Sliding piece puzzles with oriented tiles

AbstractIn this paper, we consider n identical tiles which are placed on the n + 1 vertices of a graph and which move along the edges of the graph. The tiles come with an “orientation”, an element of an arbitrary finite group H. Moving a tile along a given edge into the empty vertex changes the orientation of the tile in a prescribed way. We study the group of oriented positions of the tiles achievable from an initial position which fix the empty vertex. It may be thought of as a subgroup of the semidirect product Hn⋊Sn or the wreath product H wr Sn