Greedy quasigroups and Wythoff Quasigroups arose out of a desire to better understand certain combinatorial games. Greedy and Wythoff quasigroups have remarkable algebraic properties. In particular, I will investigate the existence of subquasigroups and isomorphism classes. Natural generalizations of greedy quasigroups are also investigated and it is shown that the "greedy" property extends nicely to conjugates. Since Wythoff quasigroups have more structure than ordinary quasigroups, it is natural to ask whether they are an example of a variety of quasigroups. This question is investigated by introducing the idea of tri-quasigroups. Tri-quasigroups are investigated and some remarkable identities are proven. Finally, in the spirit of Conway, a greedy ring is investigated. The construction and characterization are given.