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.