Source: Techfest 2010 Puzzle Hut
Problem: It is desired to invert a set of n upright cups by a series of moves in each of which n-1 cups are turned over. Show that this can always be done if n is even and never can be done if n is odd.
Solution: Posted by Ankush Agarwal (2nd yr, CSE, IITB), Dinesh Dharme (4th yr, CSE, IITB), Nikhil (2nd yr, IITD) and Siddhant Agarwal (3rd year, EE, IITB) in comments!! For just solution, read Ankush's and Sid's comment!! Proof of bound posted by me in comments!!