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God Number

By Moloy De posted Thu April 01, 2021 11:04 PM

  
God's algorithm 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, the idea being that only an omniscient being would know an optimal step from any given configuration. The minimum number of moves is termed as God Number.

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.

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. This count 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.

An algorithm to determine the minimum number of moves to solve Rubik's Cube was published in 1997 by Richard 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, it was proven in 2010 through extensive computer calculations 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.

QUESTION I : What is the God Number for 15 Puzzle?
QUESTION II : Do Chess and Go has God Number?

REFERENCE : Wikipedia , Net 
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