enumerated the fixed polyominoes up to n = 56 – approximately 6.915. Free polyominoes were enumerated in 2007 up to n = 28 by Tomás Oliveira e Silva, and in 2012 up to n = 45 by Toshihiro Shirakawa. Symmetries of polyominoes The dihedral group D4 is the group of symmetries (symmetry group) of a square. This group contains four rotations and four reflections. It is generated by alternating reflections about the x-axis and about a diagonal. One free polyomino corresponds to at most 8 fixed polyominoes, which are its images under the symmetries of D4. However, those images are not necessarily distinct: the more symmetry a free polyomino has, the fewer distinct fixed counterparts it has. Therefore, a free polyomino that is invariant under some or all non-trivial symmetries of D4 may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are equivalence classes of fixed polyominoes under the group D4. Polyominoes have the following possible symmetries; the least number of squares needed in a polyomino with that symmetry is given in each case: 8 fixed polyominoes for each free polyomino: no symmetry (4) 4 fixed polyominoes for each free polyomino: mirror symmetry with respect to one of the grid line directions (4) mirror symmetry with respect to a diagonal line (3) 2-fold rotational symmetry: C2 (4) 2 fixed polyominoes for each free polyomino: symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: D2 (2) (also known as the Klein four-group) symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: D2 (7) 4-fold rotational symmetry: C4 (8) 1 fixed polyomino for each free polyomino: all symmetry of the square: D4 (1). In the same way, the number of one-sided polyominoes depends on polyomino symmetry as follows: 2 one-sided polyominoes for each free polyomino: no symmetry 2-fold rotational symmetry: C2 4-fold rotational symmetry: C4 1 one-sided polyomino for each free polyomino: all symmetry of the square: D4 mirror symmetry with respect to one of the grid line directions mirror symmetry with respect to a diagonal line symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: D2 symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: D2. The following table shows the numbers of polyominoes with n squares, sorted by symmetry groups. Algorithms for enumeration of fixed polyominoes Inductive algorithms