A class just starts, and the classmates are asked to do an experiment together. In one whole week, everyone must choose a role as either a truth-teller or a liar consistently.
On the first day, a student called "Doggie" met 3 classmates (A, B, and C) sitting on the terrace together.
Doggie asked A: Are you a truth-teller or a liar?
A answered with his back turned.
Doggie did not hear clearly A's answer, so Doggie asked B: What did he say?
B answered: A says that he is a truth-teller.
C then immediately said: B is lying!
Doggie was confused. In his heart, he wondered: Is C a truth-teller or a liar?
In fact, is B a truth-teller or liar?
What about A?
Can anyone helps Doggie?
[The above puzzle is extracted from a funny and interesting Maths. puzzle-solving book- Averbach and Chein: Problem Solving Through Recreational Mathematics, Dover. There are hints given which Doggie have not read. Doggie tend to believe the authors' hints perhaps may not be sufficient enough to cover and explain the situations and paradoxes. ]
在我看來,整個Puzzle Setting的開始已經是錯誤,再沒有討論的必要.核心問題是,你的目的是要問A,但A不回答或你聽不清楚,其實到這裏已經完結了.為什麼要去問B,C,D,E....呢?他們都不是A,而且你原意根本不是要去問B,C,D,E...., A才是能給你答案的人.而且用心理學來說,A turned his back 己經是給你答案了.
回覆刪除Thanks, Chris. So "assuming your opinion is 100% correct", then as you said... if A turned his back already implied that A had given his answer to Doggie, then what answer should it be? Please kindly advise. Thanks.
回覆刪除This question can have more than one version. I can think of 3:
回覆刪除Version 1:
Doggie asked A: Are you a truth-teller or a liar? Doggie may be asking A whether (A is a truth-teller) or( A is a liar). If this is the case, then Doggie would expect A to answer “A is a X”(X: truth-teller or liar). However, no matter A is a truth-teller or a liar, A would only answer “A is a truth-teller”. I suppose all you guys can understand why. Then B, by answering Doggie that “A says he is a truth-teller”, is a truth-teller. C, by claiming that “B is lying”, is a liar. Whether “A is a truth-teller” or “A is a liar” is not yet known.
Version 2:
Doggie asked A: Are you a truth-teller or a liar? Maybe Doggie was actually asking A: Are you (a truth-teller OR a liar)? If this is the case, then Doggie would expect a Yes/No answer from A. Since (a truth-teller OR a liar) must be true, A would answer “Yes” if A is a truth-teller. What is interesting is that if A is a liar, then A could offer nothing but a paradoxical answer: For (a truth-teller OR a liar) must be true, A as a liar has to answer “No”. By giving such an answer, however, A already admits that A is a liar and thus a member of (a truth-teller OR a liar). No matter what, Doggie would not expect an answer from A of the form “A is a X”(X: truth-teller or liar). As such, B, by claiming that “A says he is a truth-teller”, is a liar. C, by pointing out that “B is lying”, is a truth-teller.
Version 3:
It is not known whether the student Doggie is Doggie-self one of the classmates. If Doggie is one of the classmates having chosen a role as a liar, then perhaps Doggie's question is nothing but a lie, making HWan, who has been so truly answering Doggie's question, a stupid fool.
明顯地 文章中的Doggie (不是會友Doggiedog)問了A，一條已知答案的問題。
回覆刪除如果A是老實人，他的答案也必定是 I am a truth-teller
但如果A是說謊者，他的答案也必定是 I am a truth-teller
所以Doggie是無法從問題知道A的身份。他只能知道B 及C 而已
PART (I)
回覆刪除Thanks for the advice. I still have not read any of the authors’ hints. I write in response to feedback herein on my understanding of the problem with my own limited knowledge.
(1) We may have to ask first: What are the objectives and goals of this experiment? Are the students informed of such objectives and goals? Are there control tests?
(2) How big is the sampling size? Is it supposed to be finite (N)?
(3) Is there absolute freedom to choose (a) to act AND (b) not to act?
(4) The context of (3) (a) is NOT the same as the context of (3) (b), and therefore the impact and implication on the proportions of truth-tellers (T) and liars (L) in (N) are different. In fact, those who do not choose to act (if any), what will be their role? Are they going to be ambivalent (A), playful (P), or simply erratic (E)? They may not recognize themselves as ignoramus (I) despite their indifference and if they regard truth as an impossible, then truth-tellers to them are just liars too! Let us call these “non-participants (in the heart)” (if any) skeptics (S) who never announce their non-participation.
(5) No matter what further conclusions we may come up with our concern raised under (4) above, we still should agree that:
Under (3) (a):
T + L = N
Under (3) (b):
T + L + S = N
Now we may wonder the real intention of Doggie in asking A his question:
Are you a truth-teller (T) or a liar (L)?
(a) If Doggie is a dumb-dumb, Doggie may not know that a (T) will always answer the same as a (L) that both will answer: “Yes, I am a T.” (Answer “a”) or “No, I am not a L.” (Answer “b”) Therefore Doggie might really want to hear clearly an answer from A, even though an answer from A actually cannot be helpful or meaningful anyway.
(b) If there is such an answer: “Oh yes, I am a liar.” (Answer “c”), is such an answer meaningful and helpful? No way! Doggie could be even more confused. How can one trust a liar’s confession?!
(c) Let us first of all do not make it complicated by emphasizing “or” as “OR” (as H Wan has tried to propose), we can come up with the conclusion that there can be only three answers “a”, “b”, and “c” to Doggie’s STUPID question. No more. All 3 answers help nobody.
(to be continued)
PART (II)
回覆刪除If we follow H Wan’s assumption that the question may be interpreted as:
Are you a truth-teller OR a liar? [H Wan’s assumed situation] That you (A) are either a truth-teller OR a liar, I do know about it, right? So A, please reconfirm by saying yes, okay?
Then as explained under (4) in PART (I)
In context (3) (a), all are either T or L, that T + L = N
Then person A or any person (Pn) must answer “Yes” as an absolute truth. If there is such a person (Px) who dares say “No”, such a person (Px) must be a liar (L).
However, if in context (3) (b), an answer “Yes” or “No” helps nobody.
PART (III)
In Scenario under PART (I), whether A’s answer is heard clearly or not is irrelevant to making a conclusion that A should be a truth-teller (T) or a liar (L). As a liar CAN NEVER FUNCTION and PERFORM as a liar (L) once “L is exposed”, a liar L will always answer: “Yes, I am a truth-teller (T).” (Answer “a”) Therefore A’s answer must be Answer “a” AND B could easily reaffirm such a fact, being Answer “a” WITHOUT even truly hearing clearly from A. Hence B can NEVER be a liar (L) by saying that A says that he is a truth-teller. Hence C is liar by claiming that B is lying. The said judgment made herein is subject to the presumption that there is no crazy liar who is willing to make a confession as Answer “c”. Nobody can tell whether A should be T or L.
PART (IV)
In Scenario of context (3) (b) under PART (II), a “no” answer is possible from non-participants (S). It does not necessarily imply a confession from a liar (L). A “yes” answer from S is also possible if such S just want to make fun with the situation being “outsiders” (non-participants). If B does not hear clearly what A has said, then in this context (3) (b), his speculation that “A says he is a truth-teller” may be presumptuous as there is quite a good chance that A may be a S (non-participant) who never cares about his answer’s implication to anyone doing the experiment. Then B may possibly become a liar AND C a truth-teller. BUT how can C know whether B has heard clearly from A or not? Even if C has heard clearly, it does not mean B cannot have heard very clearly from A! That implies C’s accusation can also be subjective or speculative, in a sense, just a lie!
PART (V)-
回覆刪除More advanced Learning:
Can there be a continuous act of lying?
In other words, can C’s accusation be valid that “B is lying”?
Like an act of murder, an act of telling a lie must be successfully completed without failure otherwise “there is no murder or lie”. There has been only an attempted murder or attempted lie. Therefore a successful murder or lie implies a completed act of murder or lie in the past only accomplishing an irrevocable change- someone was killed or someone was cheated. Such a crime actor, a murderer or a liar, once being found and discovered, cannot kill or lie anymore supposedly when everyone becomes alert to their acts.
Hence it is funny when C said: B is lying.
If B is engaged in his continuous act of lying but at the same time being found by C, can B still committing his act of lying without stop, if any? If impossible, how can B still lie about anything? C could only say: B lied. True or not, such past act could be proven or denied with evidence. If C keeps saying: B is lying, then the fact can only be C is lying!
Can one act as a truth-teller? If one is truthful to oneself, is one a truth-teller? Are there truths? If no, or there is only faithful belief, or only falsifiable scientific truth, how can there be genuine truth-telling? If really not possible, then all supposedly truth-tellers are really just actors. On the other hand if they are just actors, whatever they say can only be stage-talks, no truth at all. Then doesn’t it mean all such volunteer truth-tellers who choose to act as truth-tellers are simply all liars? Then does it not mean that whatever roles the students try to choose (a T or L) these students are all liars by agreeing to participate in such experiment? Then does it mean that in context (3) (b) those “non-participants (in the heart only” (S) who do not choose to act may be probably the only truthful group but some are trying to be truthful + some do not mind lying?
In other words, only the non-voluntary guinea pigs which never realize their survival period is the period of experiments are truthful beings without lying while all voluntary participants who choose to pick up a role as assigned in an experiment are prepared to act on stage-talks implying truth-telling on the role of truth-tellers is just a conceit!
The authors' hint and guided solution touch on the same typical answers between truth-teller and liar as we already know about, and because a liar will never say "I am a liar" and so "A says A is a truth-teller" is expected by B and it should be unlikely going to be wrong. That means there should be no chance that B should be lying and so C's claim that B is lying must be wrong, and C is lying instead.
回覆刪除