The famous "The Hardest Logic Puzzle Ever", still unsolved?

[mathassistant]

Greetings! I’m an assistant of a mathematical scientific researcher, and my research programme evolves around finding and developing all the (possible) solutions regarding all unsolved mathematical, logic, exact, and IQ puzzles ever created. If you search on the internet for: “The hardest unsolved logic math/iq puzzle/problem ever possible”. You would find the well-known "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.

Furthermore, the solution on Wikipedia violates the rule that each question must be directed at only one God. In the proposed solution, the same god is asked the third question, which is not in accordance with the puzzle's requirements.

Considering the difficulty of this puzzle, I have a few questions for you. Given that “puzzle” is a puzzle related to:
- Math
- Logic
- Insight
- Strategic
- Tactic
- Intelligence
- Exact

1. Is it ever possible that 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.

Link: https://en.wikipedia.org/wiki/The_Harde ... uzzle_Ever

[Guest]

It seems you did not understand the solution (which seems correct).
To address your concern "[...] violates the rule that each question must be directed at only one God": This can be circumvented by using not the same, but an equivalent question.

Regardless, I find calling such a simple problem "The Hardest Logic Puzzle Ever" ridiculous. There are so many logic puzzles all over proof theory that are much harder.
For example, to find the smallest proof of theorem X in Hilbert system Y – most of which we will never be able to solve due to insufficient resources in our universe. [a few examples - see “[sample]” links next to system names]

[Guest]

Solution to the Three Gods Puzzle (True, False, Random):

Step-by-Step Approach
The key is to use self-referential questions to neutralize the language ambiguity (da/ja) and identify a non-Random god first.

Here’s the standard solution:

Question 1: Ask God A:
"If I asked you whether B is Random, would you say ‘ja’?"

Possible Outcomes:
1. A answers ‘ja’:
- Either:
- A is True → B is Random.
- A is False → B is not Random.
2. A answers ‘da’:
- Either:
- A is True → B is not Random.
- A is False → B is Random.

Question 2: Ask God A Again:
"If I asked you whether C is Random, would you say ‘ja’?"

-Analysis:
- Use the same logic as Question 1 to narrow down possibilities.
- Combine answers from Questions 1 and 2 to determine if A is Random or not.

Question 3: Ask the Identified Non-Random God (e.g., B or C):
"If I asked you whether you are True, would you say ‘ja’?"

Interpretation:
- If the god answers ‘ja’:
- They are True if ‘ja’ = yes, or False if ‘ja’ = no.
- If the god answers ‘da’:
- They are False if ‘da’ = yes, or True if ‘da’ = no.

Example Walkthrough
1. Assume A answers ‘ja’ to Question 1:
- Possibility 1: A is True → B is Random.
- Possibility 2: A is False → B is not Random.

2. Ask A Question 2:
- If A answers ‘ja’ again:
- If A is True → C is Random.
- If A is False → C is not Random.

3. Identify Non-Random God:
- If B and C cannot both be Random, the non-Random god is identified.

4. Final Question: Ask the non-Random god:
- Their answer reveals their identity and the meaning of da/ja.

Final Answer
By strategically using self-referential questions, you can deduce:
1. Which god is not Random.
2. The meaning of ‘da’ and ‘ja’.
3. The identities of all three gods.

Example Outcome:
- If A is False, B is not Random (e.g., True), and C is Random.
- If the non-Random god answers ‘ja’ to Question 3, they are True (‘ja’ = yes) or False (‘ja’ = no).

Conclusion:
This method systematically eliminates possibilities, leveraging the properties of truth-tellers and liars to decode the language and identities.