5 hats problem........

JSYantiss
Posers and Puzzles 20 May '07 16:41
1. 20 May '07 16:41
There are 5 hats on a table. 3 white hats, 2 red ones. Facing away from the table are three chairs, all in a straight line. 3 men sit down, one in each chair. A fourth man walks by the table and randomly chooses 3 hats and places one on each sitting man's head. He tells each man to sit still and face forward so that they can only see the chair(s) in front of them. He asks the man closest to the table "Do you know what color hat you are wearing?" He says "No." He asks the middle man the same question and gets the same answer. When he gets to the last man(who cannot see anyone, as nobody is sitting in front of him), the last man answers "Yes." What color is the last man's hat, and how does he know?
2. 20 May '07 16:49
SPOILER SPACE
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Heh, I haven't done that since my Usenet days. Anyway, the hat is white.

- If the first guy could see two red hats, he'd know his own was white. Thus he must see at least one white hat.
- The second guy knows this. So if he could see a red hat, he'd know he had to be the one with a white hat.
- Thus the second guy did not see a red hat. The third guy's hat is white.
3. 20 May '07 17:40
Originally posted by CZeke
SPOILER SPACE
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Heh, I haven't done that since my Usenet days. Anyway, the hat is white.

- If the first guy could see two red hats, he'd know his own was white. Thus he must see at least one white hat.
- The secon ...[text shortened]... th a white hat.
- Thus the second guy did not see a red hat. The third guy's hat is white.
Lol at "Spoiler Space"

You are correct. That was too easy, I guess. But hey, I tried.
4. 21 May '07 05:341 edit
Here's one similar to yours, but a little more difficult (and don't answer if you've heard this one before):

An arbiter place a hat on the heads of three men. He tells them that all hats are either white or red. None of the men know at this point what color hat they are wearing, but they could see the color of the hats the other two men. The arbiter then asks the three men to raise their hand if they see a red hat. All three men raised their hands (and they saw each other raising their hands). Still, no one knew what color hat they were wearing. After a while, one of them raised their hand and correctly guessed what color his hat was. How did he know, and what colors were each of the three hats?
5. coquette
21 May '07 06:09
Originally posted by Jirakon
Here's one similar to yours, but a little more difficult (and don't answer if you've heard this one before):

An arbiter place a hat on the heads of three men. He tells them that all hats are either white or red. None of the men know at this point what color hat they are wearing, but they could see the color of the hats the other two men. The arbiter then ...[text shortened]... guessed what color his hat was. How did he know, and what colors were each of the three hats?
still way too easy.
6. 22 May '07 00:121 edit
The reasoning.

Possibilities -

3 white hats - REJECTED - Nobody would see red and thus raise their hand.

2 white hats - REJECTED - The one wearing red would see 2 whites and thus not raise his hand.

2 red hats - Possible given immediate information at hand

3 red hats - Possible given immediate information at hand

So at least 2 hats are red.

This presents a new piece of information. I will let someone else finish the reasoning process, because the crucial point takes a bit of time to become apparant, and yet the logic is sound.

I will note, however, the above analysis assumes all three men are of at least normal intelligence and honest. However, that is not such a stretch in my mind.
7. 22 May '07 03:34
red,white is not a color,
8. 22 May '07 19:43
For the purposes of this exercise, white is a color.
9. 23 May '07 07:061 edit
still way too easy.
Well, if you insist:

There's a certain game played by a team of seven people. Each one of them wears a hat that's a color of the rainbow (red, orange, yellow, green, blue, indigo, or violet). There are no limitations regarding who wears what color or how many of each color hat are worn (they could all be wearing different colored hats, or they could all be wearing red hats, etc.). None of them can see their hat, but they can see the hats of all their teammates. They must remain silent while observing their teammates' hats, and can only speak when they guess the color of their own hat. How could the team ensure that at least one of them guesses the correct color of their hat?

Once you've gotten that, can you prove whether or not there is a way to ensure that more than one of them guesses the correct color of their hat?
10. AThousandYoung
All My Soldiers...
23 May '07 07:56
Originally posted by Jirakon
still way too easy.
Well, if you insist:

There's a certain game played by a team of seven people. Each one of them wears a hat that's a color of the rainbow (red, orange, yellow, green, blue, indigo, or violet). There are no limitations regarding who wears what color or how many of each color hat are worn (they could all be wearing different color ...[text shortened]... there is a way to ensure that more than one of them guesses the correct color of their hat?
Question for the first riddle - can the team make a code up ahead of time or are they only told the rules once they are required to keep silent?
11. 23 May '07 10:09
There is no imformation from what is seen to deduce hat color, or even to reduce the possibilities.

If they can motion, then it would be easy to set up a system to for one to inform another of the proper hat color, but I assume such is out of the question.

However, using only vision and knowledge of guesses, I can see a way to have 3 guess properly.

It involves pairing neighbors together. For each pair of neighbors, The one on the right guesses the hat color of his partner. The partner, obviously, guesses the same color. The odd man is out of luck. Using this system, 3 people are guaranteed to guess their hat color, regardless of how many colors there are..

Has nobody guessed how to finish off the previous 3 hats problem yet?
12. 24 May '07 00:17
Has nobody guessed how to finish off the previous 3 hats problem yet?

JSYantiss sent an attempt at it to me, but it wasn't correct.
13. 25 May '07 14:472 edits
Originally posted by Jirakon
Here's one similar to yours, but a little more difficult (and don't answer if you've heard this one before):

An arbiter place a hat on the heads of three men. He tells them that all hats are either white or red. None of the men know at this point what color hat they are wearing, but they could see the color of the hats the other two men. The arbiter then ...[text shortened]... guessed what color his hat was. How did he know, and what colors were each of the three hats?
The hats must all be the same color, since 2/1 red/white or white/red would allow one of them to instantly figure out the color of his own hat.
Furthermore, they are all red, since they actually raised their hands.

Assuming that all three men are pretty smart, the guy who guessed correctly reasoned the following after waiting awhile:

1. I see that my two partners (A and B) have red hats.
2. If my own hat is white, then A would see B raising his hand, and realize that his own hat is red, since B is raising his hand for seeing A's red hat, and not for my own.
3. Same argument as #2, except switching A and B around.
4. Since neither A nor B were able to deduce the color of their own hate, my own hat, therefore, cannot be white, and is thus red.