*forever*to work out the "HARD" Sudoku puzzles.

The literature of Sudoku is growing. Check out these excerpts from recent publications:

Complexity classes such as NP do not measure the difficulty of any specific problem instance but rather describe the rate at which difficulty grows as a function of problem size. If we can solve an order-n Sudoku, how much harder will we have to work to solve a puzzle of order n + 1? For problems in NP, the effort needed grows exponentially.

Most discussions of the complexity of Sudoku refer to the work of Takayuki Yato and Takahiro Seta of the University of Tokyo, whose analysis relates the task of solving Sudoku to the similar problem of completing a partially specified Latin square. The latter problem in turn has been connected with others that are already known to be in NP. This process of "reduction" from one problem to another is the standard way of establishing the complexity classes of computational problems. Yato and Seta employ an unusual form of reduction that addresses the difficulty of finding an additional solution after a first solution is already known. In Sudoku, of course, well-formed puzzles are expected to have only one solution. Yato and Seta say their result applies nonetheless. I don't quite follow their reasoning on this point, but the literature of complexity theory is vast and technical, and the fault is likely my own.

Your fault and mine, dude. From Unwed Numbers: The Mathematics of Sudoku, a Puzzle That Boasts, "No Math Required!" in the Jan-Feb 2006

*American Scientist*.

We provide a simple linear time transformation from a directed or undirected graph with labeled edges to an unlabeled digraph, such that paths in the input graph in which no two consecutive edges have the same label correspond to paths in the transformed graph and vice versa. Using this transformation, we provide efficient algorithms for finding paths and cycles with no two consecutive equal labels. We also consider related problems where the paths and cycles are required to be simple; we find efficient algorithms for the undirected case of these problems but show the directed case to be NP-complete. We apply our path and cycle finding algorithms in a program for generating and solving Sudoku puzzles, and show experimentally that they lead to effective puzzle-solving rules that may also be of interest to human Sudoku puzzle solvers.

I don't need no stinking rules. All I need is a sharp pencil, a good eraser, and a few hours of uninterrupted time, so I can hit the Sudoku Zone for a few minutes here & there. From Nonrepetitive Paths and Cycles in Graphs with Application to Sudoku, in the 20 July 2005

*Computer Science*.

More on the Mathematics of Sudoku in Wikipedia.

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