May 21st, 2017, 02:59 PM  #21 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,655 Thanks: 740 
Sigh...I quit.

May 22nd, 2017, 02:40 AM  #22 
Global Moderator Joined: Dec 2006 Posts: 18,568 Thanks: 1483  Any question could be rephrased so that it's not the same question, but has equivalent meaning. More important, though is that actually asking only one question (which must be answered with "yes" or "no") implies that the answer given is the first answer given, and so can be "yes" or "no". Any hypothetical questions considered prior to answering do not give rise to actual answers, so they don't force the answer to the one question asked to be truthful.

May 22nd, 2017, 09:43 AM  #23  
Newbie Joined: May 2017 From: England Posts: 13 Thanks: 0  Quote:
To make this simpler I'll label a yes and no from each question with a number like this: yes/01, no/01 Q1) I ask the doorkeeper: is the treasure in door A. If the answer is yes_01 and it's true, then the treasure is in door A otherwise if it's a lie, then the treasure is in door B. If the answer is no_1 and it's true, then the treasure is in door B otherwise if it's a lie, then the treasure is in door A. A question that considers a hypothetical response would be like this: Q2) If I asked you Q1 and you say yes_01 would yes_01 be a lie? In this case, if the doorkeeper responds yes_02 and it's true, then yes_01 was a lie which we already said means that the treasure is in door B otherwise if yes_02 is a lie, then yes_01 is true and the treasure is in door A. If the doorkeeper responds no_2 and it's true, then yes_01 is true and the treasure is in door A. However, if the doorkeeper responds no_2 and it's a lie, then yes_01 is a lie, but this would be impossible as the doorkeeper can only lie once. This still doesn't eliminate all the options,so we need to ask a better question. This will be the best question. Q3) If I asked you Q2 and you say yes_02 would yes_02 be a lie? If the doorkeeper responds yes_03 and it's true, then yes_02 is a lie which means yes_01 is true and thus the treasure is in door A. However, if the doorkeeper says yes_03 and it's a lie, then yes_02 is the truth which means yes_01 is a lie, but this means that the doorkeeper would lie more than once, so this is impossible. If the doorkeeper instead says no_3 and no_3 is true, then yes_2 is true and thus yes_01 is a lie, so the treasure is in door B. However, if the doorkeeper says no_3 and no_3 is a lie, then yes_2 is a lie but this is impossible as the doorkeeper can not lie more than once. Therefore, Q3 can give you the right answer. I think this works because a false yes is the same as a true no and a false no is the same as a true yes. So if the question asks whether a yes is false or true, then you obtain the information about the value for no. Last edited by skipjack; May 23rd, 2017 at 11:43 AM.  
May 22nd, 2017, 11:27 AM  #24 
Global Moderator Joined: Dec 2006 Posts: 18,568 Thanks: 1483 
It doesn't matter that "as the doorkeeper can only lie once" is part of the doorkeeper's thought process, because thinking about a lie doesn't constitute lying, and so doesn't affect whether the answer to your question can be a lie. You must ensure that both "yes" and "no" are valid answers, "yes" meaning (to you) one particular selection and "no" meaning the other selection. The doorkeeper isn't allowed to answer "I don't know" or "I don't understand". You might be able to get round this impasse by allowing a question that is directly selfreferential (by referring to its own answer).

May 22nd, 2017, 12:33 PM  #25 
Newbie Joined: May 2017 From: England Posts: 13 Thanks: 0 
So are you saying that as the question is hypothetical, the no_2 lie would be the first time the doorkeeper lied which means that it would still satisfy the rules of the puzzle? But wait, if lying and saying no is not a possible answer, then why would the doorkeeper even considering making an impossible statement? Wouldn't it swap to the more valid options if lying and saying no at the same time is invalid or impossible? How does that work? The doorkeeper is still allowed to lie, but at the same time, lying is not even an answer, i.e. by lying about if you would have lied when you are only allowed to lie once and saying no I wouldn't have lied means that you would have lied, but now you can't lie as that would mean you lied more than once. Does this mean that my logic is incorrect and if so what would happen if, let's say, a computer had to run this as a program? If I was the doorkeeper, I would think that since this response does not work and doesn't fit into the category of lie, then I would have to say the truth. BTW if there is a way to get round this impasse, please tell me. Last edited by skipjack; May 23rd, 2017 at 11:31 AM. 
May 22nd, 2017, 02:11 PM  #26 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,655 Thanks: 740 
Abe, why don't you actually try this out with a friend. He flips a coin and lies if it's tail, tells truth otherwise. You of course are not "watching" this coin flip. Get my drift? Note: you need to have a friend 
May 22nd, 2017, 02:39 PM  #27 
Newbie Joined: May 2017 From: England Posts: 13 Thanks: 0  
May 22nd, 2017, 03:38 PM  #28 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
"If you lied about the treasure being behind door A , is the treasure behind door B?" Suppose door A has the treasure , he would have to lie and say no so his response to the second part would have to be a truthful no Suppose door B has the treasure , he would have to lie and say yes so his response to the second part would have to be a truthful yes. In this scenario a no response means door A and a yes response means door B 
May 22nd, 2017, 03:57 PM  #29 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,142 Thanks: 2382 Math Focus: Mainly analysis and algebra 
He doesn't have to answer both parts separately. But you can't tell whether the answer is a comment on the logical relation between the two clauses; or a reply only to the second clause.

May 23rd, 2017, 12:07 AM  #30  
Newbie Joined: May 2017 From: England Posts: 13 Thanks: 0  Quote:
 

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