Graduate Program in Mathematics.Boysal, Arzu.Cin, Uğur.2023-03-162023-03-162020.MATH 2020 C56https://digitalarchive.library.bogazici.edu.tr/handle/123456789/15315In this thesis a variant of the chip- ring game introduced by Hopkins, McConville and Propp in [1], called the labeled chip- ring on Z, is studied. We will rst explore the basic properties and examples of this game. We will then show, how one can see this game as a binary relation on the weight lattice of Type A root system. It is then a natural step to generalize it to other root systems, which was done by Galashin, Hopkins, McConville and Postnikov in [2] and [3]. After reviewing the basics of central- ring introduced in these papers, we examine the central- ring of type A2n with initial weight 0 in Chapter 5. Finally, we study the restrictions in Lemma 12 of [1] in more detail, and conjecture that the number of permutations with maximum number of inversions allowed by this lemma is given by the Catalan numbers.30 cm.Games.ix, 34 leaves ;