[more a logic question than maths]

The king has spent many months to find a suitor for his only daughter from amongst all the eligible bachelors within his kingdom. The king’s primary criteria is that the man who will marry his daughter will be a man of intelligence.

Following a series of intelligence tests, you have made it to last three, from whom the future son-in-law of the king will be chosen.

The king has devised a final intelligence test, the first of the three remaining suitors to solve the test will win the princesses hand in marriage.


The test:

You and the two other suitors are taken into a room with no windows; drawn on the floor is an equilateral triangle – each of the potential suitors stands on a corner of the triangle – facing inwards, such that each suitor has a clear view of the other two suitors.

The king announces that the lights in the room will be turned off (it will then be dark, such that you cannot see anything), a hat will be placed on each suitors head. When the lights are turned back on, each suitor will be able to see the hats that the other two are wearing, but not the hat that they are wearing themselves (it being on top of their head).
(Note that no one is permitted to remove the hat from their head, or use some other technique to observe the hat they are wearing)

The colour of the hat on each suitor’s head will be chosen at random (being either black or white), with the only proviso that not all of the hats will be black.
[To achieve this, the king will randomly select the colour of the three hats, and if all three happen to be black, they will be discarded and another three hats randomly chosen until the criteria of not all hats being black is achieved].

Given the above criteria for the hats, when the room lights are switched on the first person to announce which colour hat they are wearing (with the correct reasoning) will win.

When the room lights are switched on, you see that the other two suitors are both wearing white hats – what colour hat are you wearing, and why (quick, quick - you must announce this first, to win).

 

If I see two black hats, I know that I am wearing a white hat, so I can announce that immediately.

If I see one white hat, then the person wearing the white hat MIGHT see two black hats (if I'm wearing black) and, since they are presumably one of the three most intelligent suitors in the kingdom, will immediately announce that they are wearing a white hat. If they don't do so, then they are probably not seeing two black hats, which means I am wearing a white hat.

If I see two white hats, then if I am wearing a black hat both of the other people are seeing one white and one black hat. They, being two of the three most intelligent suitors in the kingdom, will announce that they are wearing a white hat if the other doesn't immediately announce that they are wearing a black hat. So if they don't do so, I must be wearing a white hat and declare that.

 

If I see two black hats, I know that I am wearing a white hat, so I can announce that immediately.

If I see one white hat, then the person wearing the white hat MIGHT see two black hats (if I'm wearing black) and, since they are presumably one of the three most intelligent suitors in the kingdom, will immediately announce that they are wearing a white hat. If they don't do so, then they are probably not seeing two black hats, which means I am wearing a white hat.

If I see two white hats, then if I am wearing a black hat both of the other people are seeing one white and one black hat. They, being two of the three most intelligent suitors in the kingdom, will announce that they are wearing a white hat if the other doesn't immediately announce that they are wearing a black hat. So if they don't do so, I must be wearing a white hat and declare that.

Congratulations, you have won the princesses hand in marriage.

 

Congrats WBahn.

I hope you're not already married!

 

When the lights come on, if I see two black hats, I announce that I have on a white hat. If I see two white hats, I announce that I have on a black hat. If I see one black and one white, I walk over to the guy with the white hat, knock him out then swap hats with him. I then announce that I have the white hat!

 

When the lights come on, if I see two black hats, I announce that I have on a white hat. If I see two white hats, I announce that I have on a black hat. If I see one black and one white, I walk over to the guy with the white hat, knock him out then swap hats with him. I then announce that I have the white hat!

If you see two white hats, why do you conclude that you have a black hat?

 

Congratulations, you have won the princesses hand in marriage.

More then one princess? Interesting kingdom.

I haven't necessarily won.

Remember, if I am wearing a black hat, then BOTH of the others will correctly conclude that they are wearing white hats and, presumably, simultaneously announce that fact. If nothing else, I will be eliminated from the competition because I didn't say anything. If I am wearing a white hat, then all three of us are wearing white hats and all go through the same decision process and all three announce that we are wearing white hats simultaneously.

So, knowing this, should I just make a guess and announce it immediately before the other two have a chance to make decisions based on the silence of the others? But then, perhaps they are going to use the same tactic.

 

More then one princess? Interesting kingdom.

I haven't necessarily won.

Remember, if I am wearing a black hat, then BOTH of the others will correctly conclude that they are wearing white hats and, presumably, simultaneously announce that fact. If nothing else, I will be eliminated from the competition because I didn't say anything. If I am wearing a white hat, then all three of us are wearing white hats and all go through the same decision process and all three announce that we are wearing white hats simultaneously.

So, knowing this, should I just make a guess and announce it immediately before the other two have a chance to make decisions based on the silence of the others? But then, perhaps they are going to use the same tactic.

Although the king stated that the colour of the hats would be chosen at random; having devised the intelligence test – he deliberately chose all the hats to be white, knowing the analysis the winning suitor would have to make to be the first to announce the colour of the hat they were wearing.

 

Although the king stated that the colour of the hats would be chosen at random; having devised the intelligence test – he deliberately chose all the hats to be white, knowing the analysis the winning suitor would have to make to be the first to announce the colour of the hat they were wearing.

Doesn't get around the problem. Either the other two suitors were smart enough to figure out the analysis or not.

If they weren't, then the most intelligent suitors analysis is flawed because it RELIES on the other two suitors correctly performing the same analysis.

If they were, then all three act the same way and it comes down to the reaction speed, not the intelligence, that determines the winner.

For instance, the other two suitors, who had no idea how to determine what hat they were wearing if they saw two white hats, had each picked a color at random before things began and decided that in that case they would immediately declare that color and take their chances. Thus there is a 75% chance that the most intelligent person will lose.

Reminds me of a Foxtrot cartoon where Jason is standing there holding a snowball and you see all of these figures and equations over his head in the first frame. In the second you see him getting pummeled by a barrage of snowballs. In the third frame he says something like, "There's a reason physicists always lose snowball fights."

 

Doesn't get around the problem. Either the other two suitors were smart enough to figure out the analysis or not.

If they weren't, then the most intelligent suitors analysis is flawed because it RELIES on the other two suitors correctly performing the same analysis.

If they were, then all three act the same way and it comes down to the reaction speed, not the intelligence, that determines the winner.

For instance, the other two suitors, who had no idea how to determine what hat they were wearing if they saw two white hats, had each picked a color at random before things began and decided that in that case they would immediately declare that color and take their chances. Thus there is a 75% chance that the most intelligent person will lose.

Reminds me of a Foxtrot cartoon where Jason is standing there holding a snowball and you see all of these figures and equations over his head in the first frame. In the second you see him getting pummeled by a barrage of snowballs. In the third frame he says something like, "There's a reason physicists always lose snowball fights."

The logic of the problem is that the smartest suitor would be the first to analyse the problem as you have done in your original reply.

It would not be an acceptable answer for a suitor to randomly guess the colour of their hat and be correct – remember they had to give the correct reasoning for the colour of hat they were wearing.

 

The logic of the problem is that the smartest suitor would be the first to analyse the problem as you have done in your original reply.

It would not be an acceptable answer for a suitor to randomly guess the colour of their hat and be correct – remember they had to give the correct reasoning for the colour of hat they were wearing.

Actually, I had lost sight of the requirement for the suitor to give their reasoning and was thinking that you were just wanting US to give the reasoning behind our answer.