# Is there probability based Knight-Knave puzzles?

Have you seen Knight-Knave puzzles based on probability rather then facts?

For example a puzzle with condition like this: Knight tells the answer, probability to which according to his knowledge is large that 50%. Knave - the opposite. So to the question "Will a dice show 1 or 2?" knight answers "No" and Knave "Yes".

I understand that the same effect can be achieved with usual characters and questions like "Is probability to get 1 or 2 bigger than 50%?" and I would like to see a puzzle, which can not be solved without such a probability questions.

Can you figure out a non-trivial puzzle like this?

• This is still based on facts. Just facts about probability. As such, it would have the standard solutions. – frodoskywalker Oct 20 '14 at 14:51
• @frodoskywalker, This is still based on facts, correct. But I don't understand, which solutions to which puzzles are you talking about? And my question stays the same. – klm123 Oct 20 '14 at 15:20
• OK, your edit makes it clear what you're after. I meant that since these knights/knaves are functionally identical to regular ones, the set of puzzles and solutions are identical. I now realise you want a puzzle where such probabilistic questions are required to solve it. Maybe something with 1 knave, 1 knight and 1 that could be a knave or a knight? – frodoskywalker Oct 20 '14 at 15:49
• So, you want a puzzle where a probability-based question is needed to solve one of these problems... (like maybe it's needed to beat a question limit?) – d'alar'cop Oct 20 '14 at 19:27
• @d'alar'cop, yes. – klm123 Oct 20 '14 at 19:33

The best-known type of probability-based Knight-Knave puzzle is one where there are not only Knights and Knaves but also Jokers, who lie or tell the truth at random. A fairly standard example is your own Knight, Knave and Joker puzzle. More abnormal variants include COTO's Automatically a Knight, Knave, and Joker, my own Lies, damned lies, and statistics, and last but not least, Mike Earnest's brilliant Past, Present and Future.

For a puzzle where the questions you ask the Knights/Knaves/Jokers have to be probabilistic in nature, two excellent examples are Emrakul's You have one question to tell whether the number I'm thinking of is 1, 2, or 3 and Joe Z.'s extension Differentiate between the numbers from 1 to 5 with one single yes/no question.

Does this answer your question? (Even if it doesn't, it makes a nice collection of some fantastic Knight-Knave puzzles!)

Not that I know of... But it'd be easy to come up with that kind of thing.

"You arrive at an island one night where you find 2 types of people, intelligent, and people who often think randomly. You go up to one and ask them if you're thinking of a number between 1 and 3. How can you tell who's who after they tell you their guess?"

• I do not see how it is related to my question. Explain this explicitly, please. – klm123 Oct 20 '14 at 18:56
• Maybe I misunderstood your question, what is incorrect about my puzzle? – warspyking Oct 20 '14 at 19:06
• Incorrect? I told you - I do not see how it is related to my question. You have not explained it enough. – klm123 Oct 20 '14 at 19:08
• puzzling.stackexchange.com/questions/313/… - yeah this might the right track – d'alar'cop Oct 20 '14 at 19:42