Many combinatorial procedures (hereafter "games") permit choices along the way, and in most cases those choices affect the final outcome. However, there are games in which the final outcome is predetermined, even though there are many paths to that outcome. This phenomenon is called confluence. Major examples are the chip-firing game, the rotor-router game, and the numbers game (a generalization of the pentagon game).

In this talk I'll discuss new examples of confluence, such as sorting-with-chip-firing, that skirt the boundary between confluence and non-confluence; for these systems, confluence can hinge on arithmetic conditions on the initial state of the game, and the tool that is most often used for establishing confluence, the Jordan-Holder trick, is not available.

Participants will be encouraged to use their loose change to explore these theorems and conjectures during the talk.