Source: David Richards' Puzzle Collection
A semi-infinite chess board (vary from zero to infinity in both dimensions) with counters in the three bottom left squares, as shown below.
How to move: If the squares above and to the right are free, a counter can be removed and replaced by two counters, one in the square above and one in the square to the right - as shown below
Prove that it is not possible to leave the three bottom left squares empty.
Update (March 07 2013)
Solution posted by Sudeep Kamath (PhD Student, UC at Berkeley, EE IITB Alumnus 2008), Takaki and Rahul in comments! All solutions are essentially the same. Interesting discussion and links by Sudeep.