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]