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Old 02-04-2015, 08:08 PM
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Critical thinking

Recently there have been some good threads in this forum about logic, the scientific method and critical thinking. I was reminded of them when I found a variation of the story below. If you recognize the basic puzzle I would request that you please wait 24 hours from this OP before commenting on it. This will give those who are not familiar with it a chance to solve it on their own. If you do not recognize it, please feel free to solve it at any time and post your answer. It forced me to think about things a bit differently. It's a good exercise in logic and critical thinking.

Long ago in a land far away a Kings son was murdered. Fortunately the King found an eye witness who identified the person who did it. However, within a short time two other men were found who looked exactly the same as the first. All three men denied committing the crime and none of them had an alibi. The king, not wishing to let his sons killer go free, sentenced them all to death.

In prison the three were placed in adjoining cells. As they awaited execution, the prisoner in cell "B" confessed to the prisoners in cell "A" and cell "C" that he was guilty. Being a truly evil man, he told the other two that he would never admit to this, and that their only chance to live was to help him build a tunnel to escape. They agreed, and soon there was a tunnel connecting cell "A" with cell "B". But before any more work could be done on the tunnel the King declared that the executions would take place the following day. In addition, the King declared that he would pardon one of the three prisoners, at random, and told only the warden which cell contained that lucky man.

The innocent prisoner in cell "A" got wind of this and privately begged the warden to tell him who would be pardoned. The warden said that the king had strictly forbidden him from saying who would go free, but confided to the prisoner in cell "A" that the prisoner in cell "C" would not be pardoned.


Assuming that the prisoner in cell "A" can somehow safely and easily switch places with the guilty prisoner in cell "B" (through the tunnel), should the prisoner in cell "A" switch? Would it increase or decrease his chances of survival, or would it make any difference?
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Old 02-04-2015, 08:25 PM
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maybe I should read it again, but I'll take a quick stab at it - bed time
I prolly missed sumting,

but regardless of which cell the guy chooses to go into his chances are 50-50 either way. I think. So he might as well just stay put.maybe I should read it agian?
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Old 02-04-2015, 09:10 PM
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The probablities of staying or switching depend on what sort of a person the warden is - he could be a jerk or a mensch, or impartial. But, regardless, there is an answer that is better than the other. There is a hidden probablity trap - we want to treat non-random information as if it were random.

This is a non-trivial answer, and it's a head scratcher.
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Old 02-04-2015, 09:36 PM
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We need only assume the warden is telling the truth. Everyone in the story is telling the truth with the exception of the prisoner in cell "B" when he denied committing the murder.

If you figure it out freshstart, please keep it under wraps. You are more likely than anyone to get it first. IMO the point will best be made if people are forced to struggle a bit before seeing the answer. Thanks
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Old 02-04-2015, 11:02 PM
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OK so I am going to give it a try.
King is going to pardon either a or b (at random between the 2, c being out of the equation) so whichever cell he is in, prisoner a has a 50/50 chance of survival.Switching would be indifferent.
He can either play it that way or he can creep into the guilty evil man's cell at night and smother him.
If the king is still intent on pardoning someone from either cell a or b then a will survive by default. If the king suspects foul play and conclude (erroneously) that a having killed b must also have killed his son then he will change his mind and pardon c. A will be executed but at least c who did not kill anybody will go free.
Either way, if a uses the tunnel and kills b, at least one prisoner who did not kill the king son is guaranteed to survive and justice will have been served.


Ps: Could it be that the king knew all along that prisoner b was guilty and being very cheap and unwilling to pay an executioner he is manipulating a to do the job for him?
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Old 02-04-2015, 11:16 PM
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Originally Posted by LBrain View Post
maybe I should read it again, but I'll take a quick stab at it - bed time
I prolly missed sumting,

but regardless of which cell the guy chooses to go into his chances are 50-50 either way. I think. So he might as well just stay put.maybe I should read it agian?
I agree that it is 50-50 but I don't think he should be put.
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Old 02-04-2015, 11:36 PM
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All three prisoners know that B did it. Therefore the odds are not entirely equal because if two of the prisoners team up on the other one they may be able to sway the King to murder the third. Without the tunnel if A and C pointed and swore that B was guilty the King may favour to kill B, all other things being equal.

Now if A switches with B it could sway the odds to kill him as C would still swear B did it but A would be in B's place. So A would marginally increase his chances of being killed if he switched.
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Old 02-04-2015, 11:53 PM
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OK Carlotta, but you can't add events to the story (such as A sneeking in and killing B, as much as I like that idea). If you do, then it's changing the story, and as much fun as that might be, the discussion will get quite weird, quite fast, and I don't think that will accomplich much.

All we have to work with are the events in the story, and the only question is what should the prisoner in cell "A" do? His only options are to stay put, swap cells and a third one that it does not matter. What shall it be?

I take it you are in the "does not matter" camp.

ubntubnt, Same thing. No adding events to the story, It's all about whats the best course of action for the man in "A". Stay, switch or that it does not matter.

Just over 21 hours to go.
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Old 02-05-2015, 12:19 AM
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Shoot it downed on me, he pardoned a man at random, not a cell duhh
In addition, the King declared that he would pardon one of the three prisoners, at random, and told only the warden which cell contained that lucky man.
So if he pardoned a, whether he is in one cell or the other it does not change the outcome (even if they look exactly the same, they wear different clothes so the king might have pulled "Jo who wears a red shirt" out of the hat and happens to be in cell a.)

My vote is still "it does not matter" but the reasoning behind it changed.

I still think he should kill b LOL
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Old 02-05-2015, 12:33 AM
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Interesting question this, and I like some already are looking for tricks in the language/narrative of the problem. Thinking about people and personalities and so on.... But I suspect Freshstart is right and it may be probability trap, I look forward to getting the answer. Can I ask AWUH is it anything like how we tend to over/underestimate probabilities like the math on the probability of two people sharing the same birthday in a room of thirty, where we instinctively believe its long odds but it actually is way shorter odds (higher probability). Its evening here in Australia so I will have this mulling in my head in my dreams, I still find it difficult to get away from the detail in the story as being relevant in some way.
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Old 02-05-2015, 03:07 AM
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ok I m probably adding to the story but if they are dressed alike couldn't he tell the warden he did it then switch cells with b?
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Old 02-05-2015, 03:12 AM
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Originally Posted by Carlotta View Post
ok I m probably adding to the story but if they are dressed alike couldn't he tell the warden he did it then switch cells with b?
haha, yes he certainly could. Why didn't I think of that?
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Old 02-05-2015, 03:34 AM
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They all have a 1 in 3 chance of living the next day as the king said. None of the other info is relevant. Key words being random and would (implying the decision has not been made yet). If this "wind" is caught by said king that warden broke his vow, he's wormfood also. It's good to be the king.
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Old 02-05-2015, 03:51 AM
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I am going to guess that in the beginning each had a 1 in 3 change of being pardoned. If prisoner A decides to take a new gamble on switching cells, then he lowers his probability to 1 in 6 (1/3 * 1/2), as he now has to win two gambles. He is better off staying in his cell.
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Old 02-05-2015, 03:56 AM
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One thing which makes no sense is that if the king is so set on avenging his son he is willing to punish the innocents why would he release someone at random and risk freeing his son's murderer?
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Old 02-05-2015, 04:04 AM
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ok so C is not going to be pardoned, he's SOL. Now it's a 50/50. But B is a liar. So I think based on that info that he stays where he is.
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Old 02-05-2015, 04:30 AM
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Old 02-05-2015, 04:31 AM
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What's eating at me is how C could help build the tunnel between A and B?
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Old 02-05-2015, 05:01 AM
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okay, now that I've cleaned the driveway... it's 20 degrees and a fine snow coming down fast...

The problem is exactly the same whether the guard said, 'C' or 'B'. He has no knowledge of possible switch so saying B or C is the same to him (guard). If he said 'B' was not being pardoned of course it's a no brainer.

If he told 'A' he was not being pardoned, again , switch. His chances would still be 50/50 - not 100% death.

hey! What happened to my earlier response???? good thing I copied it before I went outside
++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++

I can't - yes I can - believe this is where I come first when I woke up.

Okay, I read it a second time. Based on all things being equal, the chances are 50/50. However, the fact that the guard confided in the guy in 'A' but could only tell him it wasn't 'C' - couldn't say who the lucky man was, he is better off staying put. If he was not to be set free, executed, the guard would have (could have) told him.

The guard basically told him it is you or him but didn't say you. STAY IN 'A".
The guard by definition could not tell him two people because by process of elimination he would telling who the winner was. So by telling him anything other than he is not being pardoned, he basically told him he is.

so yes, a lot depends on whether or not the guard is a dick. But staying in 'A' is the clear best choice. I think...


++++++++++++++++++
Looks like my two posts got intertwined/mixed up. Not only staying in his cell 'A' the best choice, it is the only winning choice.
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Old 02-05-2015, 05:09 AM
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The way I see it, there are no word tricks, the answer is not hidden in the narrative.

If I understand where awuh is going with the problem, it is simply this. There is one pardon, for the prisoner who is in cell A, B or C at daybreak. It could be a daily reprieve, given the OP, but I digress. The warden knows which cell's occupant gets the reprieve. Pardon, I mean. This language stuff is tricky.

Anyway, The warden tells prisoner 1, who is in cell A, that the king's decision means that the dude in C goes to the gallows tomorrow. Now that he knows this, should he stay in A or choose cell B instead?
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