The breakthrough, known as "descent through nested sub-groups" was found by Morwen Thistlethwaite; details of Thistlethwaite's algorithm were published in "Scientific American" in 1981 by Douglas Hofstadter. The approaches to the cube that led to algorithms with very few moves are based on group theory and on extensive computer searches. Thistlethwaite's idea was to divide the problem into subproblems. Where algorithms up to that point divided the problem by looking at the parts of the cube that should remain fixed, he divided it by restricting the type of moves that could be executed. In particular he divided the cube group into the following chain of subgroups: