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   Author  Topic: Variation of red eyed monks  (Read 2483 times)
inexorable
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Variation of red eyed monks  
« on: Jan 26th, 2008, 4:58am »
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Alice was wandering along the path of a broad forest in the Land
of Magic, when she heard some voices coming from nearby. As she was a
very curious person she climbed up the nearest tree and witnessed
the following scene...
 
Around a big round table there were 31 people. They carefully listened
to a Orator, a funny man, wearing a scarlet tunic, with a short, white beard.
He quieted everyone with a gesture and made the weirdest speech that
Alice had ever heard...
 
 My friends, logicians. We, the most disciplined and strict minds of
the Land of Magic, have gathered here for our 125-th annual
convention. We will hear the most astonishing stories, think about
things unthinkable to anyone else, we will climb the slope of the
Infinite Curiosity Mountains and the most demanding Tracks of
Intellect. But beforehand, we must make sure that no intruder is
among us.
 
Afterwards, the professor walked around the table attaching a
colourful dot to every person's forehead. When he returned to his
place, he began explaining the details of this amazing experiment.
 
- Every one of you sees the dots on your friends' foreheads, but I
made sure that you cannot see your own. Your task is to guess the
colour of the dot on your forehead.
 
- There is only one, very simple, rule. Every minute this bell will
ring. If at the moment it rings any one of you knows the colour of
his dot, he should stand up and join me in the meadow, where the
convention will continue. If, on the other hand, you don't know the
colour of your dot yet, you should remain seated.
 
- Those of you, who should have stood up but remained seated, or those
who should have remained seated, but stood up, cannot obviously be
named logicians. Such people will immediately be removed from the
convention and receive a lifetime ban.
 
The professor was just about to leave, when his attention was
attracted by the visible perplexity of the smartest novice. He
removed all his doubts with the following words:
 
- Don't worry, young lad. It is possible to solve this task, but
obviously you cannot communicate with each other.
 
The novice smiled, as the Preacher of a Gathering of the Most
Disciplined and Strict Minds of the Land of Magic cannot speak
sentences that are not true.
 
Alice, still amazed, saw that the professor left the gathering, and
the experiment began.
 
After the first bell, four people stood up and left. After the second
one, everyone with a red dot on their forehead left the table. After
the third one, nobody moved, while after the fourth one at least one
person moved. The Novice and his sister, both with dots of different
colours stood up shortly after that, but every one of them did this
before the last bell.
Alice was very tired and fell deeply asleep before the experiment
ended. Can you tell her how many times the bell rang before the table
was empty?
 
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Re: Variation of red eyed monks  
« Reply #1 on: Jan 26th, 2008, 8:17am »
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hidden:
If there were no limit on the colors available, the problem would not be solvable. Therefore the logicians know that the colors must be limited to some finite list that they are aware of. Assuming that no other information is available than that implied by the story, this list must consist of the dots that they see. Thus, by saying that the problem is solvable, the Preacher has told them that nobody has a dot whose color is not among the colors they see.
 
If anybody has a unique dot, this would not be the case. So a logician seeing only one black dot, for instance, would automatically know that his own dot must be the only other black dot there. Since 4 people left, there were only two green dots as well.
 
That the red dots did not leave with the first bell means that there were more than two. Since this is now evident, those who saw only two red dots would know that they themselves must be the third. Thus all 3 red dots left at the 2nd round. Now all dots must be in groups of four or more. But nobody moved at the 3rd bell, so the groups are all five or more. Since someone left at bell 4, there must be at least one group of 5.
 
The remainder of the problem depends on whether or not the Preacher is one of the 31 people (so 30 people are in on the test), or if there were 31 besides the Preacher. If there were 30 total test-takers, then 18 remain after the first 5 group leaves. Otherwise, there are 19 left. These must come in groups of 5 or more. We can write
 
18 = 5+5+8 = 6+6+6
19 = 5+5+9 = 5+6+8 = 5+7+7 = 6+6+7.
 
Since the novice and his sister leave at the same time after the 5 groups do, and have different colors, there are two groups of 6 or more with the same number of members. Therefore we can say 18 = 6+6+6, while 19 = 5+7+7 or 6+6+7.
 
Hence, if the Preacher is one of the 31 people, the table cleared after the 5th bell. If the Preacher was not one of the 31 people, the table cleared after the 6th bell.

 
In this analysis, I have made one unmentioned assumption.  Wink
« Last Edit: Jan 26th, 2008, 8:36am by Icarus » IP Logged

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Re: Variation of red eyed monks  
« Reply #2 on: Jan 26th, 2008, 1:09pm »
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on Jan 26th, 2008, 8:17am, Icarus wrote:
In this analysis, I have made one unmentioned assumption.  Wink

The unmentioned assumption is that there is no intruder. Your proof won't work without that.
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StallionMang
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Re: Variation of red eyed monks  
« Reply #3 on: Jan 27th, 2008, 1:13pm »
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I didn't come up with any formal analysis.  I just used a little trial & error.  Here's a result that fits the constraints of the riddle:
 
1st bell: 2 greens and 2 blues leave
2nd bell: 3 reds leave
3rd bell: no one leaves
4th bell: 5 yellows and 5 oranges leave
5th bell: no one leaves
6th bell: 7 pinks and 7 purples leave
 
The novice and his sister are a pink and a purple in this case.
 
This totals 31 people.   I think it is safe to assume that there are 31 people, not 30, as the riddle clearly states that the 31 people are all listening to the wizard, which implies that the wizard is not to be counted among them.

« Last Edit: Jan 27th, 2008, 1:14pm by StallionMang » IP Logged
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Re: Variation of red eyed monks  
« Reply #4 on: Jan 27th, 2008, 8:41pm »
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I guess it does. In any case, it was a well written puzzle. Thanks, inexorable.
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Re: Variation of red eyed monks  
« Reply #5 on: Jan 28th, 2008, 8:55am »
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on Jan 26th, 2008, 4:58am, inexorable wrote:
The Novice and his sister, both with dots of different
colours stood up shortly after that, but every one of them did this
before the last bell.

 
What does "every one of them" mean here?
I suppose it does not mean "both of them" so "them" are the 31 persons. Am I right?
 

I had problem with the start and I have read the Icarus solution the "at lest 2 dots of used color assumption". Still I had problems with 667/577 branches. And I should thing about nonlogicans among them.
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Joe Fendel
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Re: Variation of red eyed monks  
« Reply #6 on: Jan 28th, 2008, 3:36pm »
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on Jan 28th, 2008, 8:55am, Hippo wrote:

 
What does "every one of them" mean here?
I suppose it does not mean "both of them" so "them" are the 31 persons. Am I right?

 
This was a point of confusion for me also, but I interpreted "them" as meaning "everyone we've discussed thus far - the 4 who left initially, the reds, the ones who left at bell 4, the novice, and his sister.  All of them left before the last bell.
 
And leaving before the last bell (rather than on it) seems important:
hidden:

We know that the novice comes from a group of at least 6, as does his sister, but since they left before the last bell, there must be a group of at least 7 also there.
 
This accounts for 2+2+3+5+6+6+7 = 31 people (who don't include the Orator - since "they" (the 31) "carefully listened to" him.)
 
6 bells seems to me to be the answer.
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Re: Variation of red eyed monks  
« Reply #7 on: Jan 28th, 2008, 3:57pm »
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This is why I don't like long puzzles in English where little changes in understanding make such big difference Wink
 
... there was no last bell the bell will ring periodically forever. But we are to count the number of ringings before the last person left.
 
And nonlogicans will influence the behaviour drasticaly ... May be from the formulation we should obtain that logicans "ban" intruders imediately so remaining logicans would not be confused...
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Re: Variation of red eyed monks  
« Reply #8 on: Jan 28th, 2008, 4:19pm »
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on Jan 28th, 2008, 3:57pm, Hippo wrote:
This is why I don't like long puzzles in English where little changes in understanding make such big difference Wink

I disagree.  I'm rather fond of inexorable's storytelling style.  I find solving puzzles to be more fun when there is an interesting backstory to them.
 
on Jan 28th, 2008, 3:57pm, Hippo wrote:

And nonlogicans will influence the behaviour drasticaly ... May be from the formulation we should obtain that logicans "ban" intruders imediately so remaining logicans would not be confused...

I believe that the suggested possibility of "an intruder" is simply a storytelling device to explain the dots.  But it is interesting to see how such intruders would affect things.
 
hidden:
For example, any lone intruder who rises when he should stay seated is smoked out immediately by logicians of different colors, while any intruder who stays seated when he should rise is discovered immediately by logicians of the same color.  Presumably, the discovery of an intruder would stop the experiment and cause it to be reset while the intruder is removed and banned.
 
More amusing, however, is the idea of multiple intruders.  Suppose, for example, there are two intruders, the only two with a yellow dot.  If they fail to rise on the first bell, then it might go unnoticed at first (if there are no color groups with precisely three people) but everyone else will rise on the second bell!
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Re: Variation of red eyed monks  
« Reply #9 on: Jan 28th, 2008, 8:24pm »
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I interpreted the phrase "every one of them did this before the last bell" to mean that everyone eventually left. I admit it was confusing, but I don't see any other interpretation for it that would fit.
 
The Orator would make his decisions on who should have left at each bell based on what has happened at previous bells. For instance if both black dots are on impostors, who stay seated at the first bell. The Orator not only knows that they are impostors, but also expects that every real logician will arrive on the next bell, regardless of their color.
 
A much more serious flaw with this test is that there is still a significant chance that an impostor will arrive at the right time anyway. Particularly if he happens to be in one of the maximum color groups. If I were an impostor in this situation, without the ability to understand why people are getting up when they are, I might be tempted to choose a big group and simply leave when they leave. Certainly, if everyone else got up and started to leave, I'd jump up and join them, pretending to have been delayed slightly in adjusting my clothes.
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Re: Variation of red eyed monks  
« Reply #10 on: Jan 29th, 2008, 12:04am »
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on Jan 28th, 2008, 4:19pm, Joe Fendel wrote:

I disagree.  I'm rather fond of inexorable's storytelling style.  I find solving puzzles to be more fun when there is an interesting backstory to them.

 
I have not say it is not nice, it is my personal problem as English is not my native language so I have sometimes problems to find small differences in meaning ...
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