Brain Teaser: How Did He Know?

Right on the money

Cool puzzle. Congrats to comment #1 who got it first. If I may, I'd like to explain it another way just for clarity.

You have three people with marks on their heads that can be red (R) or white (W). This allows the following possible configurations (RRR),(WRR),(RWR),(RRW),(WWR),(WRW),(RWW) and (WWW).

Step 1- When all three have their hands in the air, only the following configurations are possible: (RRR),(WRR),(RWR), and (RRW) since the other configurations would only result in two or no hands in the air. So we are down to 4 possibilities.

Step 2- Since the person is seeing red marks on the other two, the 4 possibilities are down to two (RRR) and (WRR).

Step 3- If the other two people were seeing white on his forehead, they would realize that they themselves couldn't have a white mark (from step one), and would lower their hands. But they don't lower their hands. Thus he can't have a white mark on his head.

So he lowers his hand because he realizes the mark on his head must be red.

By watching the other two people. One was looking at him as the blind fold was remove and as he had a red dot he immediately raise his hand because he had to look no farther. If he had a white dot eye movement from this person to the other before he raised his hand would have given that away.

There are many mathermaticians in India. It was told that arab number 1,2 ,3... was original from india? true or not?

You may be studing in middle school. Do you know math induction method? which would be studied in 3 grade in our China.

if so, the solution appears.

Not heard of any Mathematicians from China. And do not forget the Zero. And what is that Math induction Theory. Asked you a simple question and the answer to which I told you is not hard to find and you mouth some jargon that is probably valid in China. If you learn all this in Kindergarten what did you do with it and what did you do in class 4. Unified Field Theory?? Go check with US universities. No Math Dept. is complete without an Indian.

haha your English is not bad. I wish you could speak as well as an English Gentalman does.

grade 2 students may be not a level II worker.

Once you would be a true student of Einstein. you might know chinese mathematician and scienists.

Could you read works of Dr. J,Needham who was a famous scholar in Cambridge Univ.?

You will know all.

In ventin of 0 is an important milestone in math history. Could you describe again here ? I like to listen to again.

I havnt been in that USA. I dont know how many prof. in that school. and I dont know if your words is true or right, if it s really so, I wonder why so many Indian work in USA instead of serve for thwir own country and people?

You should also find out the difference between "3 grade" and grade 3. Probably a Freudian slip and you really meant "3rd grade"

well. I make a mistake.

grade 2 (not 3) , this is equal to 11 grade in some countries.

"It was told that arab number 1,2 ,3... was original from india? true or not?" Try Wikipedia.

I am offended by your denigrating language! Everyone has their strengths and their weaknesses. I have been known by friends and peers for over half a century as one who is well above average in using mathematics. Yet I find this puzzle difficult!

I am painfully aware of weaknesses in the teaching of math and science in the USA, (having taught those fields for well over three decades, but retired as early as possible to retain my sanity and get some sleep!). In the USA, 3 grade (which I would understand as 3rd grade), is for children around 8 years old. I did see the second reference, which apparently means your 3 grade means something akin to third year of secondary school, or 11th grade, or something around 16 years old. In any case, I dare say it is the exceptional 16 year old who pays attention to such studies as inductive reasoning. I have not visited your part of the world, so perhaps it is very different there; I would be most impressed if that is the case.

Even after reading Roger Pink's excellent explanation (several times, by the way), I'm still not quite clear on step 3...

I am a visually oriented person - only after creating a graphic illustration do I begin to understand.

Without naming the three subjects, there are four possibilities: 3, 2, 1, and zero red dots. Clearly the 1 and zero possibilities are eliminated by a hand not up, but that leaves possibilities 3 and 2.

I still fail to see how the upper-left individual can distinguish between possibilities 3 and 2! In both cases every individual can see at least one red dot, so should raise a hand. Please explain!

Please read my comment #7, step 3. That should answer your question.

The key is that the other two people are not lowering their hands. That provides the necessary information he needs to determine which of the two possibilities you list is the actual one.

Thanks Roger!

I guess I'm slow in this area... As I indicated above, I read your original post several times before writing my first response. After reading it a few more times, I think I'm clear on it, but I keep going back to 'no, that possibility was eliminated because in that case at least one of the other two would not have raised his hand in the first place'.

Dick

This might help a bit. Look at your case 2 above: how long would it take right or bottom left to put his hand down?

Rule was that anyone seeing at least one red mark should raise hand....

So, if there is one white mark, two persons would see one red and one white mark and third one would see TWO red marks...... Therefore everyone raise hand just like in case when every person see two red marks.......

You Wrote: "I guess I'm slow in this area"

You and me both dkwarner. Although my explanation earlier in this thread is complete, it took me a while to get it straight in my head.

Another way to think of it is you need to use all the information available to solve the problem. The relevant information is:

1. What color do you see on the other two guys.

2. How many hands are raised when the blindfolds are taken off

3. How many hands are lowered once everyone has time to assess the situation

The part where you are getting stuck is on the last bit (3), which is the information you need to finally figure out you don't have white on your head.

You may find it easier to play the "what if" game. Pretend you have white on your head and go through it, it would go like this:

1. Your blindfold is removed and the two other guys have red marks
2. All three of you have your hands raised
3. The other two slowly lower their hands

You now know because they lowered their hands that you have a white mark. Why? Because they only lower their hands if they are sure of the color THEY have on their heads. The only way they can be sure of what color THEY have is if you have a white mark on your head. They know this through the process of elimination, because the only combinations that produce three raised hands are all red or two reds and a white. Since they see you have white and they see all the hands up, they know there is no way they can have white, thus they are red. When they lower their hands, you now know your situation, they could only figure all that out if you have white on your head. If you had red, they couldn't be sure that they themselves weren't the white one and would have to leave their hands up.

I don't know if that helps. It was a fun puzzle though. I'd like to see more around CR4 like this one.

It was allready explained: since only two posibilities are left, in case person with white mark what this person can see is two red marks, while other two would see one red and one white mark in front of them, and since there could be just one white mark, they would both lower hand as conclusion would be that they have red mark on head. Since nobody lower hand, that means everyone could see two red marks, and this is possible only in case where all marks are red. Smartest person therefore understood that he has to have red mark on his head, because other two concluded they cannot know which mark is on their head.....

Sorry for offending against you here. but I dont know why? Maybe 3 grade word? if so, havent you seen #9 right above your this thread? oh, I have to correct same mistake 11 grade as grade 11. the 11th grade. Now, can you understand? This equal to a guy of 17 or 18 year old. Our chinese children go to school at 6 or 7 years old.

Most of documents think of these arab number originate from Arab for a long time, but some scholars study and find it may originate from India. Who are right? As well as the imortant number 0. most of scholars think its from India at first. from Budish.

Math induction is teached in grade 11, next year, students will graduate and exam for their college study. Every country has their own way of education. you may not be familar with USA education completely, as you aref rom Ca. I was told that USA has a good education system. Students love their subjects.

Welcome to visit and examine our chinese education. as well as our oriental education.

As yuou were a senior Math teacher, welcome else to answer this question at

http://cr4.globalspec.com/thread/35736#newcomments

It seems a somewhat difficult for many people who visited it. I think you certainly give out an excellent explanation.

Check the first reply.

Got to watch out for Randall. He gave a link to the twelve ball problem too. I always thought that the fun was in solving and felt a little sad whenever I got the solution

Since all three see two red marks, no one can be sure what mark is on their head. From fact that other two concluded they cannot know their mark, he understood situation that all of them has red mark.

Explanation:

If there would be three white marks, nobody would raise hand, with two white marks and one red, one of them would not raise hand. As all raised hands, this was not true.

So only two combinations remained: one white and two red marks or three red marks.

If there would be one white mark, two of them would see both white and red marks, and they would both realize they have red mark on their head, so they would lower hand.

One that would see two red marks would not be able to know if his mark is white or red, so he would not lower hand.

Since both of other people think they cannot know their color, smart one realized that they both see two red marks, and he see two red marks also, so his mark must be red.......

That was too easy, and is similar to puzzle with red and white Hats......

We had puzzle with villager that have to bring cabbage, goat and wolf to town market to sell, but he must cross the river in a boat that can carry only him and one other person, animal or thing. He could tie goat or wolf to a bag of cabbage, or goat and wolf together. Both goat an wolf are hungry, so goat would eat cabbage and wolf would eat goat, and he want to bring all safely to town.

How would he do it?

Take the goat across. Then the cabbage but bring the goat back after bringing the cabbage across. Then bring the wolf across. Then bring the goat across again.

Hope the goat doesn't get sea sick after 3 trips in the boat.

Right!

Don't worry about goat, we are famous for our sailors and ship captains!

Only we usually bring wolf next, so he would not be tempted to turn vegetarian by being so much time with our delicious cabbage :-))

Regards from Zagreb, the capitol of Croatia, Europe!

To do it in one trip, put cabbage in bag slung around shoulders, carry the goat on shoulders, ride the wolf onto the boat.

It is not problem with size of space on the boat, but how much it can carry... Of course, if You make both wolf and goat swim then You would be able to do it in one trip, provided neither wolf nor goat drown, and there is no hungry pirana fishes in the river :-))