# The three wise men problem

Discussion in 'General Philosophy' started by arfa brane, Jan 17, 2019.

1. ### RainbowSingularityValued Senior Member

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7,447
not so
false logic
yet sounds logical
it is a ratio issue
3 showing 2 to 2 = 1 of 2
1 of 2 is the answer
50% of the total number of a needed 60% result
the trick is the 66.66'%
1 person will see 2 hats and the 1st to speak the correct result is the winner...
"not speaking" is not a solution to the observer of the speaker whom is being looked at by the one with their eyes closed.
closing eyes confirms 2nd place

3. ### SarkusHippomonstrosesquippedalo phobeValued Senior Member

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10,406
No it doesn't have to be unfair. If all three hats are of the same colour then every "wise man" is in exactly the same position as the other two with regard what they can know, and thus what they can logically conclude.

Only when they know that the hats are not all of the same colour do you introduce any unfairness into the equation, such that only one can conclude which colour they are wearing,

They have a chance if they believe the test to be fair, in that they each have the same information from which to extract the answer.
This doesn't follow... For the same reason that he has kept his eyes shut, so might one of the others, or simply are refraining from speaking for the reasons you think this one is keeping his eyes shut.
You are also assuming that he has reasoned that the King has not made all three wear the same colour hat, but on what basis do you make this assumption? Certainly nothing in the setup that you detailed.

Basically you can make up any additional assumptions to arrive at whatever conclusion you want. The "riddle" is thus pretty pointless as there is no correct answer, and no answer that can be given bas d on the information provided alone.

5. ### SarkusHippomonstrosesquippedalo phobeValued Senior Member

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10,406
If you say there is at least one red and one white then the test isn't fair, as the only person who can say with certainty which colour hat they wear is the one who sees two hats of the same colour.
He would know just from looking at the colour of the hat the other two are wearing. If he doesn't speak out then none of them do, as each could only know that one of the other two is seeing two hats of the same colour and the other is seeing one hat of each colour. But unless they can tell which is which person who is seeing two of the same colour, they can't discern the colour of the hat they are wearing.

And all the while the advantage is with the person who can know just from observing the colour of the others' hats.
So very unfair.
Not true.
Let the guy who keeps his eyes shut be A, and the other two be B and C.
Assume that B and C both wear a red hat.
B and C will both see a red hat plus the hat colour of A - so both will see either two red hats, or a red and white hat.
A will see two red hats.
Since the only proviso is that there is at least one red hat, and each sees at least one red had, irrespective of what colour hat A is actually wearing, then none of them can say with certainty what colour hat they are wearing.
The only time they can is if they also assume, or are told, that the test is fair - and the only way that can happen is if all three hats are red. And if the other two wise men know this then they will know the colour of their own hat is red simply by being told that at least one is.

7. ### RainbowSingularityValued Senior Member

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7,447
not "fairness"
Fairness is irrelevant
to be correct is the only premise of absolute position

would a wise-women live & act in assumption that life is fair
fairer for the fairer sex

8. ### billvonValued Senior Member

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21,646
Yep. I suspect OP left out a few important bits.

9. ### arfa branecall me arfValued Senior Member

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7,832
It's unfair because the fairness of each hat being the same color means each of the three has an equal chance of guessing the color of their own hat. Having been told there are two colors without additional information means they each have the same chance: 50/50 of guessing.
But they aren't told the hats are all the same color. They only know their own hat is red or white, apart from what they can see.
He reasons this because he also reasons the king isn't playing a trick on them, and he also reasons that the other two can't decide, based on what the king has said and what they can see.
I don't say that, nor does the king.
It is true.
A keeps his eyes shut and reasons thus: that at least one hat is red, and the king isn't playing a trick on three wise men. Also he assumes the other two have their eyes open.

Suppose there is one red hat, then the first man to see two white hats knows his own must be red. This gives one of the three an unfair advantage (the king wouldn't do that, he reasons). Suppose there are two red hats, then there are two possibilities: he will see two red hats or he will see a red and a white hat. Because there must be more than one red hat (otherwise it's unfair by the previous reasoning).

If he sees two red hats but is wearing a white hat, that gives the other two an advantage he doesn't have, likewise if he sees a red and a white hat then one of them is disadvantaged and the test is unfair.

Therefore he has to conclude, given the king has set a fair test and there is at least one red hat, all the hats are red.

My version removes the information "there is at least one red hat". Does it still fly and can it be a fair test? I think it still can because all three can reason the same way, eyes open or closed. They now only have that the test might not be fair and the hats are white or red. One man keeping his eyes closed "fairs up" the otherwise unfair situation.

Fairness and balance aren't the same thing. The test is fair if each contestant has the same chance, knows the same information, etc. But a balanced test needn't be fair.

So without the information "at least one hat is red", the test isn't less fair since each man has the same chance, of guessing, of reasoning with what they do know or assume (reasonably!).

Last edited: Jan 18, 2019
10. ### SarkusHippomonstrosesquippedalo phobeValued Senior Member

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10,406
You do know you have disagreed with me and then pretty much stated what I said?
It's also not a test of wisdom, which you claimed it to be, as guessing is not a matter of wisdom. If they can reason via other information not contained in the detail you provided then your answer is a non sequitur, and meaningless with regard the challenge set.

11. ### arfa branecall me arfValued Senior Member

Messages:
7,832
FFS

Three wise men are summoned by the king. He says he want to test their wisdom and the test will be equally fair for all three men.
The king tells them each will be wearing a hat which is either red or white, and they won't be able to see their own hat.
Then he tells the three that at least one of the hats worn will be a red hat.
The first wise man to say he knows the color of the hat he's wearing, wins the chocolate fish.

Is the test fair? Is the test balanced?
Do you know what the difference is?

Suppose the king doesn't tell them the last part--all they know is the test will be fair (which a wise enough man might assume without being told, given the king isn't stupid), and that the hats are red or white. How does that change the fairness and the balance, of the problem?

Each wise man is free to reason about the wisdom of the test itself--is it fair? Is it balanced?

How about this test (not necessarily of wisdom). Three men go to a fairground and pay to try shooting a target with a gun (there's only one gun, it looks like a BB gun). The gun's barrel is bent and the sights are way out. Does each man have an equal chance? Is the "test" fair?

Last edited: Jan 18, 2019
12. ### DaveC426913Valued Senior Member

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18,960
From this information alone, the wise men should be able to deduce that they must all be the same colour, including their own.

So, the answer to the OP question, why does A not open his eyes is: he doesn't need to. His rivals will both say 'red', therefore his is the same.

The King will realize that A is the wisest, having deduced the correct answer without even opening his eyes.

13. ### RainbowSingularityValued Senior Member

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7,447
it may be helpful if you draw it on paper.
since i have had practical experience watching all sorts of different types of professionals try and solve similar maths problems i have a small advantage in being able to understand where the error is.
so i know 3 things
1 the error you are making is an error
2 how the error is repeated and made(not clinically however i am just guessing, you would need to ask a social-anthropologist as well as a professor of psychiatry[very good magicians may know the psychology behind it also]).
3 it is a relatively common error when working with 2 different equations and intermixing their formula & results to get a single value answer.
it is like a left handed right handed type brain exercise, like changing hands in your mind with thinking.

when you supplant a per-conceived value as a set function into a process of an equation it tricks many peoples brains.
the trick is mostly in the person not working through the long version of the equation and jumping off to the other equation.

magicians use part of this human error process to do magic tricks.
i don't study. i only fell into by professional need with a need to resolve it quickly to 100% while not embarrassing the professional.
not easy.

the formula result of the value of your first equation set is wrong.
your adding the total value and the equation to get the sum value of hat-seerers" into a single set which is the error.
the sum value is 1 person sees 2 hats
that doesn't change even though the required need is binary to see 2 different values of only 2 values.
this is where the error sits
i see people do it all the time.

Last edited: Jan 18, 2019
14. ### arfa branecall me arfValued Senior Member

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7,832
That's more or less it. One of the men realises he doesn't need to look at the other hats, because two of them are looking. So he can demonstrate his wisdom even if the other two get the right answer.

So what if none of them are told "at least one hat is red"? That was my approach to this, I might have gotten it wrong, perhaps a wise man would reason that the king can only make the test fair if they all wear the same color hat, so they're all red or all white.
On the other hand, he might think about a random selection, in which case there are 8 combinations, only two of which are all red or all white. He might also think about why the king would make the combination random, assuming the king (no fool) has set a fair test.

15. ### arfa branecall me arfValued Senior Member

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7,832
Meaningless . . .
unless you 1. specify what the error is, and 2. explain why it is an error.

Otherwise you might as well fart loudly.

16. ### DaveC426913Valued Senior Member

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18,960

So, the test can either be fair, or it can have a random setup for each of them, but it can't be both.

Since the King declared it to be fair, they must all start with the same information.

17. ### RainbowSingularityValued Senior Member

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7,447
i have edited my post after you have replied, to add some content for you.
hope its useful.

if you cant set your ego aside and just do the math you wont be able to see the error.
also if your low in blood glucose or electrolytes(blood oxygen etc) you also wont be able to see the error.
also if your suffering from mental fatigue, you wont be able to see the error.

18. ### arfa branecall me arfValued Senior Member

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7,832
No, 3 people see two hats, unless one of the three has their eyes closed.
You say you see an error.
I say I can't see what the hell you're talking about.

19. ### arfa branecall me arfValued Senior Member

Messages:
7,832
Even if the king didn't tell them, maybe a wise man would assume a king wouldn't go to all this trouble then set an unfair test. That is, part of the test is one or more of the wise men making that assumption, so the king doesn't tell them the test is fair . . .

I see my initial response to Sarkus contains an error, a mistaken assumption. Sarkus is correct that the only fair test is all hats the same color, therefore.

20. ### DaveC426913Valued Senior Member

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18,960
Perhaps, but, with such an open puzzle, solvers shouldn't really assume that. Simply adding a 'wise' or 'benevolent' King would indicate a lack of unfair outcomes.

21. ### RainbowSingularityValued Senior Member

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7,447
you are wrong

is 1 person not wearing a hat ?

22. ### arfa branecall me arfValued Senior Member

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7,832
How many times do you need to read something?

Once again: The king tells them each will be wearing a hat which is either red or white, and they won't be able to see their own hat.

There are three men. How many of them can see the two hats the other two men are wearing if they open their eyes?