18.204: Chip Firing Games

Chip firing is a one-player game where piles start with an initial number of chips and any pile with at least two chips can send one chip to the piles on either side of it. When all of the piles have no more than a single chip, the game ends. In this paper we review fundamental theorems related to this game on a two dimensional number line, including the fact that termination and final configuration are independent of the sequence of moves made and prove the number of moves required for termination is bounded. We then extend the game to consider distinct chips also on a two dimensional number line, where chips are represented by integers and firings result in a comparison of two chips in a pile such that the smaller is sent left and larger is sent right. We prove that for odd numbers of chips some final configurations are sorted while others are unsorted and conjecture that for even numbers of chips the final configuration is necessarily sorted.