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In an island there are 3 inhabitants, one of which is a knight (who always tells the truth) and the other two are Jokers who randomly decide whether to tell truth or lie. The 3 men have the numbers 1, 2 and 3 on their t-shirt. You need to find at least one person who can give you some information, but the problem is that you do not know who is who. You can ask use one question to one of them, which must have a number as an answer, in order to determine the knight. What question will you ask?

I have never dealt with any such type of problem. Obviously the question must be something related to the numbers they have on their t-shirt.

Any help will be appreciated :)

(this is not homework - I wish I were 17, even if I had to bear with tons of homework!! Unfortunately I am 64!)

Raahithya Vemulakonda
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2 Answers2

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How about this question:

If you have decided to tell the truth for this question, please answer with the number of the knight, and if you have decided to lie, please answer with the number of one of the jokers.

Truth tellers will respond to the first clause truthfully, giving you the number of the knight, and liars will respond to the second clause falsely, giving you the number of a someone who is not a joker.

As far as I can tell, this is a "legal" question. The question certainly needs to refer to the joker's decision to be truthful; otherwise, the Joker's two possible states would give different answers, meaning you cannot determine the truthfulness of their answer.

Mike Earnest
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  • In the end, whether a strategy like this works out depends on what a "lie" is. Is a lie just ignoring the question and answering whatever comes into the mind, e.g. a random number in ${1,2,3}$? If you give two options in your question like "please answer A or B", does he have to anwer with A or B, or can he still say C. Does this count as a lie if you cannot recognize that this is neither A nor B?. – M. Winter Apr 24 '19 at 07:28
  • @M.Winter Lie means “incorrect answer,” not “random answer”. Otherwise the puzzle would have no solution, as random answers convey no knowledge. – Mike Earnest Apr 24 '19 at 14:04
  • This does not address my point. The joker could answer "cheese" to your question, which is as incorrect as you can get, and you would get no information out of it. – M. Winter Apr 24 '19 at 14:14
  • @M.Winter If such answers are possible, then the puzzle has no solution, so I think it is implicit that Jokers only answer in the range ${1,2,3}$. Rigorously, the question is a function $Q$ whose inputs are the identity of the knight, $k$, and the decision of the speaker to lie or not, $d$, and whose output is a subset of ${1,2,3}$ containing one or two elements. When you give a person a question, they compute the function, then if they are truthful they respond with an element of $Q(k,d)$, and if they are lying they give an element of ${1,2,3}\setminus Q(k,d)$. – Mike Earnest Apr 24 '19 at 15:31
  • I think this is the only reasonable interpretation of the question, namely the only one for which the puzzle is soluble (and which avoids paradoxes and self-referential questions). – Mike Earnest Apr 24 '19 at 15:32
  • I would say that if $Q$ depends on $d$, then the question is self-referential. And it's clearly impossible to avoid this (otherwise Joker could imagine that he is Knight and answer as Knight would answer on his place). – mihaild Apr 24 '19 at 22:20
  • @MikeEarnest I agree that this is the only meaningful interpretation. I just feel unfomfortable that the question is self-referential. There is not much to do about that, but answers like this require a much more carefully stated riddle. – M. Winter Apr 25 '19 at 07:29
  • @All: I agree that Lie means incorrect answer and not random answer, just to confuse us. The answer must be a number but not necessarily in the range {1,2,3}. However, this was also my own interpretation. – Raahithya Vemulakonda Apr 25 '19 at 17:03
  • @M.Winter You seem to have a different conception of self-referential than me. My question only refers to the state of mind of the person being asked, i.e, whether they have decided to be truthful or lie. A self referential question is one which refers to its own answer, like “Will you answer no to this question?” – Mike Earnest Apr 25 '19 at 17:04
  • @mihaild Self-referential means that the question refers to itself, so my question is not self referential. $d$ is not related to my question, is related to a choice made in the mind of the person being asked. An example of self referential is “This sentence is false,” where the part in bold is the sentence referring to itself. The reason self referential things should be avoided is because they can lead to paradoxes, like a knight cannot answer the question “Will you answer no to this question?”. My question avoids such circular problems. – Mike Earnest Apr 25 '19 at 17:10
  • @RaahithyaVemulakonda My solution can be adapted to the situation where the jokers can answer with any number. Replace "answer with number of a joker" with "answer with any number except the number of the knight." – Mike Earnest Apr 25 '19 at 20:12
  • The "state of mind" is where self reference appear. What does it mean for an answer to a question to be true if correct answer to the question depends on your answer been true? – mihaild Apr 25 '19 at 21:58
  • @mihaild I take your point, but the way I see it, the joker flips a coin before answering and is truthful if it is head and lies if it is tails. I assert my question is referring to the flip of the coin, and not referring to itself. – Mike Earnest Apr 25 '19 at 22:02
  • @mihaild I challenge to write a question of the form $Q(k,d)$ which leads to some sort of self-referential paradox. – Mike Earnest Apr 25 '19 at 22:55
  • As you formulated it, the answer is a solution for $a \in_d Q(k, d)$, where $\in_\top = \in$ and $\in_\bot = \notin$, and your requirement $Q(k, d)$ been non-empty ensures such solution exists both if we fix $d = \top$ or $d = \bot$ (I think we can allow it to be ${1, 2, 3}$). What significant difference will it make if we ask a question of form $Q(k, d, a)$, ensuring that for any $k$ the equation $a \in Q(k, \top, a)$ is solvable (so everyone will be able to answer it)? – mihaild Apr 25 '19 at 23:06
  • @mihaild I agree that it is OK to allow self-referential questions if your are careful about it, as you described in your last comment. I just wanted it to be clear that my solution was not self-referential. – Mike Earnest Apr 25 '19 at 23:15
  • It depends on definition of self-reference. I would say that any question that depends on answer to it, truth value of answer and so on - anything but already fixed constants (the only one in this case is the number of knight) is self referential. Also, what do you mean by "self referential paradoxes" in this case? If it is possibility of somebody unable to answer then it can happens even if question doesn't depend on anything - like Knight can't answer question "name a number from $\varnothing$". – mihaild Apr 25 '19 at 23:19
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Assuming even Jokers answer in the way we can consistently assign truth values to their answers, we can ask, for example,

Let $u$ be the number of Knight. Let $v$ be the number you will answer to my question with. Let $x$ be defined as $$x = \begin{cases} u, & \text{if your answer to this question will be true}\\ v, & \text{if your answer to this question will be false} \end{cases}$$ What is $x$?

Assume person we ask will decide to provide true answer (they are Knight or Joker who answers true). Then $x$ is number of knight. As answer is true, the answer will be $x$ - ie number of knight.

Assume person we ask will decide to provide false answer, and their answer will be $v$. Then $x = v$, and their answer will be true, not false! So they can't consistently provide false answer.

So both Knight and Joker will answer to this question with number of Knight.

mihaild
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  • mihaild can you please explain your answer? I don't understand it. Or maybe to give an example? Thank you! – Raahithya Vemulakonda Apr 22 '19 at 05:06
  • Added a bit of explanation. If it's still too unclear - can you please point to the first unclear part? (is it what the question means? what happens if answer is true? what happens if answer is false?) – mihaild Apr 22 '19 at 12:42
  • Apologies but your answer doesn't make any sense. Either give a clear example with justification, or consider rephrasing it. – Sal.Cognato Apr 23 '19 at 12:15
  • @Sal.Cognato I believe that it should be possible to at least point to the first part / sentence of the answer that "doesn't make sense". Is it question itself? Is it what happens if answer is true? Is it what happens if answer is false? – mihaild Apr 23 '19 at 12:30
  • What does "and the number you will answer to more question otherwise" mean??? Please consider writing some examples. – Marius Stephant Apr 23 '19 at 17:10
  • Sorry, it was a typo: it should be "answer to my question". Rewrote it in hopefully more clear way. I am afraid I can't think about any example not already included in this reasoning - no matter whom we ask, they will answer with number of knight, and that's all. – mihaild Apr 23 '19 at 17:46