The New Sudoku Players' Forum

[rjamil]

A king wants to find the wisest person in his land. He sits three philosophers in a circle, facing each other. He shows them five hats: three red hats and two blue hats.

He then blindsfolds the men, places a red hat on each of their heads, and hides the blue hats. When he removes the blindfolds, each man can see the other two men's hats, but cannot see his own.

The king says: "The first person to logically deduce the color of their own hat wins.

"For a long time, the room is completely silent. Suddenly, one wise man stands up and correctly says: "My hat is red.

"How did he know?

R. Jamil

[Leren]

If one of the 3 Wise Man sees 2 blue hats on the other 2 Wise Men he would immediately exclaim with 100% certainty - My hat is red !

Why does he hesitate ? Because he does not see 2 blue hats. The probability of this being the case is 1 - (2/5)*1/4) = 90 %.

The same is true for the second Wise Man. So the probability of two Wise Men hesitating in this way is 90% * 90% = 81 %.

So the third Wise Man deduces from the hesitation of the other two is that his hat is Red with a probability that he is right being 81%. Not a certainty but more likely right than wrong.

Of course if the 3 Wise Men were equally Wise and Courageous they would all answer My hat is red at the same time.

This sort of probabilistic "logic" is used in games of chance like poker, horse racing etc.

[blue]

Suppose I'm looking at two red hats, and wondering if mine is blue.
If it was blue, the others would be looking at one red and one blue hat.
Either one, seeing a blue hat, could deduce from the silence of the other, that his own hat couldn't be blue, and must be red.
After a while, if neither speaks, I know(?) that my own hat isn't blue, and must be red.

I wouldn't bet my life on it, since I could still have a blue hat, and the others might be "slow".
But they're philosophers, right. Could they both be that slow ? ... How long do I wait ?

Cheers,
Blue.

[Leren]

There is a second way to answer this question. It assumes that the king will arrange things such that each of the three Wise Men has exactly the same information necessary for them to deduce the colour of their hat.

There is one and only one way that this can be arranged. The hats are not picked randomly, only three red hats will bring this about (which appears to have been the case in the question).

But in this case the three Wise Men don't even have to put on the hats. They can even all be blind. All that has to be done is that the king tells any blind Wise Men that there are 3 red hats and 2 blue hats.

Since the number of red hats = the number of Wise Men, they can instantly say " My hat will be red", even if they are blind and don't know what red actually is.

So the real question is, what is the question ? That is, what can we assume is True about the circumstances of the question ?

You definitely can't have 2 blue hats dealt out, because the Wise Man who could see those 2 blue hats would have an unfair advantage.

It seems that the whole thing with the blue hats and so on is just one great big red (or perhaps blue) herring.

Leren

[champagne]

Suppose I'm looking at two red hats, and wondering if mine is blue.
If it was blue, the others would be looking at one red and one blue hat.
Either one, seeing a blue hat, could deduce from the silence of the other, that his own hat couldn't be blue, and must be red.
After a while, if neither speaks, I know(?) that my own hat isn't blue, and must be red.

I wouldn't bet my life on it, since I could still have a blue hat, and the others might be "slow".
But they're philosophers, right. Could they both be that slow ? ... How long do I wait ?

Cheers,
Blue.

This looks like my answer long ago to this very old question.
To the question " how long can I wait", the answer is depending on
"how wise are the others in my opinion"
"how many can give the answer"
but I would say that usually some seconds are enough

[dobrichev]

If there is a predetermined (by the king) winner, then my answer doesn't matter. Otherwise, we all need to be put on a level playing field, and the only way to do that is to put red hats on each of us.

[rjamil]

It is fascinating how closely thinking, learning, and personal growth are linked!

Here is the answer in detail with logical reasoning:

The winner figured it out by realizing what the other two men didn't do. He used a process of elimination:

Scenario A (One Red, Two Blue): If the winner was wearing a blue hat, and another man was also wearing a blue hat, the third man would instantly see two blue hats. Since there are only two blue hats in total, that third man would have immediately jumped up and known his own hat was red. But nobody jumped up.

Scenario B (Two Red, One Blue): If the winner was wearing a blue hat, the other two men would see one red and one blue hat. If either of them thought, "What if my hat is blue too?", they would realize it would trigger Scenario A (someone else would have guessed instantly). Because neither of them jumped up after a few minutes, the winner realized his hat could not be blue.

The Conclusion: Since his hat being blue would make the answer obvious to the others, and nobody was answering, the winner realized his own hat must be red.

R. Jamil

[Hajime]

To keep the wise men in line, a timestamp needs to be added. Like 1 minute thinking and you have answer or not. Goto next minute.

It is like the brown eyes/blue eyes problem on an island, I could not find a suitable URL
You don't need a wizard or guru.

[rjamil]

I could not find a suitable URL

Well, is this or this link explain the Blue Eyes puzzle?

You don't need a wizard or guru.

I am using Google AI to show me some really tough riddles.

R. Jamil