May 20th, 2017, 12:41 PM  #11 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  
May 20th, 2017, 01:20 PM  #12 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra  
May 21st, 2017, 03:29 AM  #13 
Global Moderator Joined: Dec 2006 Posts: 20,969 Thanks: 2222 
The problem is how to determine the correct door by using only one question. That doesn't imply that the doorkeeper is incapable of answering more than one question. However, he could not answer with a lie on two separate occasions.

May 21st, 2017, 11:30 AM  #14 
Newbie Joined: May 2017 From: England Posts: 13 Thanks: 0 
OK everybody, here is the solution, but first side note when I meant that you are only allowed to ask one question, I meant that a question can only be asked once in other words you are not allowed to ask the same one question multiple times. Also, this was kind of implied, but you were only allowed to ask yes or no questions. Now I'll give the solution, but correct me if my logic is wrong Solution WARNING: very confusing! First imagine how it would be if you were not restricted to one question, how would you do it? Well, you would first ask the doorkeeper if its door A/B, in our case lets ask the doorkeeper if its door A. If the doorkeeper says "yes" then that yes is either the truth or a lie. The same is true if the doorkeeper says "no" as the no would either be the truth or a lie. Let's imagine he said "yes". Next, you would ask him: 'was that "yes" a lie'. In this case, the doorkeeper will respond either yes or no. If he responds "no" and its the truth then the original yes was the truth. However, he can't say no and lie because he'd be essentially saying "no I didn't lie" and be lying about it. Thus he lied more than once, which is not allowed, so this situation is impossible. If he says "yes", then that "yes" is either a lie or a truth. If its the truth, then he lied in the first question when he said "yes" and thus its door B. However, if he is lying when he says yes to this question then the yes from the first question was the truth, thus it would be door A. As there is still ambiguity, we'll have to try for one more question. Again, imagine he says yes. This time you will ask the doorkeeper if the "yes" from the second question was a lie. If he says "no" and its a lie, then that "yes" from the previous question would have been a lie, but this would mean that the doorkeeper would have lied more than once, so this situation is impossible. However, if he says "yes" and lies, then that yes from the second question would have been the truth, but that yes from the last question confirmed a lie in the first question, so this situation is impossible. This means by the third question, the doorkeeper is only allowed to tell the truth as lying would be impossible, so lets go through the scenario. "Yes" the second question where I said yes was a lie, which means that when I said yes to the first question it was the truth, thus the treasure is in door A. "No" the second question where I said yes was the truth, which means that when I said yes to the first question it was a lie, thus the treasure is in door B. This is great, but we still want to make it just one question, so to conclude what you would actually ask is... If I were to ask you the following question:"If I were to ask you the following question: 'If I were to ask you if the treasure is in door A and you said yes, would that be a lie?' and you said yes, would that be a lie?" AND there you have it folks, apologies if its confusing. Let me know if I got something wrong, I either explained it wrong or my logic is wrong. If anyone can confirm my answer and notify me, that'll be great 
May 21st, 2017, 11:35 AM  #15  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Quote:
2+2 = 4 No! My second 2 is really a 3: 2+3 = 5 You've lost everybody's time.  
May 21st, 2017, 12:09 PM  #16  
Newbie Joined: May 2017 From: England Posts: 13 Thanks: 0  Quote:
 
May 21st, 2017, 12:24 PM  #17 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  
May 21st, 2017, 12:49 PM  #18  
Newbie Joined: May 2017 From: England Posts: 13 Thanks: 0  Quote:
If you ask a question that considers the hypothetical response of another question, you narrow down the valid responses available to the gatekeeper until he is only allowed to tell the truth.  
May 21st, 2017, 01:22 PM  #19 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
So: limitless questions can be asked same question cannot be asked more than once only one lie can be told Then it's easy: pick an object in the room let's use the clock, which is black in color questions: is the clock black? is the clock brown? is the clock pink? ...continue until lie is told then: is the treasure behind door A? Get it? 
May 21st, 2017, 01:54 PM  #20 
Newbie Joined: May 2017 From: England Posts: 13 Thanks: 0 
No, not quite there yet. I'm sorry for the poor explanation, but what I mean is that only one question is allowed to be asked. So let's say I ask what is 2+2... this will be the only question I would ask and I won't be allowed to ask this repeatedly. The only difference between this analogy and my problem is that the question must be a yes or no. Once again, I'd like to emphasise that it's one unique question that can only be asked once. The solution to the problem is to ask a really long question that contains hypothetical questions embedded within it so that you are still only asking one question, but still receiving information about the problem. Last edited by skipjack; May 22nd, 2017 at 01:24 AM. 

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