A prison with an infinite number of cells
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A prison with an infinite number of cells
I'm not sure if this is possible. A friend told me this problem, but I'm starting to believe he's got it wrong. Anyway.
You are the warden of a prison with an infinite number of cells. In each cell there are two prisoners with numbers #a and #b (in cell 6, the prisoners are 6a and 6b). Due to new regulations you need to redistribute the prisoners, so there will be one prisoner per cell.
You make a radio announcement to the guards (whose number is also infinite) telling them what to do.
What should the radio announcement be?
Example announcement: "Move each bprisoner to cell n+1."  with this order, one prisoner will remain in cell 1 but there'll be two of them in the rest.
That "An" has been driving me nuts. jr
You are the warden of a prison with an infinite number of cells. In each cell there are two prisoners with numbers #a and #b (in cell 6, the prisoners are 6a and 6b). Due to new regulations you need to redistribute the prisoners, so there will be one prisoner per cell.
You make a radio announcement to the guards (whose number is also infinite) telling them what to do.
What should the radio announcement be?
Example announcement: "Move each bprisoner to cell n+1."  with this order, one prisoner will remain in cell 1 but there'll be two of them in the rest.
That "An" has been driving me nuts. jr
It's only when you look at an ant through a magnifying glass on a sunny day that you realize how often they burst into flames
Re: An prison with an infinite number of cells
At last, one of these I can get!
Edit to add bonus sophomoric answer: The radio message should be boosted at intervals. Otherwise it'll never reach all the guards.
Spoiler:
Spoiler:
Edit to add bonus sophomoric answer: The radio message should be boosted at intervals. Otherwise it'll never reach all the guards.
 t1mm01994
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Re: An prison with an infinite number of cells
@Jona: Nope.
Spoiler:
Re: An prison with an infinite number of cells
t1mm01994 wrote:@Jona: Nope.Spoiler:
I see your point.
I guess that
Spoiler:
 t1mm01994
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Re: An prison with an infinite number of cells
@jona:
Spoiler:
Re: An prison with an infinite number of cells
Regarding jona's proposed solution,jonachin wrote:t1mm01994 wrote:@Jona: Nope.Spoiler:
I see your point.
I guess thatSpoiler:
Spoiler:
Re: An prison with an infinite number of cells
Spoiler:
Please be gracious in judging my english. (I am not a native speaker/writer.)
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Re: An prison with an infinite number of cells
alright, how about something dead simple?
Spoiler:
Re: An prison with an infinite number of cells
do we need to work out the sequence of moving prisoners as well?

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Re: A prison with an infinite number of cells
Im pretty sure it can't be done.
Spoiler:
Re: A prison with an infinite number of cells
c4k3m4s73r wrote:Im pretty sure it can't be done.Spoiler:
I think you got something wrong:
Spoiler:
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Re: A prison with an infinite number of cells
Naturally, any process for reassigning rooms of an infinite number of prisoners will be tricky and lead to some unsatisfying answers in the short run, but look at it this way:
Spoiler:

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Re: A prison with an infinite number of cells
you're correct lorb. I was too hasty. something in my mental execution of the scenario made me imagine .
Spoiler:
Re: A prison with an infinite number of cells
The obvious evil expansion of this:
Version 1:
You're responsible for infinitely many prisons, Prison 1, Prison 2, etc. Each Prison K has infinitely many cells, K1, K2, etc. each of which has one prisoner. You receive word that all but Prison 1 are to be closed down, and everyone must be relocated to Prison 1. Each prisoner must go to a cell, and each cell must have exactly one prisoner. You're patched in to all the PA systems at once. What do you say to the guards?
Version 2:
Same as Version 1, except there are a variable (but finite) number of prisoners in each cell and you're allowed to have empty cells after the transfer.
Version 1:
You're responsible for infinitely many prisons, Prison 1, Prison 2, etc. Each Prison K has infinitely many cells, K1, K2, etc. each of which has one prisoner. You receive word that all but Prison 1 are to be closed down, and everyone must be relocated to Prison 1. Each prisoner must go to a cell, and each cell must have exactly one prisoner. You're patched in to all the PA systems at once. What do you say to the guards?
Version 2:
Same as Version 1, except there are a variable (but finite) number of prisoners in each cell and you're allowed to have empty cells after the transfer.
Re: A prison with an infinite number of cells
HonoreDB wrote:The obvious evil expansion of this:
Version 1:
You're responsible for infinitely many prisons, Prison 1, Prison 2, etc. Each Prison K has infinitely many cells, K1, K2, etc. each of which has one prisoner. You receive word that all but Prison 1 are to be closed down, and everyone must be relocated to Prison 1. Each prisoner must go to a cell, and each cell must have exactly one prisoner. You're patched in to all the PA systems at once. What do you say to the guards?
Spoiler:
HonoreDB wrote:Version 2:
Same as Version 1, except there are a variable (but finite) number of prisoners in each cell and you're allowed to have empty cells after the transfer.
Spoiler:
Re: A prison with an infinite number of cells
Your solution to version 2 isn't the kind I intended, since it requires coordination on the part of the guards.
Re: A prison with an infinite number of cells
Altering it to remove the requirement of knowing the highest number of prisoners per cell, then:
Spoiler:
Re: A prison with an infinite number of cells
Spoiler:
"Please only print this post if you really need to"
...hmm....I wonder how much extra energy is required to generate that request...We need a cost/benefit analysis, STAT!
...hmm....I wonder how much extra energy is required to generate that request...We need a cost/benefit analysis, STAT!
Re: A prison with an infinite number of cells
Well, that's only Hilbert's hotel with name changed...
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
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Re: A prison with an infinite number of cells
Jeff_UK wrote:Spoiler:
This is begging the question, i.e. assuming that a solution exists. (Since the sets you work with are countably infinite, the solution works  but you didn't prove that it's the case.)

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Re: A prison with an infinite number of cells
There's a city (with an infinite population, of course) whose inhabitants enjoy organizing themselves in clubs: every possible finite set of citizens forms a club, and no two clubs are alike. The clubs are not named; it's been decided that each club should be named after one of the citizens. (No two clubs can be named after the same person.) Can this be done?
If so, then what if the restriction that no club can have an infinite membership were lifted? If every possible set of citizens, finite or infinite, formed a club, can they be named using the same rules as above?
Hint:
If so, then what if the restriction that no club can have an infinite membership were lifted? If every possible set of citizens, finite or infinite, formed a club, can they be named using the same rules as above?
Hint:
Spoiler:
 jestingrabbit
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Re: A prison with an infinite number of cells
Mike Rosoft wrote:Jeff_UK wrote:Spoiler:
This is begging the question, i.e. assuming that a solution exists. (Since the sets you work with are countably infinite, the solution works  but you didn't prove that it's the case.)
And in fact, you could easily run out of cells, and be left with any number (including a countable infinity) of prisoners without a cell. It could even be such that the asymptotic density of the prisoners who end up with cells was 0 before the change.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
Re: A prison with an infinite number of cells
Mike Rosoft wrote:There's a city (with an infinite population, of course) whose inhabitants enjoy organizing themselves in clubs: every possible finite set of citizens forms a club, and no two clubs are alike. The clubs are not named; it's been decided that each club should be named after one of the citizens. (No two clubs can be named after the same person.) Can this be done?
If so, then what if the restriction that no club can have an infinite membership were lifted? If every possible set of citizens, finite or infinite, formed a club, can they be named using the same rules as above?
Hint:Spoiler:
Yes. No.
For the first question:
Spoiler:
For the second:
Spoiler:
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Re: A prison with an infinite number of cells
ConMan wrote:For the first question:Spoiler:
What if the persons can't be wellordered?
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 jestingrabbit
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Re: A prison with an infinite number of cells
ConMan wrote:For the second:
It seems like you're assuming that the infinitude of the citizens is countable, rather than anything more troubling. There's a way to deal with the more general setup, and you arrive at the same conclusions.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
Re: A prison with an infinite number of cells
jestingrabbit wrote:ConMan wrote:For the second:
It seems like you're assuming that the infinitude of the citizens is countable, rather than anything more troubling. There's a way to deal with the more general setup, and you arrive at the same conclusions.
I admit that while my puny monkey brain is capable of imagining countably infinitely many people, it baulks a bit at uncountably many. However, let me see how far I can get with an arbitrarily infinite set of people.
Spoiler:
pollywog wrote:I want to learn this smile, perfect it, and then go around smiling at lesbians and freaking them out.Wikihow wrote:* Smile a lot! Give a gay girl a knowing "Hey, I'm a lesbian too!" smile.

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