  Edge-matching problems, also called edge matching puzzles, are abstractions
of placement problems with neighborhood conditions. Pieces with colored edges
have to be placed on a board such that adjacent edges have the same color. The
problem has gained interest recently with the (now terminated) Eternity~II
puzzle, and new complexity results. In this paper we consider a number of
settings which differ in size of the puzzles and the manipulations allowed on
the pieces. We investigate the effect of allowing rotations of the pieces on
the complexity of the problem, an aspect that is only marginally treated so
far. We show that some problems have polynomial time algorithms while others
are NP-complete. Especially we show that allowing rotations in one-row puzzles
makes the problem NP-hard. The proofs of the hardness result uses a large
number of colors. This is essential because we also show that this problem (and
another related one) is fixed-parameter tractable, where the relevant parameter
is the number of colors.
