ravel wrote:Hm, for me there is no doubt, that Carculs method to find the solutions is not (equivalent to) brute force. Otherwise he would still be sitting before the hard puzzles trying to solve them.

So, your logical statement is:

If Carcul is not still sitting before the hard puzzles trying to solve them, then Carcul's method must not be "brute force." I do not get that kind of respect even from my family

. Since I do not know Carcul that well, I will stick with the branch of mathematics that was developed to answer questions like this.

ravel wrote:I do not think, that you can apply the Algorithmic Information Theory in that simple way to decide this. You have to take into account

- the enourmous number of theoretical possibilities a solver can exclude in short time (by his experience) and

- all the shortcuts he has by applying known techniques.

Also note that not only the complexity of single steps has to be considered, but also the order and efficiency of the steps. This gives a much higher complexity than you talked of.

We are only talking about the merits of the SIN method here. If Carcul chooses to incorporate other theoretical methods, such as uniqueness rectangles, in his chains, then the brute force comparison is allowed to incorporate those methods as well. You don't get a free ride on the coat-tails of another theoretical method.

We are only worried about advancing the puzzle from a state A (just before the application of the method in one of the SIN steps) to a state B (just after the SIN step). If you compare the number of operations in the SIN method and compare it to the number of operations for brute force to assume the value and follow it along until it crashes, you will find that the delta is always zero at best. Why do we know this? Because SIN has actually documented one possible brute force crash for us which uses the same number of steps, and it might not have even been the quickest crash. (I hope this answers ronk's question as well.)

I will grant that both Carcul and RW's experience enables them to make better educated guesses for which candidates will provide quick crashes or more valuble results for solving the puzzle. These are worthy ideas, especially if you measure success in terms of solving speed or solving without pencil marks. They can also be chock full of logic. But that does not make the methods employed any less brute force.

RW wrote:Of course the ideal would be to define theories that can solve any situation, but I don't think we'll ever get there. The hardest puzzles will always require some amount of brute force.

If you have visited the website and read the article, then you know that this is a definite possibility. With all of the advancements we have made in coming up with theoretical methods for solving sudokus in the past few months, I don't think it is time to throw in the towel just yet, though.