Imagine a sequence of 16 coins in a circle, alternating heads and tails. Your mission is to change this sequence into one with as many heads showing as possible using only the following two rules: You take any three in a row of the form HTH and change them to THT, and likewise you may change THT into HTH.
If the case of 16 is too hard, the analogous problem with 10 coins is also a worthwhile exercise.
(So that you will know when you are done, I will tell you that with 10 coins you can get 8 heads and with 16 you can get 13 heads, and these are optimal.)
Source: This game is called Blet, from a recent Amer Math Monthly (Oct) article by Villegas, Sadun, and Voloch.