TITLE: is the Sudoku puzzle NP-complete?
QUESTION [7 upvotes]: In general Sudoku on $n^2 \times n^2$ boards of $n \times n$ blocks is NP-complete. 

Is the common Sudoku on $9 \times 9$ board NP-complete?

REPLY [10 votes]: Rather than strict NP-completeness, what I think what you're trying to ask is whether there is a transition similar to the one between 3-SAT and 2-SAT where the problem goes from being NP-hard to easy and in P.  There is such a transition, but 9x9 Sudoku is on the wrong side of it.  Solving a Sudoku puzzle is equivalent to deciding whether there is a valid graph vertex coloring using $k$ colors, where $k = n^2$ in your $n^2 \times n^2$ Sudoku instance.  The graph coloring problem is known to be NP-complete for values of $k$ > 2, so 9x9 Sudoku is still hard.