Source: Australian Mathematical Society Gazette Puzzle Corner
Problem: There are coins of various sizes on a table, with some touching others. As often as you wish, you may choose a coin, then turn it over, along with every other coin that it touches. If all coins start out showing heads, is it always possible to change them to all tails using these moves?
Update (Nov 15, 2010):
Solution: Solution from the gazette author posted by me in comments! Interesting linear algebra solution posted by Siddhant Agarwal (EE, Senior Undergraduate, IITB) in comments!