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Dipsy tells the truth in 1 day and lies in the next day alternately.
If today he tell the truth, tomorrow he will lying, 2 days later he will tell the truth, and so on.

Lala tells the truth in 2 days and lies in the next 2 days alternately.
If today, and tomorrow he tell the truth, 2 days and 3 days later he will lying, and so on.

Poo tells the truth in 4 days and lies in the next 4 days alternately.

You do not know which one is Dipsi, which one is Poo, and which one is Lala.
You only can ask a Yes/No question with only Yes/No answer.
Every question you ask, to each person cost 10 dollars.
Waiting for a day to ask, cost 2 dollars.
If you ask a non Yes/No question you will be fined 100 dollars, and you will not get any answer.

How many dollar minimum you have to prepare, before asking,
to identify, which one is Dipsi, which one is Poo, and which one is Lala.

Note :

I have my own answer, But I'm not sure whether my answer is the best answer or not.

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  • $\begingroup$ @ArbitraryKangaroo You do not know in what day you are. $\endgroup$ – Jamal Senjaya Oct 1 '16 at 6:46
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    $\begingroup$ Obviously, I don't know, otherwise the answer is trivial; but I'm saying if all of their count start from the same day (i.e Dipsy, Poo and Lala starts following the pattern from the same day ?) $\endgroup$ – user27395 Oct 1 '16 at 6:48
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    $\begingroup$ @humn yes you can $\endgroup$ – Jamal Senjaya Oct 1 '16 at 7:15
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    $\begingroup$ @JamalSenjaya As you've accepted an answer which is not the optimal solution (in fact all other current answers use fewer questions), perhaps the question needs editing (i.e., stricter conditions) if the other answers are not applicable? $\endgroup$ – Arkku Oct 1 '16 at 17:21
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    $\begingroup$ There's no 'maybe', @JamalSenjaya. The optimal amount is not given by the accepted answer, neither of your comments on the optimal answers are a refution of them, even if the comments were correct (and I'm convinced they're not). $\endgroup$ – Nij Oct 2 '16 at 5:14
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If the questions you ask can be somewhat "meta", one can ignore the time aspect entirely to reach a worst case of 3 questions asked on the same day (30 dollars):

1) Ask A: “Are you Lala, or lying today, but not both?”

If the answer is yes, A is Lala, move to question 3, else:

2) Ask A: “Are you Dipsi, or lying today, but not both?”

If the answer is yes, A is Dipsi, else they are Poo.

3) We now know who A is. Let N be one of the remaining identities.
Ask B: “Are you N, or lying today, but not both?”

If the answer is yes, B is N and C is the remaining one, else C is N and B is the remaining one.

Explanation

The question gives the correct answer about the identity regardless of whether they are lying or not, since the exclusive-or (either, but not both) inverts the answer for liars only (“lying today” is true). An alternative way to phrase the question would be “is exactly one of the following true: a) you are Lala, b) you are lying today”.

For telling the truth today:

  • Lala = no, lying = no -> neither is true, so "no"
  • Lala = yes, lying = no -> exactly one is true, so "yes"

For lying today, if they only lie about the whole statement:
  • Lala = no, lying = yes -> one is true but not both, so lying "no"
  • Lala = yes, lying = yes -> both are true, so lying "yes"

For lying today, if they lie about every part separately:
  • Lala = no, !Lala = yes, !lying = no -> one is true but not both, so lying "no"
  • Lala = yes, !Lala = no, !lying = no -> neither is true, so lying "yes"

Hence the question gets the true answer no matter whether they are lying today or not, and no matter whether they lie about every part of the question or only the answer as a whole.

Looking at it graphically, consider a classic Venn diagram;

the truth table for “A or B, but not both” looks like:

but if B is “Are you lying today?”, then the person’s answers in the B circle are lies (shown in red):

so the answer to the compound/meta question will be “Yes” if and only if A is true, regardless of whether the person is lying.

The worst case of 3 questions is also optimal for yes/no questions, since there are 6 permutations of identities and each question can extract only one bit of information. Using a “no answer” question where the lack of answer is a distinct case from yes/no could theoretically help, but due to the $100 fine it is not worthwhile in this context.

Waiting for tomorrow is also not worthwhile since it conveys no information without asking a question, and it's cheaper to ask the questions without waiting.

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  • $\begingroup$ If today lala is lying, Lala can not answer "are you lying today", because a liar will never say I am lying. It will be a paradox. $\endgroup$ – Jamal Senjaya Oct 1 '16 at 18:00
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    $\begingroup$ @JamalSenjaya See edited answer for expanded explanation. It is important to note that there is no paradox in asking a liar if they are lying today: they will simply answer "no" (i.e., lie about lying). And the question that is being asked, is not even "are you lying today", but a compound statement with that as one part, so they never admit to being a liar. In fact, the answer does not tell us whether they are lying! (edit: removed spoilers from comment) $\endgroup$ – Arkku Oct 1 '16 at 22:29
  • $\begingroup$ (1) Clever?  Yes.  Perhaps too clever.  (2) Original?  AFAICT.  (3) As long as you’re going meta, why not stick with the standard “If I were to ask you ‘Are you Lala?’ would you answer ‘Yes’?” $\endgroup$ – Peregrine Rook Oct 3 '16 at 16:06
  • $\begingroup$ @PeregrineRook I personally don't like the "If I were to ask you" construct so much, and in this particular setting you would need to qualify it with "If I were to ask you today". $\endgroup$ – Arkku Oct 3 '16 at 17:46
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For an initial attempt:

40 dollars

Ask the first one (A) "Are you Dipsy? Are you you lala? Are you Poo?" They will have 2 answers the same and one different. The odd one out is the true answer. Ask A about the second person ("is this lala?). You know from your first set of questions whether A is currently a liar, and have B's identity narrowed to 2 possibilities, so 1 question will suffice. By elimination you know who C is.

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    $\begingroup$ You should probably move the questions into a spoiler block, and move the cost out (i.e., make it readily visible). $\endgroup$ – Peregrine Rook Oct 1 '16 at 21:16
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answer:

3 questions, $30 total, which is the minimum possible, since the number of possible configurations is 3! = 6, and ceiling(log(base 2, 6)) = 3

method:

you can always get the truth with nested question, e.g: 'if I asked you if 1+1 =2, would you say yes?'. This will result in 'yes' whether the person is a liar or truth teller at the time. Given that there are 6 possible orderings, you can ask the first person whether the correct ordering is one of the first 3 of these possibilities with a nested question: 'If I asked you whether the three people are either dipsi, poo and lala respectively, or dipsi, lala and poo respectively, or poo, lala and dipsi respectively, would you say yes?'. If the answer is yes, then you continue in the same vein: 'If I asked you whether the three people are dipsi, poo and lala respectively, would you say yes?'. If the answer is then no, you ask the final question: 'If I asked you whether the 3 people are dipsi, lala and poo respectively, would you say yes?'. Ask corresponding questions for the other cases.

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  • $\begingroup$ Numbers of possible configuratoms are 8, not 6. $\endgroup$ – Jamal Senjaya Oct 1 '16 at 17:46
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    $\begingroup$ How so? The only way I can see there being more than 6 configurations is if offsets are considered, e.g. is Dipsy lying today and truthful tomorrow or vice-versa, etc. But in that case it would be more than 8 $\endgroup$ – astralfenix Oct 1 '16 at 18:10
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    $\begingroup$ @JamalSenjaya There are only 6 configurations of names (3×2×1), which is what the question is asking about. If we needed to figure out also whether each is telling the truth on a given day, there would be 6×2×2×2 = 48 configurations, and if we needed to figure out the specific day in their cycle, I believe there would be 6×2×4×8 = 384 configurations. (That being said, since 2³ = 8, if there were 8 possible configurations, three questions would still be the optimal solution.) $\endgroup$ – Arkku Oct 1 '16 at 20:34
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The answer is

40 Dollars

Let the persons be labelled A,B and C
First question for "B"
I randomly choose B, and ask "Are you a Dog".
Any person who tells the truth will definitely say "NO". So from the above result we came to know that whether B is telling truth.
Assume B answered "NO". From this we conclude that B is telling truth today. (If "B" says YES, i will take opposite of what B says from now).
Second question for "B"
Are you dipsi?
Third question for "B"
Are you lala?
For the second and third question(maximum questions required here = only two), we can conclude that "B" is either dipsi, lala or polo.
Assume "B" is conformed as polo.
Fourth question for "B"
I pointed "A" and ask whether he is dipsi?
From this answer, we can conclude that "A" is dipsi or lala. This also conforms whether "C" is dipsi or lala. so we need atleast 4 questions that would cost minimum of 40 dollars to find who is Dipsy, lala or polo

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  • $\begingroup$ @Jamal Senjaya Is my answer ok? $\endgroup$ – KSR Oct 2 '16 at 9:25
  • $\begingroup$ yes, but I realized now, we just need 30 dollars. $\endgroup$ – Jamal Senjaya Oct 2 '16 at 14:08
  • $\begingroup$ I just saw above answer.Need only 30 dollars.. $\endgroup$ – KSR Oct 2 '16 at 14:17
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To be told the truth you need to ask:

If I asked you today if you are Lala/Poo/Dipsi, would you answer yes?

This will tell you the name of the first person with at most 2 questions, or just one if you're lucky.

Then ask another person just one more question to know who they are, and who the third person is.

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