Some Notes on Game Bounds

<p>Combinatorial Games are a generalization of real numbers. Each game has a recursively defined complexity (birthday). In this paper we establish some game bounds. We find some limit cases for how big and how small a game can be based on its complexity. For each finite birthday N we find the smallest positive number and the greatest game born by day N as well as the smallest and the largest positive infinitesimals. As for each particular birthday we provide the extreme values for those types of games these results extend those in [1 page 214]. The main references in the theory of combinatorial games are ONAG [1] and WW [2]. We'll use the notation and some fundamental results from WW---mainly from its first six chapters---to establish some bounds to the size of the games.</p>