  This paper extends the work started in 2002 by Demaine, Demaine and Verill
(DDV) on coin-moving puzzles. These puzzles have a long history in the
recreational literature, but were first systematically analyzed by DDV, who
gave a full characterization of the solvable puzzles on the triangular grid and
a partial characterization of the solvable puzzles on the square grid. This
article specifically extends the study of the game on the square grid. Notably,
DDV's result on puzzles with two "extra coins" is shown to be overly broad:
this paper provides counterexamples as well as a revised version of this
theorem. A new method for solving puzzles with two extra coins is then
presented, which covers some cases where the aforementioned theorem does not
apply. Puzzles with just one extra coin seem even more complicated, and are
only touched upon by DDV. This paper delves deeper, studying a class of such
puzzles that may be considered equivalent to a game of "poking" coins. Within
this class, some cases are considered that are amenable to analysis.
