Jailbirds

[Silversunset]

I don't know if you've seen this one before, but a guy at work posed it to me and it was a bit of a stumper at first.

A group of prisoners are trapped in jail. These prisoners would attempt an escape on any positive probability of success. The prisoners are monitored by a guard who has only one bullet in his gun, but who also has perfect marksmanship skills (he never misses). A maintenance technician needs to repair the gate keeping them inside the jail, and so for one second, the gate is open. How can the guard still keep all the prisoners detained?
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as you wish....

[Icarus]

Well, since you don't have a number of prisoners, etc, you're not looking for a mathematical solution.

The guard shoots the technician.

Or, the guard shoots the first prisoner to reach the gate just before reaching the gate. All the remaining prisoners trip over that first prisoner, and by the time they get up, the gate is now closed.

[Chaz]

With due diligence!

(Alternatively, the group of prisoners is only one prisoner.)
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The enemy's base is down.

[raekuul]

My first thought, fix it at night when they're asleep wouldn't have worked. We have a non-singular number of prisoners, a single shot, a single door, and a single moment in time in which we must repair the door.

Wait. I understand that this solution isn't the most humane, but the alternative is a breakout. If they haven't escaped yet despite the door being broken, why does the door need to be fixed? If they could have escaped because the door was broken, what was the guard doing to keep them from escaping?

Why not just leave the door broken?

[Chuck]

The guard could tell the prisoners that he'll shoot the first one who tries to escape.

[Chaz]

What if we look at as a logic circuit?
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The enemy's base is down.

[Zag]

The guard lines up the prisoners on the farthest point from the gate. They are all told to lie down on the ground, face down, and put their hands behind their heads. Once they are all in position, the technician does his stuff. No one can possibly get up from that position and cross the compound in one second.

[Icarus]

I guess the first thing we should ask - is there a serious answer to this, or is it a trick, joke answer. Well, considering the gate is only open for one second, and there is only 1 gate mentioned, we know this isn't a real situation, therefore we shouldn't expect a "real" answer. There simply is no way 1 second allows anyone a chance to get through the gate.

So the next question is, are we looking for a trick answer?

Trick answers can be anything silly you can think of. Shoot the technician; booby trap the ground; line up the prisoners single file, the bullet will pass through every prisoners head; etc. etc.

If this is the case, then there really isn't an answer, as any of these will do.

So if there is a serious answer, than it's got to be something practical.

How about locking up the prisoners in their own cells before working on the gate?

Or, tell the prisoners the technician is going to be working on the back fence. They'll all be looking at the back of the jail while the technician opens the front gate for 1 second.

Or - tell the prisoners the gate is going to fixed every day for a year, but never actually fix the gate. Then one random day, actually fix the gate.

[Silversunset]

Chuck: close but not *quite* the answer

I probably could have worded it differently - this was the way it was given to me which is why i think it's a stumper.

These are the "clarifications" (aka hints) i was given when i tried to solve it.
1. the 'jail' is one big open room with a single gate, so you can't lock anyone up in their cell
2. there is nothing in the room except the guard, the prisoners, and the gun the guard holds
3. there can be any number of prisoners. For the sake of argument, say 5 prisoners
4. the "one second" is an arbitrary number. raise it to 5 minutes if it helps.
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as you wish....

[Jack_Ian]

The problem as I see it:
Even if the prisoners know that the first prisoner through the door will be killed, a group of prisoners can conspire to push another through the door to take the bullet before making their escape. Given that there is only one bullet, there will always be some scenario that results in a non-impossible chance of escape once the gate opens. The only solution I see therefore, is for the gate to remain closed.
I guess he could let them go and then declare himself outside the prison and then lock the door keeping them "detained".

[Silversunset]

Not quite - i've been back and forth with my work-friend on this, and there is a way to solve this that will prevent *any* prisoner from escaping
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as you wish....

[Chuck]

I suppose the guard could threaten to shoot the first prisoner who moves or talks, but they might have previously planned for this possibility and would all rush for the door at the same time. Of course, if I were such a prisoner I'd wait until someone else was shot instead of moving with the rest.

[Silversunset]

chuck you're so close i could smack you

there is a super difficult 'mathmatic' version of the answer. and then there is a non-mathgeek answer. I'll wait a little bit before i fill in the blanks
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as you wish....

[Ctorj]

The guard tells the prisoners that if anyone attempts to escape while the door is being fixed, he will
-kill their closest family member?
-kill the person next to them?

[Chuck]

The guard could number the prisoners if they don't already have unique prisoner numbers and promise to shoot the lowest numbered prisoner who tried to escape. Then for sure the lowest numbered prisoner won't try to escape, and the second lowest would know this and also not try to escape, and so on through the group.

[Zag]

[Silversunset]

yaaaaaaaaaay ding ding ding ding.

although the numbering doesn't have to be a "number" per-se, just assign any value. when i answered it i pulled the "shoot the tallest person to move" - you have to do the 'tried to move' bit or else there is nothing preventing the group from beating someone up throwing them outside getting them shot and then running away themselves. if you tell them the tallest to move, then they won't budge at all which wipes out any chance of them forcing someone else outside.
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as you wish....

[Icarus]

[Zag]

Except that it doesn't work any better than shooting the first person to move, for the reason I was hinting at above. I was referring to the well-known riddle discussed in The Unexpected Hanging and Other Mathematical Diversions by Martin Gardner.

In the riddle, the warden tells the prisoner he will be hanged one day this week, and, on the morning of the hanging, he will not know that it is coming that day. From this, the prisoner uses your reasoning that he can't be hanged at all. He can't be hanged on Saturday, of course, because, as the last day, he would expect it on that day if they made it to Saturday. So he can't be hanged on Friday, since that would leave only Saturday which we already showed to be a logical impossibility. Continuing backwards through the days, it is clear that he can't be hanged on any day.

... and he was hanged on Wednesday, and it was a surprise.

[Silversunset]

Zag; which one doesn't work better?
(i've heard the riddle you're referring to)

if the prisoners are ranked tallest to shortest (5 feet, 5.25 feet, 5.5 feet, 5.75 feet, and 6 feet for the sake of argument) and the guard shoots the tallest person who moves:

- The person who is 6 feet tall will not move because he is the tallest.
- The person who is 5.75 feet tall will not move because he knows that the 6 foot tall person isn't going to move.
- the person who is 5.5 feet tall will not move because he knows that the 5.75 and 6 feet tall people will not move.
- the person who is 5.25 feet tall will not move because he knows that the 5.5, 5.75 and 6 feet tall people will not move.
- and lastly, the person who is 5 feet tall will not move because he knows that the 5.25, 5.5, 5.75 and 6 feet tall people will not move.

Their best bet would be for two (or more) to move at the same time, which wouldn't happen because the taller of the pair wouldn't move, and the shorter one would figure it out and then not move either.

i'm not sure how you'd end up with a surprise in this one.
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as you wish....

[Zag]

With few enough people, it would work, as would just saying you'll shoot the first person who moves.

If there were 100 prisoners, then a bunch of them in the 75-100 range in ranking are very likely to move, I think. Many of them would think that there is at least a 50% chance that someone taller than he will also move, based on the same logic. Or one of them will surreptitiously poke a tall one with a pin, counting on you not catching the slight movement of poking him.

Also, say you're the guard, and the shortest guy moves a tiny bit. Do you shoot right away? How do you know a taller person isn't going to move a quarter of a second later? ... a tenth of a second? ... a millisecond? Since it's impossible for someone to be perfectly still, how much movement is enough? What if it's just a sneeze?

We could remove the ridiculous judgment issues by placing it in Puzzlania, each prisoner is in a cell, and all are told that the cell doors will be open for 5 seconds, then closed again. Anyone can choose to leave their cell, but the tallest person to leave the cell will be killed. Everyone else who left their cell will be freed, and all who do not leave their cells will be imprisoned forever.

Let's say you're the shortest person alive, and you know that there are N prisoners. For what number of N will you leave? A million? yes, easily. Four? probably not. It's somewhere in the middle.

[Icarus]

I formally declare "PROTEST"!!

How is any solution different than this:

[groza528]

If the prisoners start walking out one by one, from shortest to tallest and stopping at an arbitrary time, then the guard would not know which prisoner was the tallest to move until several had already escaped.

[Chaz]

I want to agree with Icarus (and in principal I do agree with Icarus.)

However.

In the case of "first to move" they can all move at the same time, so there's a good chance (4/5 chance) that they will escape. In the case of "shortest one who tries to escape", they would all have a 0/5 chance of escaping if nobody shorter than them tried to escape as well. They can't all decide to be the same height, but they can decide to move at the same time.

Since shortiest has a 0/5 chance no matter what, he'll stay put. And so on up the scale.

[Silversunset]

Chaz -
but what happens if two guys beat up the third, throw him out and the guard shoots - then everyone is free to run out.
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as you wish....

[bgg1996]

"Also, if you try to throw somebody over the gate I will shoot you. If you do it at the same time, I will shoot the taller one."
"Also, if you try to manipulate somebody's limbs to make them throw somebody over the gate, I will shoot you. If you do it at the same time, I will line you up, and try to shoot both of you, tallest man first.

[Chuck]

If the guard shoots somebody, all of the other prisoners still in the cell will escape. As long as he has his bullet, any remaining prisoners might be intimidated and not try to escape. One prisoner might realize this and walk out, figuring that the guard won't shoot him and lose all of the other prisoners. But then the other prisoners except for the last two might have the same thought and also walk out. I figure that the guard can keep two prisoners because with two left, he can't lose more than one whether he shoots or not so his threat to shoot the next one who tries to leave should keep the last two from trying.

[Chaz]

Stand them all five feet apart, and threaten to shoot the first person who's foot leaves the ground (or the tallest of a group who's feet leave the ground.)
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The enemy's base is down.

[Chaz]

I still like Icarus' original solution.

Shoot the maintenance man.
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The enemy's base is down.

[Icarus]

[Chaz]

So shoot them alphabetically. Or, as Chuck suggested, give them each a number.

It doesn't really matter, as long as you have some preset order in which they'll die if two of them make a run at the same time.
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The enemy's base is down.

[Silversunset]

True, it doesn't matter what the value is as long as they are all assigned something.

and it would keep them in, if you knew the "tallest" person would be shot and you were the tallest you wouldn't move, and so-on down the line. The tallest person wouldn't be stupid enough to move (in this riddle anyways) so youd never have the chance where more than one person would try to escape at the same time.
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as you wish....

[bgg1996]

How do they know for sure that they will be shot if all you do is tell them that they will be shot. You could be lying.

[Chaz]

I think the more important question is: where do I hire this maintenance guy? That one second exchange is effing amazing!
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The enemy's base is down.

[Icarus]

Let's use numbers since they can be unique.

And let's say there are 10 prisoners.

I get assigned 9, you get 10.

Very well, you are #10, therefore I know you aren't going to budge.

The gate is open for 5 minutes.

I'm #9, I am walking toward that gate. How exactly does the gaurd know to shoot me?

All I have to do is tell the guard, hey, once you shoot me, then what's keeping #10 from leaving? And by your logic, you said you would shoot the Highest # to attempt escape.

What if #10 intends to wait until 4 minutes after the gate was opened, and then tries to escape?

At what point is the guard able to determine the highest/lowest # prisoner escaping?

[Chaz]

First to move OR highest number if two or more move at the same time.

No matter what, 10 knows he will die if he tries to go for the door.
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The enemy's base is down.

[Chaz]

And if 10 knows that he'll die, then 9 knows that 10 knows this, so 9 knows that he'll die instead... etc.
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The enemy's base is down.

[groza528]

[Sentran]

The guard grabs himself by the collar with one hand, while pointing the gun to his chin with the other. He then claims, in a very serious voice, that if anybody moves towards the gate, the guard gets it. If the jail is in Blazing Saddles, this will have a 100% chance of success.
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Sentran
"Speaking of double negatives, I haven't read greylab yet today." - Lifeinmomland

[Jack_Ian]

Everyone surrounds the shortest guy and brings him through the gate.
Everyone, including the shortest guy has a chance of escape.