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A guard has to persuade 5 prisoners to stay inside a circle drawn on the ground. To discourage them, he can use his one-bullet gun. To always watch them, the guard stays outside the circle. What public threat can he announce to prevent any of the prisonners from escaping?

  • The prisoners are brave (they try to escape if they think they have at least one chance in a billion).
  • The prisoners are not suicidal (they stay if they are 100% sure that an escape attempt will kill them).
  • The prisoners know that the gun has only one bullet.
  • The prisoners know each other and can communicate privately.
  • The prisoners can act simultaneously and collaborate on a common strategy if: each of the collaborators staying thinks at least one prisoner will espcape, and each of the colloborators trying to escape thinks he/she has a chance.
  • Most importantly, the prisoners are all rational and trust that the guard will behave in the way he announces.

A similar question has been closed as too broad and has been given an answer which is not completely satisfying. I have reformulated the question with more details about the hypothesis. Be as logical as the prisoners.

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closed as too broad by Deusovi, xnor, manshu, Aza Feb 19 '16 at 23:18

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Possible duplicate of 100 prisoners and 1 bullet $\endgroup$ – Deusovi Feb 19 '16 at 16:48
  • $\begingroup$ @Deusovi but that one's been closed for being too broad - this one is significantly better-defined $\endgroup$ – question_asker Feb 19 '16 at 16:50
  • $\begingroup$ @question_asker: Not really. They're pretty much the same but with a different number of prisoners. $\endgroup$ – Deusovi Feb 19 '16 at 16:58
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    $\begingroup$ @Deusovi OK. You've already made up your mind. $\endgroup$ – question_asker Feb 19 '16 at 16:58
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    $\begingroup$ So if I am a prisoner, I need to be 100% sure I will be shot in order to not act right? If I think I have a 10% chance at survival, I will take it right? $\endgroup$ – Trenin Feb 19 '16 at 18:00

10 Answers 10

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Simple rule similar to that of Solocutor, with a patch in case of simultaneous escapes:

The guard publicly assigns a number to each prisoner and says, "If any prisoners try to escape, I will kill the prisoner among them with the smallest number."

Reasoning is by induction:

The prisoner with the smallest number, call him/her 1, surely won't try to escape. The prisoner with the second-smallest number, call her/him 2, realizes that 1 cannot attempt anything. Trying something is thus surely suicidal for 2 also. And so on for all prisoners. Defining a total order is the only solution: if all try to escape the rule announced in the threat must designate a precise prisoner so that he/she is sure to die, call him/her 1. Reasoning similarly if all but 1 try to escape, the rule must designate a prisoner among the four others, and so on. Making sure that all prisoners know this ordering is then sufficient.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Aza Feb 20 '16 at 1:27
  • $\begingroup$ Your answer to your own question looks good. Has anyone been able to refute it effectively? (I didn't see any effective refutations of it in the chat.) $\endgroup$ – Lawrence Feb 20 '16 at 3:51
  • $\begingroup$ Surely the flaw in this answer is the ambiguity of 'try to escape' if 2 grabs 1 and runs for it, is there a 1/1b chance the guard will think 1 is 'trying to escape' $\endgroup$ – Ewan Feb 23 '16 at 1:05
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Stated briefly:

The guard says "I will shoot the first person who tries to escape." No prisoner will move unless another prisoner has already started to move. So they will all stay in place.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Aza Feb 20 '16 at 1:26
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My attempt:

State that "I will shoot the first person who fails to get hold on the person in front of him/her." Assume they formed a circle with hands placing on the shoulder on the person in front of them. Each person will try their best to grab / hold the person in front of them thus no one will be able to escape.

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  • $\begingroup$ Nice one, but while in the circle around the guard, they may still try to move all together to move as a circle and "untangle" from the guard, all while not leaving the hold. Or not? I mean like trying to pass over the guard / passing the guard under their armpits $\endgroup$ – Diego Martinoia Feb 19 '16 at 17:18
  • $\begingroup$ It really doesn't have to break though: they may just "run over" the guard while in a circle and then trample it, without never letting go of each other. While in an awkward movement angle against the unconstrained guard, it's still 5 on 1 $\endgroup$ – Diego Martinoia Feb 19 '16 at 17:22
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    $\begingroup$ What if none of them comply? He can't shoot them all so there is no reason for them to grab on. $\endgroup$ – Trenin Feb 19 '16 at 17:23
  • $\begingroup$ deleted my first comment as I missed the 2nd part of Diego's comment $\endgroup$ – Alex Feb 19 '16 at 17:24
  • $\begingroup$ @Trenin ah good thinking, i'll edit my answer. Needs to be careful when consider they "do no comply" vs "The prisoners are not suicidal". I have added a word in the guard's statement - "first" person. $\endgroup$ – Alex Feb 19 '16 at 17:26
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Well, the obvious has already been stated, but let's suppose that the prisoners would, in that case, try to get someone else killed and then run (such as pushing them out of line or similar).

In that case, they solution lies in a threat that will invalidate everyone's escape if triggered.

So here we go with a possibly slightly lateral option:

The guard handcuffs himself to each prisoner independently (there are 5, so maybe he could do 1 per arm / leg and one at the guard's neck). Then he threatens to kill himself if someone tries to escape. If the guard is dead, the prisoners (unless they are VERY strong) are constrained by a) moving together and b) carrying around the weight of a dead adult at least until finding tools to get rid of the cuffs, which likely means not being able to escape in time and also being accused of the guard's death, thus resulting in a worse sentence, likely death sentence, so escape is more likely to get them killed than succeding

A slightly less violent option:

Same rationale, but chain them all together. The guard threatens to shoot the key away (by putting it into the chamber before shooting), rather than itself. Less effective (bc no body to carry around), but the guard lives.

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First solution

I assume I have a bit of time before they coordinate anything. So the first thing I do is:

"First person to talk or try to run gets shot!"

That should keep them quiet and close by for a bit until they realize

if they all talk/run at the same time, I can't shoot anyone.

So, while they are pondering that, I will then pick a prisoner and:

"Close your eyes and lie on your stomach or I will shoot you in two seconds if you don't comply!"

Repeat until all prisoners are

on their stomachs with their eyes closed.

If I am quick enough and threatening enough, they will comply quickly without communicating to each other (via sign language?) any type of coordinated attack/movement.

Even if

Some of them try to run while I am dealing with the others, I can call out to one of them running "You in the blue! Down now or I shoot!". And repeat with all the ones running.

So long as they don't leave my effective hand gun range, I should be ok.

Then, I will yell and berate them indefinitely, constantly reminding them that I can kill them at any time so they better not try anything funny.

Another solution:

"If I think any of you are trying to escape, I will close my eyes and spin around. When I open my eyes, I will shoot the prisoner closest to my line of sight."

None of them will want to try to escape because they may be the one targeted.

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  • $\begingroup$ If you can do multiple instructions this is good (eyes/ears closed -> no information, and threaten to shoot the first one who tries to get some) $\endgroup$ – Diego Martinoia Feb 19 '16 at 17:40
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    $\begingroup$ the "Another solution" part is simplest to analyze and strictly wrong - according to the puzzle setup, the non-suicidal part means that they will avoid actions where they are guaranteed to die, but if they believe that they may be the one targeted, they will attempt to escape as they may be the one not targeted and able to escape safely. Thus, if two prisoners have a good reason to believe that you'll shoot one of them (50% chance to die, 50% chance) then you lose, because they both try to escape and one of them succeeds. $\endgroup$ – Peteris Feb 19 '16 at 21:17
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Tell each prisoner privately this:

“If any prisoner escapes, I will kill YOU”

Then once they are all together they won’t communicate this to each other, as that would be suicide because they don’t know what you told all the others. And all of them will try to keep each other for attempting to escape.

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  • $\begingroup$ Should work too ,but you need more hypothesis : being able to tell something to each prisoner privately. $\endgroup$ – init55 Feb 19 '16 at 23:20
  • $\begingroup$ I have rephrased the question with broader hypothesis about collaboration, this answer does not work: a prisoner could collaborate if he/she thinks others can escape and he/she will not die. If I am sure I cannot escape I tell it to others to help them. $\endgroup$ – init55 Feb 20 '16 at 4:55
  • $\begingroup$ @init55 But if I am told that I will be killed if anyone escapes, then why would I say anything? If the guard told me the truth, then me telling another person I would be shot is suicide; that person will know they are 100% safe and run for it. So why would anyone communicate anything here? $\endgroup$ – Trenin May 3 '16 at 12:28
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Simple:

The guard tells them that there are 20 trained gunmen watching. If they hear a gunshot, they will execute everyone.

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  • $\begingroup$ Let's say they are in desert. How would you better formulate the way they believe what the guard says ? $\endgroup$ – init55 Feb 19 '16 at 19:28
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    $\begingroup$ Mines hidden underground. $\endgroup$ – Aggie Kidd Feb 19 '16 at 19:43
  • $\begingroup$ Drones. Camouflaged sharpshooters. Sand worms. $\endgroup$ – Ian MacDonald Feb 19 '16 at 20:26
  • $\begingroup$ WMDs and BLTs... who needs one bullet when we have the cavalry! $\endgroup$ – Daedric Feb 19 '16 at 23:19
  • $\begingroup$ satisfied all 6 conditions $\endgroup$ – wilson Feb 22 '16 at 2:58
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Handcuff them all together hand to foot in a circle. Tell them that the first person to try to move from their stop gets shot and you will kill them rest of them using the gun.

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My answer:

The guard tells the prisoners "If any of you tries to escape, I will kill the man to his right." This way the prisoners are regulating themselves in their private communications. I think this works with rules 5 and 6.

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Simple:

The guard should say: "If anyone tries to escape i will shoot a random person" No one will try to escape because of rule #2

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    $\begingroup$ This doesn't make them 100% sure an escape will kill them, it makes them 20% sure. If you use this strategy, all prisoners will immediately attempt to escape, knowing that they each have an 80% chance of survival. $\endgroup$ – Ajedi32 Feb 19 '16 at 20:52
  • $\begingroup$ Well, the prisoners are a bit suicidal then. Playing Russian roulette is suicidal IMO. If you allow for them to be partially suicidal they can just run away to different directions without any worries :) $\endgroup$ – Myslius Feb 19 '16 at 20:58
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    $\begingroup$ @Myslius the whole point of this puzzle is to find a way that would prevent that from happening even if they're willing to play russian roulette. $\endgroup$ – Peteris Feb 19 '16 at 21:18

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