The Google problem[S]

This problem is said to have appeared in Google's entrance exam.
Problem:
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1

what is the next line ? [S]

Colouring Problem

Problem : Six sides of a die are painted with six different colours.In how many ways can this be done ?
See comment for clue

Divsion problem[S]

Problem : find the maximum value of 'n' such that 18^n divides 181!
See comment for clue[S]

Bug split

A bug either splits into two perfect copies of itself or dies. If the probability of splitting is p (and is independent of the bug's ancestry), what is the probability that a bug's descendants die out?

Weighing problem[S]

There are 101 coins among which one is counterfeit. we do not know whether the counterfeit coin is lighter or heavier ( than the rest of the coins ) which has to be determined within three weighings.how do you do it ?[S]

A problem from cryptography[S]

This field deals with converting plain text into cipher text ( un understandable ) using a mathematical construct to keep the information secure.A monoalplabetic substitution of +3 applied on the plain text "ram" gives cipher text "udp".Here 'mono alphabetic substitution' is the algorithm and '+3' is the key.
by knowing the algorithm but not the key can we retrieve plain text from cipher text ? [S]

A simple inequality[S]

For real nums a,b,c prove that
(i) (a+b)(1/a +1/b)>=4

(ii)(a+b+c)(1/a +1/b +1/c) >=9

(iii) well, if this seems trivial prove in general for reals nums a_1,a_2,...a_n prove that
(a_1 +a_2 +...+a_n) (1/ a_1 +1/ a_2 +...+1/ a_n)>=n^2

(iv) well you may still go further proving (1+1/2 +1/3 +....) is not convergent.[S]

child and knives

There are ten knives kept on the table with their sharp edges pointing in one direction.

i i i i i i i i i i

a child looks at this and reverses the orientation but maintaining the neighbouring knives in opposite direction.one way child could do this is
i ! i i i i ! i i !
find the total num of ways in which child could do ?

A research problem

There are 'n' homes on the circumference of a circle.In each home there reside husband and wife.People of each home know only their neighbors (on both sides) and no one else.once,all the people of ten homes meet at a party.men dance with women.what is the probability that people do not know whom they danced with?
try out for n=5.

triplet problem[S]

(a,b,c) form a triplet if all a,b,c are natural numbers and a^2=b^2+c^2
prove that (1)one among the triplet is even.(2)one among the triplet is a multiple of 3.
[S]

treasure problem[S]

you are in 2d space and you may call you original location as (0,0).I shall tell you at what dist you are from the treasure, but not the direction.you have to guess where the treasure is and you will be transfered to that point.if you find the treasure there well and good.else I shall tell you the dist again.this constitutes one move.write an algorithm to find out the treasure.prove that in worst case you can do it in 3 moves.[S]