Ramsey Theory is an intricate field of pure mathematics, but it also has applications in computer science, where graphs are commonlu used as data structures. Every pair of "forbidden" graphs has a Ramsey number, which indicates when one of those graphs must occur. A Ramsey number for a pair of forbidden graphs is the smallest number of points R for which every edge-coloring of the complete graph on R points includes one of the forbidden graphs in its respective color. In this paper we use two original methods to find and prove new Ramsey number formulas for certain tadpole graphs based upon known Ramsey number formulas for arbitrary cycle graphs. A tadpole graph Qn,t is formed by connecting one end of a path graph on t points (called the “tail”) to a cycle of size n (called the “head”). A pan graph Qn is defined as a tadpole graph with t=1. We first use an intuitive counting method to prove exact Ramsey number formulas for all “near-diagonal” pan graphs, i.e. those pairs of pan graphs where the maximum cycle size of the pair is no more than 5/2 times the minimum cycle size. Then, we demonstrate a number-theoretic method to find exact Ramsey numbers for arbitrarily far-from-diagonal tadpole pairs with arbitrarily long tails. However, the second method requires that the tadpoles meet number-theoretic constraints (rather than size constraints). For example, in the diagonal cases, n-1 must be relatively prime to n+t, where either n-1 has an odd order (mod n+t) or the additive inverse of n-1 has an even index relative to n-1 (mod n+t). These “scattered” on- and off-diagonal results provide upper bounds on the Ramsey numbers for other tadpole pairs that fail to meet the number-theoretic constraints.