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Passionately Curious
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2008 Dwarves
« on: Oct 13th, 2008, 10:24am » |
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You visit an island that is inhabited by 2008 dwarves, all of which have blue eyes (though they do not know it). The dwarve's are very friendly and tell you all about their community. They seem like an ordinary group of people but you found one thing very strange: If any of the dwarve's find out the s/he has blue eyes, then, at midnight, s/he will jump off the highest cliff on the island into the lake, thus perishing in the waters below. There are no mirrors (or any other reflective surfaces) on the island, and because the dwarves are friends and have great respect for each other, none of the dwarves will tell any of the other dwarves what colour their eyes are. Therefore, none of the dwarves can see their own eyes and none of the dwarves will be told what colour their eyes are so none of the dwarves know if they have blue eyes or not and have lived happily on the island for many years. Upon your departure, all 2008 dwarves go to the harbour to bid you farewell. You tell the dwarves that at least 2008 -sorry, meant 1234- of them have blue eyes and then you sail away. If the dwarves are infallibly logical, will anything interesting happen after your departure?
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| « Last Edit: Oct 13th, 2008, 10:31am by Passionately Curious » |
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rmsgrey
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Re: 2008 Dwarves
« Reply #1 on: Oct 13th, 2008, 12:05pm » |
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on Oct 13th, 2008, 10:24am, drdyl111 wrote:You visit an island that is inhabited by 2008 dwarves, all of which have blue eyes (though they do not know it). The dwarve's are very friendly and tell you all about their community. They seem like an ordinary group of people but you found one thing very strange: If any of the dwarve's find out the s/he has blue eyes, then, at midnight, s/he will jump off the highest cliff on the island into the lake, thus perishing in the waters below. There are no mirrors (or any other reflective surfaces) on the island, and because the dwarves are friends and have great respect for each other, none of the dwarves will tell any of the other dwarves what colour their eyes are. Therefore, none of the dwarves can see their own eyes and none of the dwarves will be told what colour their eyes are so none of the dwarves know if they have blue eyes or not and have lived happily on the island for many years. Upon your departure, all 2008 dwarves go to the harbour to bid you farewell. You tell the dwarves that at least 2008 -sorry, meant 1234- of them have blue eyes and then you sail away. If the dwarves are infallibly logical, will anything interesting happen after your departure? |
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They'll get together with the red-eyed monks and come after you...
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Passionately Curious
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Re: 2008 Dwarves
« Reply #3 on: Oct 16th, 2008, 7:24am » |
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Thank you.
Does that mean that all the dwarves will kill themselves on the 775th night?
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towr
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Re: 2008 Dwarves
« Reply #4 on: Oct 16th, 2008, 8:50am » |
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on Oct 16th, 2008, 7:24am, drdyl111 wrote:| Does that mean that all the dwarves will kill themselves on the 775th night? |
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Yes, that's what I get as well.
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gotit
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Re: 2008 Dwarves
« Reply #5 on: Oct 16th, 2008, 9:20am » |
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I was just wondering whether the result will be affected if one or more dwarves die a natural death during this period.
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towr
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Re: 2008 Dwarves
« Reply #6 on: Oct 16th, 2008, 10:30am » |
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on Oct 16th, 2008, 9:20am, gotit wrote:| I was just wondering whether the result will be affected if one or more dwarves die a natural death during this period. |
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I think it would set the mass suicide date back a day. If they all know that at least X dwarfs have blue eyes, and then one dies, they know that at least X-1 dwarves have blue eyes.
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gotit
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Re: 2008 Dwarves
« Reply #7 on: Oct 16th, 2008, 10:48am » |
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But the total number of dwarves will also decrease by 1. So if we consider the figures given in the problem and one of them die, then the rest know that at least 1233 dwarves have blue eyes, but the total number of dwarves are now 2007 and the difference is still 774. 
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towr
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Re: 2008 Dwarves
« Reply #8 on: Oct 16th, 2008, 11:04am » |
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on Oct 16th, 2008, 10:48am, gotit wrote:| But the total number of dwarves will also decrease by 1. |
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True, but it's one of the ones that everyone* know that everyone* knows (etc) has blue eyes.
*) everyone that's not dead
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| « Last Edit: Oct 16th, 2008, 11:06am by towr » |
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rmsgrey
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Re: 2008 Dwarves
« Reply #9 on: Oct 17th, 2008, 10:01am » |
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on Oct 16th, 2008, 11:04am, towr wrote:
True, but it's one of the ones that everyone* know that everyone* knows (etc) has blue eyes.
*) everyone that's not dead |
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That's why the current count decreases by 1 - they now know that at least X-1 (surviving) dwarves have blue eyes - but it also decreases the target number to which they need to count by 1, so the suicides will still happen on the same day.
If you kill 1 dwarf publicly on the final day, then all the others will still suicide at midnight - they already know they have blue eyes. So if you'd killed one the day before, it still wouldn't have made a difference - each dwarf knows that, if he doesn't have blue eyes, then all the blue-eyed dwarves would already know they had blue eyes, so would suicide that night - when they don't, the mass suicide goes ahead on schedule. And so on...
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Grimbal
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Re: 2008 Dwarves
« Reply #10 on: Oct 17th, 2008, 1:14pm » |
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I think that if n dwarves die before day n, then they are saved. Well, at least those who survived.
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rmsgrey
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Re: 2008 Dwarves
« Reply #11 on: Oct 19th, 2008, 9:09am » |
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on Oct 17th, 2008, 1:14pm, Grimbal wrote:| I think that if n dwarves die before day n, then they are saved. Well, at least those who survived. |
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Depending on how you count days.
If, on day 1 (the day you make the announcement), you announce publicly "there are at least 1234 blue-eyed dwarves on the island" then shoot 234 (blue-eyed) dwarves, then it's common knowledge that there are at least 1000 (living) blue-eyed dwarves...
If, instead of day 1, you call the day it becomes common knowledge that there are at least 1234 blue-eyed dwarves day 1234 (and number from there) then killing n dwarves not later than day n will save the rest (you need enough dead that it's not common knowledge that there are any blue-eyed survivors)
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Grimbal
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Re: 2008 Dwarves
« Reply #12 on: Oct 19th, 2008, 2:45pm » |
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Uh ... right. What I said would be for the case where the traveler announced that he saw at least 1 blue-eyed dwarf on the island. if the announcement is there are 1234 of them, I guess n+1233 must die (or leave the island) before day d to save the rest.
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| « Last Edit: Oct 19th, 2008, 2:46pm by Grimbal » |
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