The whole difficulty of solving puzzles like sudoku is doing it without guessing. Well, either without any guessing whatsoever or with little enough that a human can still make retractions and come to a solution.
But what is guessing? How do we define it? People intuitively feel it but don't really have a precise definition. And that's what this question is about.
Let's start with a statement (albeit I'm not 100% sure that it's true) that one doesn't guess when there's only 1 possibility. For instance, if there's only 1 cell in a sudoku board where a certain number can go, one can do that move with certainty and it's not guessing. So, as an implication, guessing would be when there is more than 1 possibility and one makes a move without knowing if it's correct or not (in the context of whole board).
But I want to make something clear here - making guesses in your mind (instead of on paper) doesn't change anything - it's still guessing. And whether we guess 1 step ahead or 100 - there's no difference.
For example, let's talk about finding the common part. By that I mean that we taking a part of the whole board, analyzing its possible solutions, writing down all of them and taking the common part (numbers that are the same in all of them). For instance, let's say we have only 2 possible numbers for a given cell, we try one and see that it implicates a number 6 for another cell. Then we try the other number and see that it also implicates the number 6 for that another cell. So we can safely use that number 6 in our solution as we know it's the only possibility. So it this guessing?
On one hand, we know we're making a valid move (the only valid move) but on the other, we had to make a guess, see where it leads to, make another, see where it leads to and so on...
And if it's not pronounced enough yet, in the particular case the part of the board that we apply this technique to might be the whole board. And that's just brute force.
But this is only one of many issues (or solving techniques) to consider.
Solving puzzles like sudoku is mostly about propagating the limitations. E.g. pencil marking in sudoku. We know numbers given in some cells pose limitations on other cells and we have to somehow propagate this information to other cells. But what's legal here and what's not (in the context of guessing)?
I know my question might create broad discussions and we might not even know the answer to it. But on the other hand, I do believe there is one precise answer. Either way, I'd like to get closer to it and establish at least some ground rules.
