The puzzle Topspin is a sliding number game consisting of an oval track containing a random arrangement of numbered discs, and a small turnstile within the track. A game is of the form [t,n] if it has n total discs, and a turnstile with t discs. Using concepts of group theory, the solvability, or ordering, of the discs is determined or conjectured for all values of t and n. Furthermore, if a game is not solvable, its attainable subgroup is determined or conjectured for all values of t and n. Several notations are used in the proofs of these theorems to help the reader follow visually as well as mathematically. Solvability is difficult to prove, but in the puzzle [t,n] where t and n are both even, we reveal the complex series of flips and shifts needed to prove the solvability of the game. Finally, using the results of the [t,n] games, the solvability is determined or conjectured for multiple turnstile games.