page_title: Up to
section_title: Eight queens
hierarchical_section_title: Up to / Examples / Eight queens
caption_attribution_description: Dansk: Symmetrisk løsning på 8 dronninge problemet. English: Symmetrical solution of the eight queens problem.
context_page_description: In mathematics, the phrase up to is used to convey the idea that some objects in the same class — while distinct — may be considered to be equivalent under some condition or transformation. It often appears in discussions about the elements of a set, and the conditions under which some of those elements may be considered to be equivalent. More specifically, given two elements, " and are equivalent up to " means that and are equivalent, if criterion, such as rotation or permutation, is considered. In which case, the elements of can be arranged in subsets known as "equivalence classes", sets whose elements are equivalent to each other up to. In some cases, this might mean that and can be transformed into one another—if a transformation corresponding to is applied.
  If is some property or process, then the phrase "up to " can be taken to mean "disregarding a possible difference in ". For instance, the statement "an integer's prime factorization is unique up to ordering" means that the prime factorization is unique—when we disregard the order of the factors.
context_section_description: 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 identical, and one usually says "there are 92 () unique solutions up to permutations of the queens", or that "there are 92 solutions mod 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 Eight queens puzzle#Solutions).
caption: A solution of the eight queens problem
exif: {}