Riddles

[Firemeboy]

No, this new riddle isn't a political joke.  If you want to, you can change it to prime minister or king or czar or somethin.

And yes, you are wearing a hat, and no, you may not peek at it, but if you are logical, you can figure out what color hat you are wearing.

[Eric James Stone]

I know the hat one because I've heard it before.  Am I allowed to answer?

[Firemeboy]

Well, it doesn't look like anybody else is taking a stab at it...

[Legion]

I have now seen the error of my logic....sorry for all the grive I have been given you about this I can be a little thick skulled at times.....

[Eric James Stone]

You and two other very smart people were brought before the President.  He want's to see which of you is the smartest, so you can figure out how to lower gas prices.  

You are led blind-folded into a small room, and seated around a table.  The president describes the test.

"Upon each of your heads I have placed a hat. You are either wearing a blue hat or a white hat. You don't know which, but I will tell you this; at least one of you is wearing a blue hat. There may be only one blue hat and two white hats, there may be two blue hats and one white hat, or there may be three blue hats. But you may be certain that there are not three white hats."

The president explains that when the blindfolds are taken off, the first to correctly announce the color of his hat shall be his advisor.  

With that, the president uncovers your eyes and you see that your two competitors are each wearing blue hats. You see from the look in their eyes that they are thinking, "What is the color of my hat?"

For hours nobody speaks, then finally you stand up and say, "The color of the hat I am wearing is..."  

What color is your hat?  And how do you come to that conclusiong?

My hat is blue and here's why:

There are three possible configurations of what each of the three contestants could see.

1. 2 white hats.  If you see this, you would know instantly that you had a blue hat, because there cannot be more than two white hats, so you would announce the answer and win the contest.  Since someone did not do this, we know nobody saw this configuration.

2. 1 blue, 1 white.  If you see this, then you have to ask yourself what color your own hat is.  If it were white, then the guy in the blue hat would see configuration #1, and so he would announce his answer and win the contest.  Since he doesn't do that, your hat must be blue.  So you would announce the answer and win the contest.   Since someone did not do this, we know nobody saw this configuration.

3. 2 blue.  If you see this, then you have to ask yourself what color your own hat is. If it were white, then the other two contestants would see configuration #2, and  one of them would announce the answer and win the contest.  Since neither of them do that, you know your hat must be blue.

Since I see two blue hats, if my hat were white, then one of the other two very smart people would have already won this contest, instead of us sitting around here for hours.  So my hat must be blue.

[Eric James Stone]

Legion, you're not alone.  The first time I heard the puzzle it took me quite a while to understand why it paid to switch.

[Firemeboy]

That is correct Eric, you're up...

[Eric James Stone]

On a certain island there are two tribes of people.  The people of the Truthteller tribe always answer questions truthfully.  The people of the Lieteller tribe always answer questions by lying.

There's also a group of outcasts from both villages.  The outcasts sometimes tell the truth and sometimes lie, without any consistent pattern.

You have landed on this island, and you need to get to the village of the Truthtellers, but you don't know whether it's the east village or the west village.

On the road between the two villages, you encounter three people.  Because they each have different tribal markings, you know that one is a Truthteller, one is a Lieteller, and one is an outcast.  You don't know which is which, but at least their tribal markings look like an A, a B, and a C, so you can easily tell them apart.

You can ask a total of two yes-or-no questions.  Each question will be directed to only one of the three people, and only that one will answer.  You may direct both questions to one person or direct the first question to one and the second question to another.

What two questions do you ask (and of whom) in order to correctly determine whether the east village or the west village belongs to the Truthtellers?

[Firemeboy]

Well, the castout throws a wrench into the works.   You could pick somebody at random and say, "If I were to ask this person (again, pick somebody at random), "does the truthtellers village lies to the East", what would they say?"

Let's assume that the truthtellers village does lie to the east.  If you asked the person who always lies, they would say, "He would say it does not lie to the East".  And if you asked the person who always tells the truth, he would say, "He would say it does not lie to the East."  

Or if it did not lie to the east, the liar would say, "He would say it lies to the east."  And the truth teller would say the same thing.  So you would go the opposite way of whatever is said.

And it wouldn't matter if you picked the castout, because again, their answer would be the same.

The only tricky part is I don't know how the person would answer if you picked a castout as the 'other person'.  Because the first person wouldn't know how that person might answer.  

So maybe I will pose that question to you.  If I were to pick the liar, and ask him how this person would answer that question (and happened to pick the castout), what would they answer?

Am I on the right vein here?

[Eric James Stone]

Let's assume that the truthtellers village does lie to the east.  If you asked the person who always lies, they would say, "He would say it does not lie to the East".  And if you asked the person who always tells the truth, he would say, "He would say it does not lie to the East."

Well, they're supposed to be yes-or-no questions, but the principle works.  "If I were to ask him if the Truthtellers village lies to the east, would he say yes?" If there were only the Truthteller and the Lieteller, either of them would reply "No" when asked about what the other would say, and you would know that answer was false, and that the village truly was to the east.

If I were to pick the liar, and ask him how this person would answer that question (and happened to pick the castout), what would they answer?

Hmmm. Interesting question.  If you pointed at the outcast and asked the Truthteller, "If I asked that man if the Truthteller village is to the East, would he answer 'Yes'?" then the only truthful response would be "I don't know."

So, I guess the only thing the Lieteller could do would be the opposite: "I know."

Am I on the right vein here?

It's not the vein of the answer I'm familiar with, but it may lead to a correct answer.

[Firemeboy]

If that is the case, then the two questions I would ask are as follows...

I would pick one person at random, and ask, "If I asked him (pointing to a fellow), if the truth village lies to the East, would he say yes?"

If he answered "I don't know, or I know," Then I would ask the same fellow the same quetsion, but about the next man.  Then I would ascertain the answer, go the opposite way, and find the village.

This way you might find the answer in one question.

But, it sounds like you are looking for something else.  I'll have to think about this one some more.

[Eric James Stone]

You haven't dealt with what you do if the person you ask first was the outcast.

If you're pointing at the Truthteller, and the outcast tells the truth, then the answer will be "Yes."  

If you're pointing at the Truthteller, and the outcast tells a lie, then the answer will be "No."

If you're pointing at the Lieteller, and the outcast tells the truth, then the answer will be "No."  

If you're pointing at the Lieteller, and the outcast tells a lie, then the answer will be "Yes."

Thus the answer to the first question cannot tell you whether the village is in the east.

[Eric James Stone]

That said, the first question you have proposed does elicit sufficient information for you to solve the puzzle, if you ask the right second question of the right person.  It's not the same first question I would have used, but in analyzing it, I'm pretty sure it works.

[Firemeboy]

Ah...  I see.  Yes, it is that pesky unknown that causes a problem.  But assuming that you ask two people, you can be assured that one of them is not the outcast.

But I'm still curious about the solution you are thinking of, so I'm going to keep thinking.  I'm assuming there is some question that allows you to isolate or identify one of the players.

[Firemeboy]

Let's see. You could ask the question, "If I ask him a question, will he always tell me the truth?

If you got the answer yes, then you would know either
A - Outcast, telling the truth about the truth teller
B - Liar telling a lie about the outcast.
C - Outcast telling a lie about the liar

If you got a no:
A - Outcast telling the truth about the liar
B - Truth teller telling the truth about the liar
C - Truth teller telling the truth about the Outcast
D - Outcast telling a lie about the Truth Teller
E - Liar telling a lie about the truth teller

Which tells you absolutely nothing.  Though if the answer if yes, you know for certain that the person you just asked is not the 'truth teller'.