  We analyze the computational complexity of several new variants of
edge-matching puzzles. First we analyze inequality (instead of equality)
constraints between adjacent tiles, proving the problem NP-complete for strict
inequalities but polynomial for nonstrict inequalities. Second we analyze three
types of triangular edge matching, of which one is polynomial and the other two
are NP-complete; all three are #P-complete. Third we analyze the case where no
target shape is specified, and we merely want to place the (square) tiles so
that edges match (exactly); this problem is NP-complete. Fourth we consider
four 2-player games based on $1 \times n$ edge matching, all four of which are
PSPACE-complete. Most of our NP-hardness reductions are parsimonious, newly
proving #P and ASP-completeness for, e.g., $1 \times n$ edge matching.
