Random's responses are completely random. The goal is to determine the identities of A, B, and C by asking three yes-no questions, with each question directed at only one god. The gods respond in their own language, where the words for yes and no are da and ja, in some order, and we do not know which word corresponds to which answer.

End quote.

The proposed solution on Wikipedia assumes that one of the gods must answer a factual question truthfully, leading to the conclusion that "ja" corresponds to "yes" and "da" corresponds to "no." However, this assumption is not valid within the constraints of the puzzle, as Random's responses are completely random, and there is no guarantee that a factual question will elicit a truthful response.

In order to make this puzzle even harder, a restriction could be imposed on it, such that you can only ask one question per God. Meaning that that it would still be an unsolved puzzle.

Considering the difficulty of this puzzle, I have a few questions for you. Given that “puzzle” is a related to:

- Mathematics

- Logic (boolean algebra)

- Insight

- Strategy

- Tactics

- Intelligence

- Exact

1. Does a harder, unsolved puzzle compared to "The Hardest Logic Puzzle Ever" exists? If so, what makes it more challenging?

2. Is there a definitive solution to "The Hardest Logic Puzzle Ever"? Are there any alternative valid solutions? Because based on all our research, the “solutions” available are all the same type (which are all false because of violations of the rules or assumptions).

3. If there is a solution, can a valid truth table be constructed to represent the possible answers of the gods and their identities?

I would greatly appreciate your insights and any additional information you can provide regarding the puzzle. Your contributions will aid our ongoing research into unsolved mathematical, logical, and IQ puzzles. Thank you in advance.

Answer 1

This is a cool problem, and it _is_, indeed, quite hard! I hadn't seen it before.

If we look at the primary solution offered in the Wikipedia article, it does not have the fatal flaw you think it does. The solution does assume, as you say, that "one of the gods must answer a factual question truthfully". That's fine, as that is part of the statement of the problem: "True always speaks truly", and we know that True is one of the gods. But nowhere in the rest of the solution is there the conclusion or assumption that "ja" corresponds to "yes" and "da" corresponds to "no." It's an intricate and complicated solution, and so it can be hard to follow, and I totally empathize with perhaps misreading it.

I don't think I can write up in this space another presentation of that solution that's better or clearer than the one in the Wikipedia article to help show that it's never assigning a specific meaning for "ja" or "da". But if you'd like to schedule a session with me, I'm happy to walk through the solution together and show you how it works and address questions.

I can't really help with questions about whether there are harder, unsolved puzzles. There are plenty of unsolved, interesting mathematical questions out there, but I'm not sure they're puzzles, per se. You might be interested in Hilbert's problems, if you're not already considering those: https://en.wikipedia.org/wiki/Hilbert%27s_problems

Your questions 2 and 3 are basically answered by the Wikipedia article, if you accept that it actually does solve the puzzle: it's a valid solution, and it even has a truth table!