Colour of your HAT ???

Originally posted by Irax
Whiterose gave the correct answer. It's a pure logic problem, and not a problem of probability.

a) If someone saw 2 black hats, then they'd know for sure that their hat was white.

b) If someone saw 1 black hat. They'd be able to work out that their hat was white, because nobody used the logic in part a

c) If someone saw 0 black hats (2 white hats), th ...[text shortened]... also wise, and they have the required motivation, you know they're not going to lie to you.

I called it a logic problem but you are quite right I treated it like a probability problem. I wasn't too convinced by whiterose's explanation but yours removes any shadow of a doubt. I like those ones particularly when they produce an answer that is counter-intuitive.

**Sigh** Yet more sucky, mechanical, hacneyed thinking: I appear to be going through the motions recently.

Nice answer though, thanks Irax, and whiterose.

But if the other two princes are also wise, why didn't they reach the same answer, since the situation is symmetric?

Originally posted by Irax
Whiterose gave the correct answer. It's a pure logic problem, and not a problem of probability.

a) If someone saw 2 black hats, then they'd know for sure that their hat was white.

b) If someone saw 1 black hat. They'd be able to work out that their hat was white, because nobody used the logic in part a

c) If someone saw 0 black hats (2 white hats), th ...[text shortened]... also wise, and they have the required motivation, you know they're not going to lie to you.

To be completely fair, this logic puzzle is pretty poor, as in any of the situations, the other two people can work out the colour of their hat. We have to make the stupidly precise assumption that if your hat is black, they work out the colour of theirs, but if you're hat is white, suddenly they aren't as clever as you?

Originally posted by doodinthemood
To be completely fair, this logic puzzle is pretty poor, as in any of the situations, the other two people can work out the colour of their hat. We have to make the stupidly precise assumption that if your hat is black, they work out the colour of theirs, but if you're hat is white, suddenly they aren't as clever as you?

The whole point is for the king to find out who is the smartest and the bravest. If all the hats are white, nobody knows for sure what colour their hat is(unlike if one hat is black, is which case at least one person can be certain of their own hat colour). Therefore, the bravest person will guess first, risk everything, and get the princess.

Just because that this is an EQULE Chance to all the three princes (to make this competetion a "fair" one), equal probablities would be given to all the 3 princes to 'guess' the color of their hats.

This is possible only when All the 3 princes are given white color hats.
Else, probability of 'guessin' the color of their own hats will be UNEQUAL.

So, the color of the hat is WHITE.

Originally posted by GregM
But if the other two princes are also wise, why didn't they reach the same answer, since the situation is symmetric?

Because the puzzle states that they are all intelligent. It doesn't say that they are all equally intelligent, but rather assumes that one would have to be more intelligent than the other two in order to win.

Originally posted by Irax
Whiterose gave the correct answer. It's a pure logic problem, and not a problem of probability.

a) If someone saw 2 black hats, then they'd know for sure that their hat was white.

b) If someone saw 1 black hat. They'd be able to work out that their hat was white, because nobody used the logic in part a

c) If someone saw 0 black hats (2 white hats), th ...[text shortened]... also wise, and they have the required motivation, you know they're not going to lie to you.

Unless of course, they are even more intelligent than you and decide to deceive you by not acting appropriately (although I understand the solution and have seen this before).

Game theory always dictates that in order to win you must act in a manner that puts your opponents at the greatest disadvantage, or give yourself the biggest advantage. As you cannot gain any advantage in this problem, the logical extension would be to not react, even if you know the colour of your hat. This would eliminate your opponents.

FYI- Here's another version of the problem:

The king blindfolds the 3 princes. He then puts a black cap on each of them... he then removes the blindfolds and says to them "Each of you now wears either a black or a white cap. Each of you should raise your hand if you see a black cap, and lower your hand as soon as you know the color of your own cap". All hands remained raised for a couple of minutes... then, one of the princes eventually lowers his hand and correctly says, "my cap is black".

By working out the logic of the different combinations, the smartest prince correctly deduced that putting black caps on all three princes is the only combination whereby it would be impossible for any of the princes to IMMEDIATELY deduce the color of his own cap.

Originally posted by Gastel
As you cannot gain any advantage in this problem, the logical extension would be to not react, even if you know the colour of your hat. This would eliminate your opponents.

Come again?

This is an old problem and is actually phrased incorrectly.

In the original problem which I heard years ago the first prince (philosoper, wiseman, etc) says he doesnt know, the second says he doesnt know. The third deduces the answer because of the first two 'passes'.

It is ONLY possible to get the answer after 2 princes have said they dont know.

Originally posted by wolfgang59
This is an old problem and is actually phrased incorrectly.

In the original problem which I heard years ago the first prince (philosoper, wiseman, etc) says he doesnt know, the second says he doesnt know. The third deduces the answer because of the first two 'passes'.

It is ONLY possible to get the answer after 2 princes have said they dont know.

This is how I heard it, too.

Originally posted by Palynka
Come again?

The game falls apart a bit because everyone can figure out their hat colour once somebody speaks. (Person A says, I know my hat, and therefore gives information to person B and 3.) If you want others to make a mistake you have to not give the information you have.

Originally posted by Gastel
The game falls apart a bit because everyone can figure out their hat colour once somebody speaks. (Person A says, I know my hat, and therefore gives information to person B and 3.) If you want others to make a mistake you have to not give the information you have.

Yes, but if you know what colour is your hat the best strategy is obviously to say it. Game over, you win.

Am I missing something here?

Originally posted by Palynka
Yes, but if you know what colour is your hat the best strategy is obviously to say it. Game over, you win.

Am I missing something here?

You are relying on your opponents to let you win (like a helpmate). But a wrong answer might mean death, so to win you might decide to act incorrectly and let the others die, therefore allowing you an easy win. (Its called an alternative solution, and I like it because it does not rely on the play of others to give you a method of play.)

Originally posted by Gastel
You are relying on your opponents to let you win (like a helpmate). But a wrong answer might mean death, so to win you might decide to act incorrectly and let the others die, therefore allowing you an easy win. (Its called an alternative solution, and I like it because it does not rely on the play of others to give you a method of play.)

But then you don't know what the colour of your hat is, if you're not sure.

Let's try to write up the strategy (symmetric across opponents):

1) As long as you don't value the death of your opponents, then you'd say the colour of your hat immediately when you deduce it.
2) If you're not sure, then you shouldn't say anything.

Since this is valid for all players, this means that you can rationally conclude by the others silence that they must be in point 2). This doesn't solve the problem of who will deduce it first, but I don't see how the best strategy is always to keep silent.

Edit: I'm not trying to be pedantic here, I'm interested to see this through since I like game theory.