Deductive thinking is thinking that leads us to a conclusion with certainty. Logic puzzles and problems (like the one below) require deductive reasoning. See if you can use your powers of deductive reasoning to figure out what color the front genius’ hat is. Post not only your answer, but your thought process, and remember—this is not based on probability. Probability is an inductive process.

Three geniuses stand in a file (one behind the other). Each can see only to the front, so the rear person can see the middle and the front, the middle person can see the front, and the genius in the front cannot see anyone.

You have five hats. Two are white, and three are red. You blindfold the three geniuses, who are utterly truthful, and put a hat--at random--on the head of each. Then you hide the other two hats and remove the blindfolds.

You then ask each genius to name the color of his hat (which he cannot see).

The rear one says "I don't know." The middle on says, "I don't know." Then the front one says, "I know."

WHAT COLOR IS THE FRONT GENIUS' HAT??

Red.

We know at least one of the hats is red (since you started with only 2 whites). The last guy in the line doesn't see 2 whites on the guys in front, because if he did then he would know he must have red. So when he says "I don't know" everyone now knows that he sees either two reds or one red and one white.

If the middle guy now saw that that the front guy has white, he would know that his own hat must be red. Since he says "I don't know," that must mean that he sees a red hat on the front guy.

So the front guy now knows he has red.

He ignored the "random"

Since this is a published home work assignment I will only make this a brief explanation.

Without being given more information, it is impossible to logically conclude the true color of the hat the genius in front is wearing.

The "--at random--" brings emperically many means of randomness and without knowing scientifically how the hats were sorted before they were dispursed at random.

Moreover, there is no description of the room or what the 1st genius immediately saw after the blindfold was lifted.

Do this experiment at home to see this.

Stephanie 123 - please explain why my answer is wrong. As I see it, there are 7 possible combinations. Listing them in order from front to back, the 7 possibilities are:

1) RRR
2) RRW
3) RWR
4) WRR
5) WWR
6) WRW
7) RWW

Am I missing something here? Are there any other possibilities?

When the guy at the back of the line says he doesn't know his own color, that eliminates possibility 5. This assumes that the geniuses know that there were 5 hats to choose from, and only 2 were white (if the geniuses didn't know this, then the problem is indeed unsolvable, and the genius in front is really not such a genius). So now everyone knows that the first two guys in line must either each have a red or one has a red and the other a white.

When the middle guy says he doesn't know, that eliminates possibilities 4 and 6, because if he sees white on the first guy, he would know he must has a red hat. So the only combinations that are left are 1, 2, 3, and 7. All of these combinations involve a red hat on the first guy, so he knows that he has a red hat. There are no other possibilities. Now, I'm not claiming he knows the colors of the other guys'' hats, but that wasn't the question.

So again, where is the error in deductive thinking?

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Originally Posted by ebaines

When the middle guy says he doesn't know, that eliminates possibilities 4 and 6, because if he sees white on the first guy, he would know he must has a red hat.

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How would the middle guy know that he has a red hat? There are TWO white hats not one. The middle guy could have another white hat. How would it eliminate 4 or 6? It doesn't because he could still have a red or a white and so could the guy behind him. If the back guy said he knows what color the 2 in front has (which ONLY if they two whites would he know what color he had) then the others would know what color they had too. I do not see how the front would know how what color his hat would be. The reason I say this is because we know its not WWR because the last one didn't say he knew. If the color is WR? Or RW? Being that he didn't know I have no idea how the front person would know.

Enless... the first person is indeed in a file line BUT he is faced the other way that the other two aren't (looking back at the 2nd and 3rd person). This would make that HE could see what the 3rd persons hat was. The first two are WR or RW OR RR we know because of the 3rd person saying I don't know. The 1st person would know what color hat he has just by looking at the 2nd and third person IF the 2nd and 3rd had both W hats on. I say this because if the 2nd person has a W hat on and the 3rd has a R hat on that still leaves the W or R possibility open for the 1st. If the 2nd person has a R and the 3rd has a R that still leaves the possibility of the 1st having a R. If it is ?RW that still leaves both open again. So IF the 1st is looking back at the other two AND the 3rd and 2nd both have Ws, this would be the only way that he would know.

*EDIT* ignore this because I'm stupid lol not taking it off though so you can see it too! Lol

*2EDIT* what's funny is I disproved what I said in the first paragraph in my 2nd paragraph ahah it also proves what ebaines says is right >.<

I'm with ebaines on this, they're asking each genius what colour their own hat is, not the others hats - so they have to work ou twhat there hat is from the other's hat colours.

The other two genius' are saying that they can't work out what their own hat is - meaning that the front genius knows which colour his is!

How would he know what color his is hunters? You don't explain :(.

Right, if the guy at the back would have said 'I know my colour' the only possible way would have been able to tell this is for the front two to both have white hats on. Because he doesn't know, the front guy know that at least he, or the middle guy MUST be wearing a red hat.

Now from the back guys answer, the middle guy knows that either he, or the guy in front of him MUST be wearing a red hat! If the front guy WAS wearing a white hat, then the middle guy would know that his was definitely red... but he's staring at a red hat on the front guys head... so he doesn't have a clue what his is!

Now the front guy knows that the only possible conclusion is that he's wearing a red hat!

Any clearer!

Ahhhhh, OK I got what you mean now :) nice conclusion. I didn't even think of using the 2nd guy much in my deduction which was my mistake xD. I see what ebaines was talking about now. Don't think I read through all of his comment though lol or misread it or something =/ Don't know how I missed this one!









Ok... enough excuses I'm just stupid ahah

Here's the answer:

1) The man in the back can see two hats, but he doesn't know what color his hat is. This means that the two hats are either red/red, or red/white. If they were both white, he would know his was red.

2) The middle man is the key. From the back man, he can deduce the two options, red/red or red/white. He can see only 1 hat (the one front man's hat). If that hat were white, he would know he was in the red/white combo, and would therefore know that his must be red. But he did NOT know.

3) From this, we can deduce that the front man's hat is red.

(Just noticed that 'HuntersBar' had answered the question too, but I hope mine was a little clearer... Anyhoo, fun riddle :)

You're welcome :)

Since #3 did not know the color of his hat, that means that at least one of #s 1 or 2 had a red hat. Since there'd been only 2 white hats, if #s 1 and 2 both had white hats then #3 would've known that his hat was red. Now, if #1's hat had been white, then #2 would've known that his hat was red. However, since #2 did not know the color of his hat, it then stands to reason that #1 had a red hat, thereby not allowing #2 to deduce with certainty the color of his hat.

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Originally Posted by NannaLinda

Deductive thinking is thinking that leads us to a conclusion with certainty. Logic puzzles and problems (like the one below) require deductive reasoning. See if you can use your powers of deductive reasoning to figure out what color the front genius’ hat is. Post not only your answer, but your thought process, and remember—this is not based on probability. Probability is an inductive process.

Three geniuses stand in a file (one behind the other). Each can see only to the front, so the rear person can see the middle and the front, the middle person can see the front, and the genius in the front cannot see anyone.

You have five hats. Two are white, and three are red. You blindfold the three geniuses, who are utterly truthful, and put a hat--at random--on the head of each. Then you hide the other two hats and remove the blindfolds.

You then ask each genius to name the color of his hat (which he cannot see).

The rear one says "I don't know." The middle on says, "I don't know." Then the front one says, "I know."

WHAT COLOR IS THE FRONT GENIUS' HAT??????

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Why is deductive logic also called symbolic logic and mathematical logic.

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Originally Posted by NannaLinda

Deductive thinking is thinking that leads us to a conclusion with certainty. Logic puzzles and problems (like the one below) require deductive reasoning. See if you can use your powers of deductive reasoning to figure out what color the front genius’ hat is. Post not only your answer, but your thought process, and remember—this is not based on probability. Probability is an inductive process.

Three geniuses stand in a file (one behind the other). Each can see only to the front, so the rear person can see the middle and the front, the middle person can see the front, and the genius in the front cannot see anyone.

You have five hats. Two are white, and three are red. You blindfold the three geniuses, who are utterly truthful, and put a hat--at random--on the head of each. Then you hide the other two hats and remove the blindfolds.

You then ask each genius to name the color of his hat (which he cannot see).

The rear one says "I don't know." The middle on says, "I don't know." Then the front one says, "I know."

WHAT COLOR IS THE FRONT GENIUS' HAT??????

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What you presented is not an example of deductive logic for the very simple reason that color is wholly subjective. Deductive logic is a sham.

“Before the 17th century, science was mostly a process of deductive logic. Consider the following line of reasoning used by certain CAM proponents:
1. We know through scripture that God is the creator of the universe and that God is perfect.
2. Therefore creation must be perfect.
3. If illness were real, then creation would not be perfect.
4. Therefore, illness cannot be real.
This is the basic reasoning behind the denial of illness’s reality by Christian Science as well as some forms of New Thought. Before the 17th century, this type of deductive logic was science. Some characteristics of scientific principles under this definition are those of ‘immutable law’ and ‘absolute certainty.’ We know them by logical reasoning; empirical testing is superfluous. This approach reads like a bumper sticker: ‘My mind is made up. Don’t confuse me with facts.’
Many opponents of CAM, while well educated in empirical science, are ignorant of metaphysics. They fall into the same trap but from the other side. Consider this line of reasoning:
1. We know through science that the only real powers in the universe are molecules in motion and the four forces of physics.
2. The mind and/or the spirit possess none of the four forces of physics.
3. If the mind and/or the spirit do not possess any of the four forces, they can have no aetiological effect on the physical universe.
4. Therefore, the mind and/or the spirit can neither cause nor cure disease.
Their basic modus operandi is the same as that of the Christian Scientists: logically deriving a conclusion on the basis of a metaphysical assumption. We don’t know that the forces of physics discovered to date are the only aetiological forces any more than we know that God is perfect. No experiment has ever been designed to test either proposition. Yet this seems to be the gist of Relman’s argument to Weil against mind-body medicine as cited earlier.”

Unsnarling the CAM-knot

Quote:

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Originally Posted by NannaLinda

Deductive thinking is thinking that leads us to a conclusion with certainty. Logic puzzles and problems (like the one below) require deductive reasoning. See if you can use your powers of deductive reasoning to figure out what color the front genius’ hat is. Post not only your answer, but your thought process, and remember—this is not based on probability. Probability is an inductive process.

Three geniuses stand in a file (one behind the other). Each can see only to the front, so the rear person can see the middle and the front, the middle person can see the front, and the genius in the front cannot see anyone.

You have five hats. Two are white, and three are red. You blindfold the three geniuses, who are utterly truthful, and put a hat--at random--on the head of each. Then you hide the other two hats and remove the blindfolds.

You then ask each genius to name the color of his hat (which he cannot see).

The rear one says "I don't know." The middle on says, "I don't know." Then the front one says, "I know."

WHAT COLOR IS THE FRONT GENIUS' HAT??????

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Well my deduction is that at least the one in the front can actually see,therefore deducting that the rest can see as well,when the one in the back says that he doesn't know there are 2 posibilities the other 2 guys are using red hats or one white and the other a red one,then the one in the middle coming to the same conclusion,if he sees the one in the front with a white hat then he would come to the conclusion he is in fact wearing the red one but he saw a red hat therefore not knowing wheather he is wearing the white one or a red one.by the time the one in the front is asked .he has in his mind that he is wearing a red hat.they always say the truth so no matter if is true or not the one in the front said what he though he was wearing that in fact is a red hat.

The front one had mind powers, was joking, or was standing in front of a mirror. But I also got something in the back of mind that I can't really explain, the front one knows his hat color because the other two don't.?

Ok so first off the only reason I'm answering this is because my girlfriend has a stupid business class which asked her this question as an assignment. I am 23 years old and a high school drop out... but don't pass me as an idiot! I find it funny that even when the instructions say that this is not a question of probobility, that some people are still trying to solve for X. If this were a question of probobility there would be more than one answer to "X". However we do not have to worry about that because this is a riddle! Read the question carefully. The idea of "deductive" reasoning is to logically take away from the situation until you have a procise answer. Which this question does have a procise answer! If you look real closley the deduction has already been done for you in the question when it says that YOU HIDE THE LAST 2 HATS! Bingo! Slam dunk! The only thing you need to worry about now is what color hat is on the front mans head! Lets look at the question for a minute. It asks you what color hat is on the mans head right?. So if you are the one who put the hats on at random... then you ought to know what color is on his head! (dummy moment) you put the hat on the heads.. you saw it! Therefore you know!. so tell me what color hat is he wearing? What has been done here is not calculus! They are trying to throw you off in the question when they have each of the "geniuses" tell weather they know what color hat they have on or not. When the first 2 say "I dont know" and the one in front says "I know". The question is trying to get you to believe that you need to answer for the genius and explain his logic for how he knows. But look closley... the question proposes that you ask them what color hat they have on... 2 of them admit they do not know... and one admits he knows... however he does not explain! But the final question is to you!. not to them!. So what color hat do they have on?. only you know! If you were looking for the geniuses answer... you need to ask the genius. So the question in the story is left unanswerd as well as completley separate from the final question which is asked to you directly. I am curious to know how the front genius knows! The idea of asking me this question is insulting for I already know what hat I have put on... The hats I hid are the deduction. And the reason is because those were the hats left over! Remember this is a puzzle... a riddle... not a mathmatical equation. The reason this is confusing for some is because both the story and the question are relevant.

Well that certainly is a VERY different way of looking at the situation! :)

I like what you're thinking, but it's not exactly correct. Deductive thinking is where you know the premise (or premises) of a subject, and thus you can come to a conclusion. A good example of this is;

All apples are fruit.
Some apples are red.
Therefore some fruits are red.

(Thank you wikipedia! Ha ha!)

This question is also about probabilty, as each genius has a different chance of getting a different coloured hat. This probability changes as the hats are handed out.

The whole point of this question is to find out the colour of the front guys hat, and explain how you got to that conclusion. Although saying "It's red and I know because I put it on!" would probably get a few laughs, it doesn't show any deductive thinking.

I don't understand why you're banging on about hiding the two hats though?

Christ, I can't believe this question is almost 18 months old! Ha ha!

Right, in case anyone else comes here with any doubts, or any strange answers, here is a conclusive, consice and very simple to understand!

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A tricky one, but I have it sorted out for you. You start by considering the ways in which the front and middle geniuses could be wearing hats. There are four possibilities:-
1) Front: white, Middle: white.
2) Front: white, middle: red.
3) Front: red, Middle: white.
4) Front: red, Middle: red.
Now consider each statement in turn. If the rear genius could see two white hats then he would know that his hat was red. Therefore you can exclude possibility 1. The middle genius now knows this as well, so if he can see a white hat he would know that his hat was red. Therefore you can exclude possibility 2. The front genius now knows this as well and as both the remaining options have him wearing a red hat he now knows the colour of his own hat.

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Borrowed from this link!

Please - let this one go! Ths question was asked and answered over a year ago! See post #2.