Well, It's certainly possible to solve a Sudoku puzzle without trial and error and guessing. Here's the proof: you can pose the solution of Sudoku as a problem in satisfiability (SAT): specifically you can have one variable for each pair (square,number) (so there are 729 variables). You then write down clauses which do the following: 0) You have 1-clauses, corresponding to the initial placement of the numbers that you're given: (s,n), means that number n is in square s. 1) 2-clauses: for each pair (s_i,n_i) (square and number) that "conflicts" (i.e. can't be present at the same time), you have a clause of the form (not (s_1,n_1)) or (not (s_2,n_2)) 2) For each square on the board (all 81 of them) you have a clause of the form or_i (s,i) You can solve any solvable SAT problem via resolution (which is like Gaussian elimination), and back substitution. This doesn't involve any guesswork. However, it could be exponential (since resolving on a variable can create a large number of new clauses). I'm not saying that this is the best way to do it though! However, if you wish to do it this way, it is certainly best to first do resolution on the subset of 1-clauses and 2-clauses (which is very fast -- there is a linear time algorithm for this). This corresponds to making the standard initial deductions. Victor