Brilliant crazy thought section

I think Brilliant.org should have a seperate section for some interesting and crazy questions whose answers are simple but yet thought provocative. These help to open up your thinking power which ultimately helps you in solving tough problems of olympiads.

Note by Iam Mangod96
6 years, 4 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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Cannibals ambush a safari in the jungle and capture three men. The cannibals give the men a single chance to escape uneaten.

The captives are lined up in order of height, and are tied to stakes. The man in the rear can see the backs of his two friends, the man in the middle can see the back of the man in front, and the man in front cannot see anyone. The cannibals show the men five hats. Three of the hats are black and two of the hats are white.

Blindfolds are then placed over each man's eyes and a hat is placed on each man's head. The two hats left over are hidden. The blindfolds are then removed and it is said to the men that if one of them can guess what color hat he is wearing they can all leave unharmed.

The man in the rear who can see both of his friends' hats but not his own says, "I don't know". The middle man who can see the hat of the man in front, but not his own says, "I don't know". The front man who cannot see ANYBODY'S hat says "I know!"

How did he know the color of his hat and what color was it?

- 6 years, 4 months ago

This is actually a really cool problem. I remember solving it when I was little.

The man in front is wearing a black hat.

Why? Well, the two men in the back are actually giving us more information than we notice at first.

Let's say that the man in front is not wearing a black hat. We're going to eliminate such scenarios where the man in front wears a white hat. That gives us two cases.

Case $1$ : The man in the middle is wearing a white hat.

What would happen if the man in front and the man in the middle both wore white hats. Well, the man in the back would think: 'I see two white hats in front of me. I know that there are two hats. So obviously, I must be wearing a black hat.'

So the person in the back would be able to successfully discern the color of his hat. We know that didn't happen.

Therefore, Case $1$: eliminated!

Case $2$: The man in the middle is wearing a black hat.

What would happen if the man in front wore a white hat and the man in the middle wore a black hat? Now the the man in the back wouldn't be able to guess [with 100% accuracy] the color of his hat. So he would say: 'I don't know'. The other men hear this.

Now let's move on to the man in the middle. Let's try to get in his head. What is he thinking?

"Well, I see the person in front of me is wearing a white hat. If I were wearing a white hat as well, the man behind me would be able to discern the color of his hat [see case 1]. Well, that didn't happen. So, I must be wearing a black hat. Aha!"

So in this case, the person in the middle would be able to successfully find out the color of his hat. He wouldn't say: 'I don't know'

Therefore, Case $2$: eliminated!

In other words, if the man in front is wearing something other than a black hat, either the man in the back or the man in the middle will correct guess the color of their hats.

That means, the man in front must be wearing a black hat.

- 6 years, 4 months ago

IF all of them are allowed 1 guess each, they can actually use the pigeon hole principle by all guessing black. One of them MUST be wearing black.

- 6 years, 4 months ago

Yes, at least one of them is wearing a black hat. But that doesn't help you solve the problem. You can't just guess. You can answer only if you know what your color is. That is the point of the problem.

If everyone says black, at least one of them will be correct. But no one is allowed to guess unless they know for sure. This problem actually deals with the fact that the two persons in the back being unable to discern their own colors forces the person in the front to have a black hat.

If the two person behind you don't know what their colors are, you have to be wearing black!

- 6 years, 4 months ago

Nice Question

- 6 years, 4 months ago

The man in the back doesn't know, implying that the two people in front don't both have white hats. The man in the middle doesn't know either -- but he knows that it can't be both of them with white hats -- and since he doesn't know, then that means the third guy must have a black hat (and he knows this). If the third guy had a white hat, then since the guy in the middle knows that both of them can't have a white hat, then the guy in the middle would have known that he had a black hat. But since he didn't know, then the third guy knows he has a black hat.

Quite weird cannibals!

- 6 years, 4 months ago

I think the front man was wearing a black hat. Bcoz as the rear man said he didnt know that means the two others were not wearing both wihite hats. Also as the middle man didnt know bcoz if the front man was wearing white and he himself was waring black from rear one's saying.

- 6 years, 4 months ago

Call a black hat $B$ and a white hat $W$.If the last man does not know the color of his hat, then at least one of the two people in front is wearing a black hat,example $B,W$.So the first two people must be wearing either $W,B$,$B,W$ or $B,B$.In the first case,the second person would know that at most one of them is wearing a white hat(from the last man's statement),so he would know that his hat is black.However he says "I don't know",so the person in front must be wearing a black hat.

- 6 years, 4 months ago

Since the man in the rear couldn't guess his hat's colour, the two men in front are not BOTH wearing white hats. Thus either:-

-1-One of them is wearing black and other white OR -2-Both are wearing black.

Now if the second person sees that the frontman is wearing white, he would know that his is black.

Therefore both of them are wearing BLACK.

- 6 years, 4 months ago

Hello all,

If we were to create a "separate section" to host "interesting and crazy questions with simple and provocative answers," what should the name of it be. In my opinion "Crazy thoughts" is a little too vague (there are many ways for thinking to get crazy). But I am struggling to come up with a better genre name for the kinds of problems you guys are discussing here. I feel like "Brain teasers" or "Riddles" would imperfectly work, but am not sure if those terms are too idiomatic.

Staff - 6 years, 4 months ago

- 6 years, 4 months ago

I was thinking about 'Math Gym'. But I turned to agree with you as my opinion materializes these images .

- 6 years, 4 months ago

How about critical thinking skills? :) I saw the hats question in an a-levels critical thinking subject.

- 6 years, 4 months ago

it could be named as the ''RECREATIONAL SECTION'

- 6 years, 4 months ago

How about 'Brilliant Brain Stimulation' section? That is the first thing that came to my mind.

- 6 years, 4 months ago

- 6 years, 4 months ago

BRAIN IQ

- 6 years, 4 months ago

NEW QUESTION : A man working as a security guard dreamt that the aeroplane in which the boss of the company was travelling crashed and the next day the plane actually crashed. When the manager of the company came to know about it he quickly fired him. Why?

- 6 years, 4 months ago

Security guards aren't supposed to sleep [at night]! The guard wasn't doing his job. That's why he got fired.

- 6 years, 4 months ago

It wasn't specified that the guard was sleeping at night though.

- 6 years, 4 months ago

The guard was dreaming, thus it can be inferred that he was sleeping.

- 6 years, 4 months ago

That's a funny solution! Lol

- 6 years, 4 months ago

Jack tore out several successive pages from a book. The number of the first page he tore out was 183, and it is known that the number of the last page is written with the same digits in some order. How many pages did Jack tear out of the book?

- 6 years, 4 months ago

This problem may look simple at first glance. However there's a subtlety involved.

As the first page number was $183$, the last page number is one of these: $318, 381, 813, 831$ according to the problem [i'm not listing $138$ because it's obvious that that's not the case].

Now you would think all of these are possible. No, they're not.

If you take a book and tear out page number $183$, you're tearing out page $184$ as well. So if you tear some pages out, and the first page happens to be an odd number, then the last page number has to be an even number [I advise you to try it with an actual book but I'll not be held responsible for anything that happens because of that :)]. So, that just leaves $318$ and Jack tore out $136$ pages [or $68$ pieces of paper. I'm now confused about what a page is. Can you ever tear out one page?] out of the book.

- 6 years, 4 months ago

The book pages could be like napkins, when you fold it all the way, you can rip it in half since there are two layers of the napkin.

- 6 years, 4 months ago

there is a train in which only objects of dimensions cannot be more than 1 m. You have a sword of length 1.5m and you have to take it in the train and travel. What should you do now?

- 6 years, 4 months ago

Are you posting problems from 'The Art & Craft of Problem Solving' by Paul Zeitz [great book!]?

Answer: Try putting your sword along the diagonal of a $1\times 1\times 1$ meter box. The diagonal of that box is equal to $\sqrt{1^2+1^2+1^2}$ meters which is greater than $1.5$ meters. So the sword should fit. And as you can see, the dimensions of the box is not more than $1$ meter.

- 6 years, 4 months ago

Nice answer. I thought it had to do something with relativity with near light speed motion of the objects :)

- 6 years, 4 months ago

However Yash, this is a problem in real life, with possible actions. Accelerate the train to a great percentage of c is a little bit of unnecessary, if you can understand me.

- 6 years, 4 months ago

Even such relegation never exists in real life. (That's why we adorably call them 'crazy problems'!). Crazy problems require realistic solutions. I am really passionate about these problems. I hope they launch this beautiful section soon.

- 6 years, 4 months ago

Put the sword in a box of dimension $1m \times 1m \times 1m = 1m^{3}$.The diagonal of the box is $\sqrt{1^{2} + 1^{2} + 1^{2}}m = 1.732m.....$

- 6 years, 4 months ago

Where do you guys get those "crazy" questions? I'm interested about them.

- 6 years, 4 months ago

You are in the downstairs lobby of a house. There are three switches. all in the “off” position. Upstairs, there is a room with a lighibulb that is turned off. One and only one of the three switches controls the bulb. You want to discover which switch controls the bulb, but you are only allowed to go upstairs once. How do you do it? (No fancy strings. telescopes, etc. allowed. You cannot see the upstairs room from downstairs. The lightbulb is a standard 100-watt bulb.)

- 6 years, 4 months ago

well it has to do with the thermal heat that each bulb gives out, I'd say put on one switch wait for about 15 minutes (maybe less) then turn it off. Turn on the other Switch wait for about 10 minutes (maybe less too) then go upstairs and feel the light bulb, if its not hot, then its the third switch, if its really hot then it is the second, if it warm (Not that hot) then its the first switch....

- 6 years, 4 months ago

You don't need to wait for the second switch. If the bulb is on, then it's the second.

- 6 years, 4 months ago

If the bulb is off and cold, it's the third. If it's off and hot, it's the first.

- 6 years, 4 months ago

Lets name the switches A, B, C. Turn any two of them ON(say A & B) for sometime. Now turn one of them OFF(say B). If:- 1>-The bulb is on, its A. 2->The bulb is OFF and HOT or WARM then its B. 3->The bulb is OFF and at room temperature its C.

- 6 years, 4 months ago

Brilliant man! I would never think in such clever solution. I thought it was more difficult.

- 6 years, 4 months ago

Speaking of difficult solutions, here's a funny article about how Feynman would solve the problem:

- 6 years, 4 months ago

This article very funny San Ying! I think Feynman has a little bit of Sheldon's personality lol.

- 6 years, 4 months ago

Would life be possible if there was no friction?

- 6 years, 4 months ago

Probably yes. Just to introduce the discussion of this question, without friction, animals and bacteria would be unable to move. This would cause a big problem.

- 6 years, 4 months ago

I remember a really cool question on stackexchange that went something like this:

A group of three people are each assigned a distinct number from the set {0, 1, 2} (And they know that they each are assigned a distinct number from this set). How can the guy with the 1 figure out what number the other two has (of course without asking "do you have 0?" or something stupid like that). I think that's how it went (If someone knows what I'm talking about and can remind me that'd be greatly appreciated)

There were a lot of cool answers.

- 6 years, 4 months ago

I'm not going to repost the whole discussions there, but here are two interesting questions: 1 question to know if the number is 1, 2 or 3 How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen? I think the first one is similar to yours.

- 6 years, 4 months ago

Sorry Michael, but what kind of questions can we ask in this problem? Because a thought to ask things like: you got an even number? How much is the sum of your number with the other one?...

- 6 years, 4 months ago

Would be good for some brain racking....... count me in

- 6 years, 4 months ago

Are there different terms for 0/0 and any number n/0 ??? I mean I have read somewhere that 0/0 is not defined where as n/0 (where n is not equal to 0) is infinity... Is this correct or both are not defined or both are infinity? Also explain me ...

- 6 years, 4 months ago

Lakshaya, here's the deal:

1) I cannot explain you with details, but 0/0 certainly is not defined.

2) According to the Calculus approach, the limx->0 (1/x) tends to infinity, as it's states in L' Opital theorem I think. You can understand easier like it follows: You have an object and you are willing to divide it in halves. So you obtain 2 pieces. For dividing it in thirds, you get 3 pieces, and so on.

- 6 years, 4 months ago

They are both undefined.

- 6 years, 4 months ago

You have my vote for this.

- 6 years, 4 months ago

And mine

- 6 years, 4 months ago

what is the factorial of 33.

- 6 years, 4 months ago

Zero("0") is what ? Even or Odd ?

- 6 years, 4 months ago

$0$'s even.

Even numbers are numbers that leave a remainder of $0$ when divided by $2$. In other words, even numbers are numbers in the form $2n$ when $n$ is an integer. Note that $0=2\cdot 0$.

Between two consecutive integers, exactly one of them is odd and the other one is even. Take $0$ and $1$. They are consecutive integers. $1$ is odd. So $0$ has to be even.

- 6 years, 4 months ago

Would life exist if $1 + 1 = 3$? Does dividing by zero cause the world to collapse onto itself?

- 6 years, 4 months ago

This is just about science ;-) 1 man + 1woman = 2 adult + 1 child = 3 person XD

- 6 years, 4 months ago

Well life would exist but if $1+1=3$, then trust me each mathematics question(Olympiad or otherwise) would have infinitely many solutions ;)

- 6 years, 4 months ago

i think ,that 1+1 =2 is a matter of definition ,meaning that 2 is by definition equals 1+1 ,asking the question "what if the word apple means beans ?" is meaningless because the word apple by definition is not beans but "apple"....right ?

- 6 years, 4 months ago

What would happen if every time you put together 2 apples you magically got 3 but when you separated them they became 2 again?

- 6 years, 4 months ago

Have any of you read Gödel, Escher, Bach? One of the diagrams in his book would look different if 13 was not a prime number.

- 6 years, 4 months ago

If $1 + 1 = 3$ then life would exist ,but the definition of each number would be different,and dividing by zero does not cause the world to collapse onto itself.$\frac{1}{0}$

- 6 years, 4 months ago

You're right. Absolutely nothing happens when you divide by ze- Implodes spectacularly

- 6 years, 4 months ago

What you suggest is that you could generate energy from literally nothing, unless there was a formal trigger for this random generation of energy - i.e. the idea of combination (if we approach your question philosophically) things would get terribly confusing - perhaps to the extent of constant reactions as violent as the big bang, which would be pretty difficult for life to evolve out of [self-replicating macromolecules?]. And no, 1/0. ;)

- 6 years, 4 months ago

This is treating 1 and 3 by the current definition they hold now

- 6 years, 4 months ago