Fixed point theorem

Source: Nash Equilibrium's derivative also, in Martin's Gardener My best Mathematical and Logical Puzzles


Problem:  One morning at 6 am, a monk began climbing a tall mountain, which happened to only have one path to the top. He ascended the path at his leisure, taking some stops along the way. He reached the top at 8 pm.

The next morning at 6 am, the monk descended the mountain along the same path. He took several breaks along the way, and reached the bottom at 8 pm.

The amazing result: there is some spot on the path that the monk occupied at precisely the same time of day for both trips. Why is this?

This is true even if the monk took breaks to meditate, chatted it up with a fellow traveler, or if he took some time to massage his feet. You get the point: it really does not matter what he did during the trips going up or down.

Beauty Contest

Source: Some Basic Game theory books


Problem: 


Pick a number from 0 to 100. The winner is the person who chooses the number closest to 2/3rds of the group's average response. 
What is the rational answer?

Soccer fan's dream

Source: Moscow Puzzles

Problem:
A soccer fan upset by the defeat of his favorite team slept restlessly. In his dream ,he saw the goalkeeper was practicing alone in a large room, tossing the ball against a wall and catching it.
But the goalkeeper got smaller and smaller and changed into a ping pong ball, and the soccer ball swelled into a huge cast iron ball.
The iron ball circled around madly, trying to crush the ping pong ball which darted desperately about .
Could the ping pong ball find safety without leaving the floor?

The Traveler

Source: Martin Gardener's -My Best Mathematical And Logical Puzzles

Problem: A traveler comes to a fork in the road which leads to two villages. In one village the people always tell lies and in the other village the people always tell the truth. The traveler needs to conduct business in the village where everyone always tells the truth. A man from one of the villages is standing in the middle of the fork, so there is no indication of which village he comes from. The traveler approaches the man and asks him just one question. From the man’s answer, he knows which road to follow. What is the question?

Solution : - in comments below

Birthday problem or a paradox

Source: wikipedia

Problem: There is a big line of people waiting outside a theater for buying tickets. The theater owner comes out and announces that the first person to have a birthday same as someone standing before him in the line gets a free ticket. Where will you stand to maximize your chance.

Chameleons

Source: Some Maths Blog

Problem: On an island live 13 purple, 15 yellow and 17 maroon chameleons. When two chameleons of different colors meet, they both change into the third color. Is there a sequence of pairwise meetings after which all chameleons have the same color?

Insect in 3D

Problem: An ant crawls from one corner of a room to the diametrically opposite corner
 along the shortest possible path. If the dimensions of the room are
 3 x 4 x 5, what distance does the ant cover?

Lazy Vera

Source: 300 Moscow Puzzles

Problem: Mother asked Vera to type a manuscript for her. Vera said she will type
20 pages a day. Vera was lazy during the first half of the manuscript and
typed at speed of 10 pages per day and the latter half of the script at 30
pages per day.

Vera said, see I made it on time half of 30 + 10 is 20 which is my average.
Mother said ‘No, You didn’t’.
Who was right and why?

Bertrand's paradox

Source: Internet

Problem:  In a village, the barber shaves everyone who does not
shave himself/herself, but no one else.
Who shaves the barber?

It’s called Bertrand’s paradox after Bertrand Russell.

Sleeping Beauty Paradox

Source: Tim Whitter's Paradoxical natures

Problem:  Its sleeping beauty paradox, quite famous.
I have not fathomed it out yet.

The paradox imagines that Sleeping Beauty volunteers to undergo the
 following experiment. On Sunday she is given a drug that sends her to sleep.
 A fair coin is then tossed just once in the course of the experiment to
determine which experimental procedure is undertaken. If the coin comes up
 heads, Beauty is awakened and interviewed on Monday, and then the
 experiment ends. If the coin comes up tails, she is awakened and interviewed
 on Monday, given a second dose of the sleeping drug, and awakened and
interviewed again on Tuesday. The experiment then ends on Tuesday,
 without flipping the coin again. The sleeping drug induces a mild amnesia,
so that she cannot remember any previous awakenings during the course of
the experiment (if any). During the experiment, she has no access to anything
 that would give a clue as to the day of the week. However, she knows all the details of the experiment.
Each interview consists of one question, "What is your credence now for the proposition that our coin landed heads?"

This problem is considered paradoxical because the answer is often given as either 1/3 or 1/2.

Think!! :)

Monty Hall Problem

Source: Various(Movies e.g. 21, Maths courses, books on basic probability)

Problem: Its Monty hall, also used in movie ‘21’,when Kevin spacey asks
Jim Sturgess in form of game show host problem.


You are on a game show and there are three doors. The presenter tells
 you that behind one of doors there is a car and behind the other two are
 goats. If you pick the car you win it. After you have picked a door the
 presenter opens a different door with a goat behind it, he then gives you
the chance to change your choice to what door you open. What should you do?

Lexicon

Source: Moscow Puzzles

Problem:

A slim crocodile living in the Nile took a child. His mother begged to have

 him back. The crocodile could not only talk, but was also a great sophist

and stated, "If you guess correctly what I will do with him, I will return

 him. However, if you don't predict his fate correctly, I'll eat him."

What statement should the mother make to save her child?

Dollar notes

Source: Bhupesh Bhargava,Algo Embedded systems

Problem: Distribute 127 one rupee notes among 7 wallets So that any integer sum
 from 1 to 127 can be paid without opening the wallets.

Twins

Source: Some Problem website

Problem:   Julius and Vincent are brothers. "We are born within the same hour," says Julius, "on the same day of the same year." "But," says Vincent, "we are no twins!"

How??

CAT 2011 question

Source: Of course CAT 2011

Problem:    Suppose you have a pond and in that pond you have some goldfish. Size of all the fish is same and they are evenly distributed in the pond.

You want to count them. You threw the net inside the pond and saw that there are 40 fish in the net. You marked all of them with blue color.

You repeated the same activity and this time you got 60 fish out of them 4 had the blue color mark on them(it means they were from the first withdraw).

What is the approximate number of fish in the pond?

a.      200
b.      400
c.       600
d.      800

uniqueness

What makes 8,549,176,320 only one of its kind? ???