It was a Pink Island. There were 201 individuals (perfect logicians) lived in the island. Among them 100 people were blue eyed people, 100 were green eyed people and the leader was a black eyed one.
Except the leader, nobody knew how many individuals lived in the island. Neither have they known about the color of the eyes. The leader was a very strict person. Those people can never communicate with others. They even cannot make gestures to communicate. They can only talk and communicate with the leader. It was a prison for those 200 individuals.
However, the leader provided an opportunity to leave the island forever but on one condition. Every morning he questions the individuals about the color of the eyes! If any of the individuals say the right color, he would be released. Since they were unaware about the color of the eyes, all 200 individuals remained silent. When they say wrong color, they were eaten alive to death. Afraid of punishment, they remained silent.
One day, the leader announced that "at least 1 of you has green eyes! If you say you are the one, come and say, I will let you go if you are correct! But only one of you can come and tell me!"
How many green eyed individuals leave the island and in how many days?
All 100 green eyed individuals will leave on the 100th night.
Consider, there is only one green eyed individual lived in the island. He will look at all the remaining individuals who have blue eyes. So, he can get assured that he has green eyes!
Now consider 2 people with green eyes. Only reason the other green-eyed person wouldn't leave on the first night is because he sees another person with green eyes. Seeing no one else with green eyes, each of these two people realize it must be them. So both leaves on second night.
This is the same for any number. Five people with green eyes would leave on the fifth night and 100 on the 100th, all at once.
Search: Monty Hall problem
Why it's important for the solution that the leader said the new information "at least 1 of you has green eyes", when they must knew from the beginning, that there are no less than 99 green-eyed people on the island? Because they cannot depart the island without being certain, they cannot begin the process of leaving until the guru speaks, and common knowledge is attained.
Search: Common knowledge (logic)
In classic mythology, there is the story of the Sphinx, a monster with the body of a lion and the upper part of a woman.
The Sphinx lay crouched on the top of a rock along the highroad to the city of Thebes, and stopped all travellers passing by, proposing to them a riddle.
Those who failed to answer the riddle correctly were killed.
This is the riddle the Sphinx asked the travellers: "What animal walks on four legs in the morning, two legs during the day, and three legs in the evening?"
This is part of the story of Oedipus, who replied to the Sphinx, "Man, who in childhood creeps on hands and knees, in manhood walks erect, and in old age with the aid of a staff."
Morning, day and night are representative of the stages of life.
The Sphinx was so mortified at the solving of her riddle that she cast herself down from the rock and perished.
Justin Case and Auntie Bellum are fellow con artists who deliver coded messages to each other to communicate. Recently Auntie Bellum was put in jail for stealing a rare and expensive diamond. Only a few days after this, Justin Case sent her a friendly letter asking her how she was. On the inside of the envelope of the letter, he hid a code. Yesterday, Auntie Bellum escaped and left the envelope and the letter inside the jail cell. The police did some research and found the code on the inside of the envelope, but they haven't been able to crack it. Could you help the police find out what the message is?
This is the code:
The message was "loose bricks in left wall." The message was put backward with words related to time in between. This is how the message looks when separated:
ll watch awtfe clock Inisk sundial cirbe timer sool
If you take out watch, clock, sundial, and timer, this is what is left:
Look at this backwards and this is what you have:
loose bricks in left wall
Auntie Bellum took out the bricks and escaped in the night. Then, she put the bricks back where they were.
You can easily "tile" an 8x8 chessboard with 32 2x1 tiles, meaning that you can place these 32 tiles on the board and cover every square.
But if you take away two opposite corners from the chessboard, it becomes impossible to tile this new 62-square board.
Can you explain why tiling this board isn't possible?
Color in the chessboard, alternating with red and blue tiles. Then color all of your tiles half red and half blue. Whenever you place a tile down, you can always make it so that the red part of the tile is on a red square and the blue part of the tile is on the blue square.
Since you'll need to place 31 tiles on the board (to cover the 62 squares), you would have to be able to cover 31 red squares and 31 blue squares. But when you took away the two corners, you can see that you are taking away two red spaces, leaving 30 red squares and 32 blue squares. There is no way to cover 30 red squares and 32 blue squares with the 31 tiles, since these tiles can only cover 31 red squares and 31 blue squares, and thus, tiling this board is not possible.
We all know that square root of number 121 is 11. But do you know what si the square root of the number "12345678987654321" ?
It's a maths magical square root series as :
Square root of number 121 is 11
Square root of number 12321 is 111
Square root of number 1234321 is 1111
Square root of number 123454321 is 11111
Square root of number 12345654321 is 111111
Square root of number 1234567654321 is 1111111
Square root of number 123456787654321 is 11111111
Square root of number 12345678987654321 is 111111111 (answer)
You are standing in a pitch-dark room. A friend walks up and hands you a normal deck of 52 cards. He tells you that 13 of the 52 cards are face-up, the rest are face-down. These face-up cards are distributed randomly throughout the deck.
Your task is to split up the deck into two piles, using all the cards, such that each pile has the same number of face-up cards. The room is pitch-dark, so you can't see the deck as you do this.
How can you accomplish this seemingly impossible task?
Take the first 13 cards off the top of the deck and flip them over. This is the first pile. The second pile is just the remaining 39 cards as they started.
This works because if there are N face-up cards in within the first 13 cards, then there will be (13 - N) face up cards in the remaining 39 cards. When you flip those first 13 cards, N of which are face-up, there will now be N cards face-down, and therefore (13 - N) cards face-up, which, as stated, is the same number of face-up cards in the second pile.
A swan sits at the center of a perfectly circular lake. At an edge of the lake stands a ravenous monster waiting to devour the swan. The monster can not enter the water, but it will run around the circumference of the lake to try to catch the swan as soon as it reaches the shore. The monster moves at 4 times the speed of the swan, and it will always move in the direction along the shore that brings it closer to the swan the quickest. Both the swan and the the monster can change directions in an instant.
The swan knows that if it can reach the lake's shore without the monster right on top of it, it can instantly escape into the surrounding forest.
How can the swan succesfully escape?
Assume the radius of the lake is R feet. So the circumference of the lake is (2*pi*R). If the swan swims R/4 feet, (or, put another way, 0.25R feet) straight away from the center of the lake, and then begins swimming in a circle around the center, then it will be able to swim around this circle in the exact same amount of time as the monster will be able to run around the lake's shore (since this inner circle's circumference is 2*pi*(R/4), which is exactly 4 times shorter than the shore's circumference).
From this point, the swan can move a millimeter inward toward the lake's center, and begin swimming around the center in a circle from this distance. It is now going around a very slightly smaller circle than it was a moment ago, and thus will be able to swim around this circle FASTER than the monster can run around the shore.
The swan can keep swimming around this way, pulling further away each second, until finally it is on the opposite side of its inner circle from where the monster is on the shore. At this point, the swan aims directly toward the closest shore and begins swimming that way. At this point, the swan has to swim [0.75R feet + 1 millimeter] to get to shore. Meanwhile, the monster will have to run R*pi feet (half the circumference of the lake) to get to where the swan is headed.
The monster runs four times as fast as the swan, but you can see that it has more than four times as far to run:
[0.75R feet + 1 millimeter] * 4 < R*pi
[This math could actually be incorrect if R were very very small, but in that case we could just say the swan swam inward even less than a millimeter, and make the math work out correctly.]
Because the swan has less than a fourth of the distance to travel as the monster, it will reach the shore before the monster reaches where it is and successfully escape.
Emily was sitting at her study table, home alone, on a cold and stormy night. Her parents had taken a flight earlier in the morning to Australia as her grandmother had passed away. She had wanted to follow her parents but she had an important English examination the next day which she could not miss. The storm was getting heavier by the minute and the wind was howling outside. All this noise made it very hard for her to concentrate. She was on the verge of dozing off when she was shaken alert by a sudden "THUD!" She dismissed it as a window which had been slammed shut by the wind.
She tried to concentrate on her books when she heard faint footsteps. Emily got out of her room and looked around when suddenly, without warning, she was grabbed by the neck. She tried to scream but it came out as a mere whimper as the intruder was pressing hard against her throat with his arm. She tried to free herself from his grip but to no avail.
"Give me all your money!" growled the man who had grabbed her from behind.
"Th-there is none h-here! Please ll-let me go!" cried Emily.
"Don't LIE TO ME!" screamed the increasingly agitated man. She felt the man strengthen his grip around her neck. She said nothing and a few seconds passed by in silence. Suddenly the phone rang which alerted both of them.
"People will get suspicious if I don't answer the phone," said Emily, with a controlled voice. The intruder let her go.
"Alright, but NO funny business, or ELSE!" said the nervous intruder. Emily walked toward the phone. She took a deep breath and calmed herself. She picked up the phone. "Hey Em! How's the revision going?" said the caller.
"Hey Anna. Thanks for the call. Hey you know those Science notes I lent you last week? Well I really need them back. It would be a great help to me. It's an emergency, so if you could give me them tomorrow it would be great. Please hurry in finding the notes. I need to get back to my books now. Bye," Emily said. She hung up the phone.
"It was wise of you not to say anything," said the intruder, although he was more than a bit confused by her conversation.
"Now TELL ME WHERE THE MONEY IS KEPT!" screamed the thief.
"It...it's...in my dad's room. The first room on the right. Third drawer," said Emily. "SHOW me!" said the man, and removed his grip around her neck. She took a big gulp of air and nearly fell. She swallowed hard and said a silent prayer. She walked slowly, in silence, toward her father's room. All of a sudden, they heard police sirens. The intruder froze in his footsteps. He ran to the nearest window and jumped out of it. Emily ran outside in time to see the intruder being escorted into the car. She saw Anna and she ran toward her and hugged her.
"Smart kids," said the policeman.
What had happened?
Emily had used the mute button during her conversation with Anna so that all Anna heard was: "call...help...emergency...please hurry".
Anna, sensing something was wrong, called the police and told them Emily's address. The police were able to come to Emily's house in time to catch the perpetrator.
You have just purchased a small company called Company X. Company X has N employees, and everyone is either an engineer or a manager. You know for sure that there are more engineers than managers at the company.
Everyone at Company X knows everyone else's position, and you are able to ask any employee about the position of any other employee. For example, you could approach employee A and ask "Is employee B an engineer or a manager?" You can only direct your question to one employee at a time, and can only ask about one other employee at a time. You're allowed to ask the same employee multiple questions if you want.
Your goal is to find at least one engineer to solve a huge problem that has just hit the company's factory. The problem is so urgent that you only have time to ask N-1 total questions.
The major problem with questioning the employees, however, is that while the engineers will always tell you the truth about other employees' roles, the managers may lie to you if they like. You can assume that the managers will do their best to confuse you.
How can you find at least one engineer by asking at most N-1 questions?
You can find at least one engineer using the following process:
Put all of the employees in a conference room. If there happen to be an even number of employees, pick one at random and send him home for the day so that we start with an odd number of employees. Note that there will still be more engineers than managers after we send this employee home.
Then call them out one at a time in any order. You will be forming them into a line as follows:
If there is nobody currently in the line, put the employee you just called out in the line.
Otherwise, if there is anybody in the line, then we do the following. Let's call the employee currently at the front of the line Employee_Front, and call the employee who we just called out of the conference room Employee_Next.
So ask Employee_Front if Employee_Next is a manager or an engineer.
If Employee_Front says "manager", then send both Employee_Front and Employee_Next home for the day.
However, if Employee_Front says "engineer", then put Employee_Next at the front of the line.
Keep doing this until you've called everyone out of the conference room. Notice that at this point, you'll have asked N-1 or less questions (you asked at most one question each time you called an employee out except for the first employee, when you didn't ask a question, so that's at most N-1 questions).
When you're done calling everyone out of the conference room, the person at the front of the line is an engineer. So you've found your engineer!
But the real question: how does this work?
We can prove this works by showing a few things.
First, let's show that if there are any engineers in the line, then they must be in front of any managers.
We'll show this with a proof by contradiction. Assume that there is a manager in front of an engineer somewhere in the line. Then it must have been the case that at some point, that engineer was Employee_Front and that manager was Employee_Next. But then Employee_Front would have said "manager" (since he is an engineer and always tells the truth), and we would have sent them both home. This contradicts their being in the line at all, and thus we know that there can never be a manager in front of an engineer in the line.
So now we know that after the process is done, if there are any engineers in the line, then they will be at the front of the line. That means that all we have to prove now is that there will be at least one engineer in the line at the end of the process, and we'll know that there will be an engineer at the front.
So let's show that there will be at least one engineer in the line. To see why, consider what happens when we ask Employee_Front about Employee_Next, and Employee_Front says "manager". We know for sure that in this case, Employee_Front and Employee_Next are not both engineers, because if this were the case, then Employee_Front would have definitely says "engineer". Put another way, at least one of Employee_Front and Employee_Next is a manager. So by sending them both home, we know we are sending home at least one manager, and thus, we are keeping the balance in the remaining employees that there are more engineers than managers.
Thus, once the process is over, there will be more engineers than managers in the line (this is also sufficient to show that there will be at least one person in the line once the process is over). And so, there must be at least one engineer in the line.
Put altogether, we proved that at the end of the process, there will be at least one engineer in the line and that any engineers in the line must be in front of any managers, and so we know that the person at the front of the line will be an engineer.