6 Colored balls

We have two red, two green and two yellow balls. For each color, one ball is heavy and the other is light. All heavy balls weigh the same. All light balls weigh the same. How many weighing on a beam balance are necessary to identify the three heavy balls?

Hole and the Board

Cut this board into only 2 pieces so that they fit into the given hole.

2n digit number

i) Consider a 4 digit number abcd. Will there exist a 'k' such that

abcd = k*ab*cd

If so, find the k and abcd?

ii) Now, consider a 6 digit number abcdef. Will there exist a 'k' such that 

abcdef = k*abc*def

If so, find the k and abcdef?
iii) Using these 2 results, what can you conclude about a existence of a 'k' for a generic 2n digit number?

Maximum and Minimum of array

Find the maximum and minimum of an array of 'n' numbers using only 3n/2 comparisons.

PS: This question was given to us in the DS class.

Color matching in concentric circles

A circle is divided into 8 equal sectors. Half are coloured red and half are coloured
blue. A smaller circle is also divided into 8 equal sectors, half coloured red and half
coloured blue. The smaller circle is placed concentrically on the larger. Prove that
no matter how the red and blue sectors are chosen it is always possible to rotate the
smaller circle so that at least 4 colour matches are obtained.