God's algorithm of the Rubik cube is a notion originating in discussions of ways to solve the Rubik's Cube puzzle, but which can also be applied to other combinatorial puzzles and mathematical games. It refers to any algorithm which produces a solution having the fewest possible moves (i.e., the solver should not require any more than this number). The allusion to the deity is based on the notion that an omniscient being would know an optimal step from any given configuration.
Scope
Definition The notion applies to puzzles that can assume a finite number of "configurations", with a relatively small, well-defined arsenal of "moves" that may be applicable to configurations and then lead to a new configuration. Solving the puzzle means to reach a designated "final configuration", a singular configuration, or one of a collection of configurations. To solve the puzzle a sequence of moves is applied, starting from some arbitrary initial configuration.
Solution An algorithm can be considered to solve such a puzzle if it takes as input an arbitrary initial configuration and produces as output a sequence of moves leading to a final configuration (if the puzzle is solvable from that initial configuration, otherwise it signals the impossibility of a solution). A solution is optimal if the sequence of moves is as short as possible. The highest value of this, among all initial configurations, is known as God's number, or, more formally, the minimax value. God's algorithm, then, for a given puzzle, is an algorithm that solves the puzzle and produces only optimal solutions. Some writers, such as David Joyner, consider that for an algorithm to be properly referred to as "God's algorithm", it should also be practical, meaning that the algorithm does not require extraordinary amounts of memory or time. For example, using a giant lookup table indexed by initial configurations would allow solutions to be found very quickly, but would require an extraordinary amount of memory. Instead of asking for a full solution, one can equivalently ask for a single move from an initial but not final configuration, where the move is the first of some optimal solution. An algorithm for the single-move version of the problem can be turned into an algorithm for the original problem by invoking it repeatedly while applying each move reported to the present configuration, until a final one is reached; conversely, any algorithm for the original problem can be turned into an algorithm for the single-move version by truncating its output to its first move.
Examples Well-known puzzles fitting this description are mechanical puzzles such as Rubik's Cube, the Tower of Hanoi, and the 15 puzzle. The one-person game of peg solitaire is also covered, as well as many logic puzzles, such as the missionaries and cannibals problem. These have in common that they can be modeled mathematically as a directed graph, in which the configurations are the vertices, and the moves are the arcs.
Mechanical puzzles
n-Puzzles The Fifteen puzzle can be solved in 80 single-tile moves or 43 multi-tile moves in the worst case. For its generalization the n-puzzle, the problem of finding an optimal solution is NP-hard, so it is not known whether there is a practical God's algorithm.
Towers of Hanoi For the Towers of Hanoi puzzle, a God's algorithm is known for any given number of disks. The number of moves increases exponentially with the number of disks ( 2 n − 1 {\displaystyle 2^{n}-1} ).
Rubik's Cube
An algorithm to determine the minimum number of moves to solve Rubik's Cube was published in 1997 by Richard E. Korf. While it had been known since 1995 that 20 was a lower bound on the number of moves for the solution in the worst case, Tom Rokicki proved in 2010 that no configuration requires more than 20 moves. Thus, 20 is a sharp upper bound on the length of optimal solutions. Mathematician David Singmaster had "rashly conjectured" this number to be 20 in 1980.
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