can be thought of as the same piece when reflected. The Tetris game does not allow reflections, so the former statement is likely to seem more relevant than the latter. To add in the exhaustive count, there is no formal notation for the number of pieces of tetrominoes. However, it is common to write that "there are seven tetrominoes (= 19 total) up to rotations". Here, Tetris provides an excellent example, as one might simply count 7 pieces × 4 rotations as 28, but some pieces (such as the 2×2 O) obviously have fewer than four rotation states. Eight queens In the eight queens puzzle, if the eight queens are considered to be distinct, then there are 3709440 distinct solutions. Normally, however, the queens are considered to be equal, and one usually says "there are unique solutions up to permutations of the queens", or that "there are 92 solutions modulo the names of the queens", signifying that two different arrangements of the queens are considered equivalent if the queens have been permuted, but the same squares on the chessboard are occupied by them. If, in addition to treating the queens as identical, rotations and reflections of the board were allowed, we would have only 12 distinct solutions up to symmetry and the naming of the queens, signifying that two arrangements that are symmetrical to each other are considered equivalent (for more, see ). Polygons The regular -gon, for a fixed , is unique up to similarity. In other words, by scaling, translation, and rotation, as necessary, any -gon can be transformed to any other -gon (with the same ). Group theory In group theory, one may have a group acting on a set , in which case, one might say that two elements of are equivalent "up to the group action"—if they lie in the same orbit. Another typical example is the statement that "there are two different groups of order 4 up to isomorphism", or "modulo isomorphism, there are two groups of order 4". This means that there are two equivalence classes of groups of order 4—assuming that one considers groups to be equivalent if they are isomorphic. Nonstandard analysis A hyperreal and its standard part are equal up to an infinitesimal difference. Computer science In computer science, the term up-to techniques is a precisely defined notion that refers to certain proof techniques for (weak) bisimulation, and to relate