This thread is intended to throw some light on aspects of sudoku that I had not found somewhere else. If you look for tips or methods for manual solving, this thread is not for you. It is purely theoretical. Nevertheless I can show some interesting experimental results especially for solving very hard sudoku problems.

Introduction: (skip if boring)

The term “solution” denotes a full set of numbers (symbols) complying with all base constraints of sudoku. Each particular sudoku problem can be unsolvable, or has one to many solutions. Sudoku is NP-complete (proof elsewhere). That translates to: The only way to solve a sudoku problem completely is to explore all possible cases. This is done by backtracking, possibly enhanced by cutting off assured dead branches. The procedure is quick and efficient and solves all sudoku with all multiple solutions if existing. So there are no remaining unsolved sudoku problems. Or in other words: Solving sudoku is trivial.

So what about all the solvers based on methods and rules discussed in this forum are doing other than solving already solved problems?

Answer (intentionally provocative): Any attempt to build a solver from a limited number of methods and rules is futile. Such solvers will either not solve all sudoku, or if one is meant to solve all, the proof is another and very large NP-complete problem.

The value of methods and rules is not solving but analysis. Unfortunately mostly the possible goals of this analysis is not clearly defined. That results frequently in misunderstanding and unproductive discussions. I do not intend to diminish the value of rules and traditional methods, but at least for me it is important to put them in the right place. In spite of everything I continue to use the term solver like anyone else.

If you have a solution for a particular sudoku, its correctness can be verified by itself. There is absolutely no need to present how the solution was found. This is different from the statement, that the sudoku can be solved e.g. with singles, pairs and triples. You have to show up with an appropriate elimination sequence to prove this statement.

There are two completely different directions of analysis goals.

- You can analyze properties of a particular sudoku and compare these with other sudoku.

Any scalable property defines a specific aspect of rating. (I come back to this important point later).

Additionally you have classes of problems with the same shared properties.

- You can analyze the scope of rules and rule sets.

This results in classes of sudoku solvable by such rule/method sets.

How to compare such classes is not obvious generally.

Sorry for the long introduction. Two more posts to come.

1) definition of universal elimination pattern independent from rules

along with an explanation why this is necessary

some general properties and resulting consequences for rating

2) experimental solving results on a large set of very hard sudoku with

application of universal eliminations at level 1 ( one candidate assumption )