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Topic: Logicians and stamps (Read 6224 times) |
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Altamira_64
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Logicians and stamps
« on: Apr 3rd, 2013, 11:53am » |
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N logicians are about to play the following game:
A moderator sticks two stamps on each logicians' forehead. He tells everyone that in the beginning of the game he had n+1 red stamps and n+1 black stamps. They do not know, however, that the moderator has stuck one red and one black stamp on the forehead of each logician, except for one to whom he has stuck two red stamps.
The logicians are sitting on a circle so that anyone can see everyone else's stamps. The moderator asks them in turn, starting from the logician who is sitting in the position Nr 1, "what color are your stamps?" The logician with the two red stamps on his forehead is sitting in the position Nr x (unknown to us).
For which values of n and x the logician with the two red stamps can guess the color of his own stamps?
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rmsgrey
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Re: Logicians and stamps
« Reply #1 on: Apr 4th, 2013, 7:39am » |
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It depends how the logicians answer when they have incomplete knowledge of their own stamps - if they simply say they don't know, then no-one gains any new information from anyone's answer.
If the logicians say what they do know about their stamps (that they don't have two red) then, unless x=1, as soon as #1 speaks, everyone else, including X, knows that #1 sees n red stamps (and n-2 black stamps) so can deduce their own stamps. When x=1, everyone else deduces their own stamps as soon as #2 speaks. In both cases, the person to give partial information about their own stamps gains no information thereafter (since everyone else knows what he sees, they can deduce their stamps without considering his, meaning their deductions tell him nothing)
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towr
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Re: Logicians and stamps
« Reply #2 on: Apr 4th, 2013, 11:55am » |
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on Apr 4th, 2013, 7:39am, rmsgrey wrote:| It depends how the logicians answer when they have incomplete knowledge of their own stamps - if they simply say they don't know, then no-one gains any new information from anyone's answer. |
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They do gain some information; for example for N=3, if x=2, player 1 and 3 will know by their second turn that they have a red and black stamp; if x=1 or 3, blue will know by his second turn that he has a red and black stamp. (Assuming I worked it out correctly.)
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| « Last Edit: Apr 4th, 2013, 11:56am by towr » |
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Altamira_64
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Re: Logicians and stamps
« Reply #3 on: Apr 4th, 2013, 4:12pm » |
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on Apr 4th, 2013, 11:55am, towr wrote:
They do gain some information; for example for N=3, if x=2, player 1 and 3 will know by their second turn that they have a red and black stamp; if x=1 or 3, blue will know by his second turn that he has a red and black stamp. (Assuming I worked it out correctly.) |
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Blue??
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towr
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Re: Logicians and stamps
« Reply #4 on: Apr 4th, 2013, 10:18pm » |
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Whoops.. That should be player 2
I work with a graph representation using colored edged to denote which worlds/states are indistinguishable to a user. So for me 1=red, 2=blue, 3=green.
You might have guessed by the fact player 1 and 3 were already mentioned and 2 was the only one left. 
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| « Last Edit: Apr 4th, 2013, 10:47pm by towr » |
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rmsgrey
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Re: Logicians and stamps
« Reply #5 on: Apr 5th, 2013, 5:50am » |
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Ah, yeah, the old red-eyed monks thing of what they know about what each other knows about...
So, with 3 people, and 4 stamps of each colour:
if x=1:
#2 and #3 both know they don't have a red pair.
#1's statement shows he doesn't see two black pairs. #2s statement shows #3 doesn't have a black pair, so #3 knows what colours he has, from which #1 can deduce that he has a pair, but not which colour. From that, #2 can deduce he's not a black pair (otherwise #1 could deduce his own colour) and knows his pair. #1 is stuck with no way to break the colour symmetry.
If x=2:
#1's statement doesn't help #3, but from there it's as above, one turn later - and X (#2) is still stuck due to symmetry.
If x=3:
#1 and #2 tell each other nothing, #3 tells them they're not both black pairs, so #2 gets there on his second turn and #1 on his third.
In general, it looks like X can never know his own stamps because he has no way of telling red from black
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towr
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Re: Logicians and stamps
« Reply #6 on: Apr 5th, 2013, 6:19am » |
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Yeah, considering the situation is completely symmetric if you change colors, the person stuck with the pair of one color can never find out which it is. Nothing he sees is different, and nothing that happens is different.
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Student2013
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Re: Logicians and stamps
« Reply #7 on: Jun 19th, 2013, 7:02am » |
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Hey Guys, i hope that i can get some help from you. I really hope i can pass my logic exam tomorrow, so that i can take a break from school, and i dont know who to ask for help.
Would you please help me? It would mean the world to me, because i am really having a hard time now.
Problem:
Assuming that Lxy is a means "x loves y" and Rx "x is rich" write logical formulas for
a) It is enough to love someone to be rich
b) being rich doesn't mean loving someone
c) you are not rich i you love no one.
d) Love is not always a symmetrical relation
and
Problem 2)
Imagine you are married and wish to open a joint credit account. Which of the following would you prefer to have printed on your checks: "John and Jane Dole" or "John or Jane Dole"? Explain your decision.
Again, if there is a good soul out there that wish to help me out, i am going to be forever thankful.
yours sincerely,
psychology student year 4.
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towr
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Re: Logicians and stamps
« Reply #8 on: Jun 19th, 2013, 10:35am » |
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It might have been better to start your own topic rather than append it to this one...
As for 1d,   (x,y: Lxy  Lyx)
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Grimbal
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Re: Logicians and stamps
« Reply #9 on: Jun 20th, 2013, 9:46am » |
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You should tell us what you think the answers are. We would be happy to tell you where you are wrong.
But it is probably too late already.
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