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.

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.

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!! :)

Probability in air

Source:Peter Winkler “Mathematical Puzzles. A Connoisseur’s Collection”,
published by A.K. Peters, Ltd. in 2004.


Problem:
100 people board an airplane with 100 seats. Each person has a seat assigned. For
Some reason, the 1st person who gets in takes her seat at random. Then the 2nd
passenger takes her seat if it is not occupied (by the 1st), and picks a seat at random
if her seat is occupied. Then the 3rd passenger takes her seat if it is not occupied
(by the 1st or 2nd), and picks a seat at random if her seat is occupied. And so on.
What is the probability that the last person will sit in her seat?

Sum to prove

Source: http://www.math.udel.edu/~lazebnik/papers

Problem:
Consider any positive integer N whose (decimal) digits read from left to right are in
non-decreasing order, but the last two digits (tens and ones) are in increasing order.
Prove that the sum of digits of 9N is always exactly 9.

Fish in the pond

Source: CAT 2011 question(I couldn't solve it)


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?

Probability Again

Source: Surprises by Felix Lazebnik


Problem:
Write two distinct integers, one on a card, and put two cards on the table face
down. You can pick any of the two, look at it, and then you have to guess whether
the other number is larger or smaller. Prove that you have a strategy to make a
correct guess with probability strictly greater than 1/2.

Egg Breaking strategy

Source: Google Interview Puzzle

Problem:
With two eggs and a building with 100 floors,
what is the optimal strategy for finding the lowest floor at which an egg
will break when dropped?

http://www.scribd.com/doc/7217370/Egg-Puzzle-Generalized-Solution-for-Any-Number-of-Eggs

Grid Computing

Source: Algorithm analysis -Coreman

Problem:

A room has n computers, less than half of which are damaged. It is possible to query a computer about the status of any computer. A damaged computer could give wrong answers. The goal is to discover an undamaged computer in as few queries as possible.