What is the answer to the riddle of the king and the three green dots?

[Howard]

The king is about to die and has no heirs to this throne. He sent messengers throughout the land seeking the 3 smartest people. The messengers return with three people. He now gives them all a task to see which one is the wisest.

He tells them, "I have seated you in an equilateral triangle so that each of you faces two others. While you are blindfolded I will paint a dot on each of your foreheads. Each dot will be red or green so that there can be any combination of red and green dots, for example, 1 red and 2 greens, or all red, etc. When I remove the blindfolds each of you must raise your hand if you see _any_ green dots, i.e. 1 or 2 dots. As soon as you have figured out what color your own dot is, lower your hand and tell me how you knew."

So he paints a green dot on all three foreheads. When the blindfolds are removed, all three hands go up. After a long pause, the wisest person says, "Your highness, I have a green dot." How did he know?

 

[Ness]

Let's name these people person A B and C.

We'll say that person A is the wisest one.

Now, since A B and C raised their hands, that must mean that at least 2 of them have green dots. If two of them were red, the person with the green dot would not raise their hand, and if all 3 were red, no one raised their hands.

So that means at least two people had green dots.

This meant that A B and C all had a view of a green dot...

Now, person A raised his hand, so he saw one of the other green dots, possibly two. He also saw that the other person with a green dot raised their hand, so we can deduce that either he has a green dot, or all the people have one.

Now, if A looks and sees a red dot on another person's forehead, let's say B, then he sees a green dot on C's forehead, he knows that He has a green dot too, otherwise C would not be able to raise his hand because neither A or B would have a green dot.

In this situation, both A and C could answer this question, but B has no way to determine, after seeing two green dots, what color he has.

 

[JohnCU]

The solution is more confusing than the riddle itself.

 

[Howard]

I'm not understanding this paragraph:

Now, person A raised his hand, so he saw one of the other green dots, possibly two. He also saw that the other person with a green dot raised their hand, so we can deduce that either he has a green dot, or both other people have one.

 

[Howard]

The other two people DO have green dots, so maybe the required 2 green dots are on the other two and not on A himself.

 

[SammyBoy]

because if only 2 of them had a green dot, one of the green ones would have known that the other green one would only raise their hand if they saw another green dot. Since 1 has a red, the other person they are seeing has to be a green too. If they were really wise it'd be a race between the 2 greens to lower their hands.

Since these are all wise people, and no one figured it out and lowered their hand quickly, the wisest one knows that they all have to be green or else 1 of the other 2 would have already figured it out.

 

[Lithium381]

that one baffled me, but logic makes sense out of the solution

 

[Ness]

Originally posted by: Howard
I'm not understanding this paragraph:

Now, person A raised his hand, so he saw one of the other green dots, possibly two. He also saw that the other person with a green dot raised their hand, so we can deduce that either he has a green dot, or both other people have one.

A saw at least one green dot, otherwise he would not have been able to raise his hand.

If he notices that one person has a red dot (B) then he knew that C would not be able to raise his hand if A also had a red dot, because you cannot raise your hand unless you see a green dot.

 

[Howard]

Originally posted by: SammyBoy
because if only 2 of them had a green dot, one of the green ones would have known that the other green one would only raise their hand if they saw another green dot. Since 1 has a red, the other person they are seeing has to be a green too. If they were really wise it'd be a race between the 2 greens to lower their hands.

Since these are all wise people, and no one figured it out and lowered their hand quickly, the wisest one knows that they all have to be green or else 1 of the other 2 would have already figured it out.

Great work!

 

[Cyberian]

Originally posted by: ness1469
Let's name these people person A B and C.

We'll say that person A is the wisest one.

Now, since A B and C raised their hands, that must mean that at least 2 of them have green dots. If two of them were red, the person with the green dot would not raise their hand, and if all 3 were red, no one raised their hands.

So that means at least two people had green dots.

This meant that A B and C all had a view of a green dot...

Now, person A raised his hand, so he saw one of the other green dots, possibly two. He also saw that the other person with a green dot raised their hand, so we can deduce that either he has a green dot, or all the people have one.

Now, if A looks and sees a red dot on another person's forehead, let's say B, then he sees a green dot on C's forehead, he knows that He has a green dot too, otherwise C would not be able to raise his hand because neither A or B would have a green dot.

In this situation, both A and C could answer this question, but B has no way to determine, after seeing two green dots, what color he has.

I was with you right up to here.
Wouldn't all 3 men have had the same information to work with if all 3 raised their hands?

 

[DrPizza]

Alternate solution:

Two of the men are much much wiser than the other.... they each see a green and a red dot. Each quickly realizes that they must have a green dot, but hate the 3rd man. By not lowering their hands, the 3rd man follows the logic above, thinks he's green, and is beheaded for not being as smart as the other two.


edit: typo

 

[DrPizza]

...continued from previous post.
The two remaining smart men then form the first co-monarchy (or would that be bi-archy??) and share the Coors twins (Diane and Elaine Klimaszewski)
Pics

 

[Ness]

Originally posted by: Cyberian

Originally posted by: ness1469
Let's name these people person A B and C.

We'll say that person A is the wisest one.

Now, since A B and C raised their hands, that must mean that at least 2 of them have green dots. If two of them were red, the person with the green dot would not raise their hand, and if all 3 were red, no one raised their hands.

So that means at least two people had green dots.

This meant that A B and C all had a view of a green dot...

Now, person A raised his hand, so he saw one of the other green dots, possibly two. He also saw that the other person with a green dot raised their hand, so we can deduce that either he has a green dot, or all the people have one.

Now, if A looks and sees a red dot on another person's forehead, let's say B, then he sees a green dot on C's forehead, he knows that He has a green dot too, otherwise C would not be able to raise his hand because neither A or B would have a green dot.

In this situation, both A and C could answer this question, but B has no way to determine, after seeing two green dots, what color he has.

I was with you right up to here.
Wouldn't all 3 men have had the same information to work with if all 3 raised their hands?

No, in order to raise your hand, you need only to see at least one green dot. The guy with the red dot would see a green dot, but the guys with the green dots would see a red and a green.

The guy with the red dot can not determine if the other two raised their hands because they saw each other, or because they saw him.

 

[fatkorean]

Originally posted by: Cyberian

Originally posted by: ness1469
Let's name these people person A B and C.

We'll say that person A is the wisest one.

Now, since A B and C raised their hands, that must mean that at least 2 of them have green dots. If two of them were red, the person with the green dot would not raise their hand, and if all 3 were red, no one raised their hands.

So that means at least two people had green dots.

This meant that A B and C all had a view of a green dot...

Now, person A raised his hand, so he saw one of the other green dots, possibly two. He also saw that the other person with a green dot raised their hand, so we can deduce that either he has a green dot, or all the people have one.

Now, if A looks and sees a red dot on another person's forehead, let's say B, then he sees a green dot on C's forehead, he knows that He has a green dot too, otherwise C would not be able to raise his hand because neither A or B would have a green dot.

In this situation, both A and C could answer this question, but B has no way to determine, after seeing two green dots, what color he has.

I was with you right up to here.
Wouldn't all 3 men have had the same information to work with if all 3 raised their hands?

This is because if one of the men had a red dot, then he wouldn't have the same information as the other two who are green.
A=red, B=green, C=green
A raises hand because he sees B,C with green
B raises hand because he sees green with C and knows A is red
C raises hand because he sees green with B and knows A is red
Now A sees that all see green but doesnt know what A is
B knows hes green because C sees green but A isn't green so B must be green
C knows hes green because B sees green but A isn't green so C must be green

Now if all three were green, the wisest man would know that since noone lowered their hands (B or C), noone was red so all must be green.

-fk

 

[Cyberian]

Originally posted by: fatkorean

Originally posted by: Cyberian

Originally posted by: ness1469
Let's name these people person A B and C.

We'll say that person A is the wisest one.

Now, since A B and C raised their hands, that must mean that at least 2 of them have green dots. If two of them were red, the person with the green dot would not raise their hand, and if all 3 were red, no one raised their hands.

So that means at least two people had green dots.

This meant that A B and C all had a view of a green dot...

Now, person A raised his hand, so he saw one of the other green dots, possibly two. He also saw that the other person with a green dot raised their hand, so we can deduce that either he has a green dot, or all the people have one.

Now, if A looks and sees a red dot on another person's forehead, let's say B, then he sees a green dot on C's forehead, he knows that He has a green dot too, otherwise C would not be able to raise his hand because neither A or B would have a green dot.

In this situation, both A and C could answer this question, but B has no way to determine, after seeing two green dots, what color he has.

I was with you right up to here.
Wouldn't all 3 men have had the same information to work with if all 3 raised their hands?

This is because if one of the men had a red dot, then he wouldn't have the same information as the other two who are green.
A=red, B=green, C=green
A raises hand because he sees B,C with green
B raises hand because he sees green with C and knows A is red
C raises hand because he sees green with B and knows A is red
Now A sees that all see green but doesnt know what A is
B knows hes green because C sees green but A isn't green so B must be green
C knows hes green because B sees green but A isn't green so C must be green

Now if all three were green, the wisest man would know that since noone lowered their hands (B or C), noone was red so all must be green.

-fk

OK - I'm friggen dense!

If all 3 raised their hands and each one had a view of the other 2, isn't A the wisest only because he figured it out first?

 

[her209]

Three hands when there are at least two people with green dots.

Two hands when there is only one green dot.

No hands when there are no green dots.

 

[fatkorean]

Originally posted by: Cyberian

Originally posted by: fatkorean

Originally posted by: Cyberian

Originally posted by: ness1469
Let's name these people person A B and C.

We'll say that person A is the wisest one.

Now, since A B and C raised their hands, that must mean that at least 2 of them have green dots. If two of them were red, the person with the green dot would not raise their hand, and if all 3 were red, no one raised their hands.

So that means at least two people had green dots.

This meant that A B and C all had a view of a green dot...

Now, person A raised his hand, so he saw one of the other green dots, possibly two. He also saw that the other person with a green dot raised their hand, so we can deduce that either he has a green dot, or all the people have one.

Now, if A looks and sees a red dot on another person's forehead, let's say B, then he sees a green dot on C's forehead, he knows that He has a green dot too, otherwise C would not be able to raise his hand because neither A or B would have a green dot.

In this situation, both A and C could answer this question, but B has no way to determine, after seeing two green dots, what color he has.

I was with you right up to here.
Wouldn't all 3 men have had the same information to work with if all 3 raised their hands?

This is because if one of the men had a red dot, then he wouldn't have the same information as the other two who are green.
A=red, B=green, C=green
A raises hand because he sees B,C with green
B raises hand because he sees green with C and knows A is red
C raises hand because he sees green with B and knows A is red
Now A sees that all see green but doesnt know what A is
B knows hes green because C sees green but A isn't green so B must be green
C knows hes green because B sees green but A isn't green so C must be green

Now if all three were green, the wisest man would know that since noone lowered their hands (B or C), noone was red so all must be green.

-fk

OK - I'm friggen dense!

If all 3 raised their hands and each one had a view of the other 2, isn't A the wisest only because he figured it out first?

No, A doesn't know what his is. To his view, everyone is the same. All three raised their hands. To A's point of view, he could be either a red or a green dot. If hes a green dot, all three raise their hands. If hes a red dot all three still raise their hands.

-fk

 

[blahblah99]

Originally posted by: her209
Three hands when there are at least two people with green dots.

Two hands when there is only one green dot.

No hands when there are no green dots.

Yup.. and how do we determine whether there are three green dots or two?

If there are two green dots and 1 red dot, then either guy with the green dot KNOWS he has a green dot because all three hands are up. The guy with the red dot doesn't have enough information to tell if he's red or green.

 

[Cyberian]

Originally posted by: fatkorean

No, A doesn't know what his is. To his view, everyone is the same. All three raised their hands. To A's point of view, he could be either a red or a green dot. If hes a green dot, all three raise their hands. If hes a red dot all three still raise their hands.

-fk

Thanks - I'm still confused, but I'll sleep on it and see if that helps!

 

[her209]

Okay, I think I got the answer:

This riddle can be proved by contradiction:

There are two cases when all 3 hands go up.
2G and 1R or 3G

Now assume that you have a red dot, also note that you see that the other two guys have green dots. In the case of 2G and 1R dot, either of the guys with the green dot will know immediately what color they have but the person with the red dot cannot know for certain what color they have. Since the two guys with the green dots do not know immediately what color dot they have, it contradicts the assumption that you have a red dot, therefore you have a green dot.

 

[bolomite]

my head herts!

 

[AkumaX]

1) A B and C all have green (3 hands)
2) A has red, B and C has green (3 hands)

now how is A supposed to determine what dot he has

 

[Adul]

he used a mirror duh!

 

[MagicianBdotCom]

Originally posted by: her209
Okay, I think I got the answer:

This riddle can be proved by contradiction:

There are two cases when all 3 hands go up.
2G and 1R or 3G

Now assume that you have a red dot, also note that you see that the other two guys have green dots. In the case of 2G and 1R dot, either of the guys with the green dot will know immediately what color they have but the person with the red dot cannot know for certain what color they have. Since the two guys with the green dots do not know immediately what color dot they have, it contradicts the assumption that you have a red dot, therefore you have a green dot.

Yeah you got it, you have to assume since they're wise, they should know immediatly what they have if you yourself had a red dot. If they didn't then you have a green. Very nice.

 

[Skyclad1uhm1]

Originally posted by: AkumaX
1) A B and C all have green (3 hands)
2) A has red, B and C has green (3 hands)

now how is A supposed to determine what dot he has

If B sees 3 hands go up, and sees a red dot on A, he knows he cannot have a red dot himself, as C would only see 2 red dots (A and B). C raises his hand, so B would have a green dot.

Since everyone raises their hand, and no one lowers theirs with the deduction that, since they see one red dot they themselves must be green, all must be green.