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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.

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  • $\begingroup$ I’m delighted that this question has been asked. Sadly, I believe that it is unclear, too broad, and primarily opinion-based — and maybe off-topic (questions about definitions of words belong on English Language & Usage). $\endgroup$ – Peregrine Rook Jan 16 '17 at 1:03
  • $\begingroup$ English language? Are you kidding me? The question isn't about the meaning(s) of the word, it's about a certain problem in puzzles like sudoku - what techniques do we consider guessing/cheating/invalid and which ones we don't. And I don't agree that this question is opinion-based. But since I wasn't getting any concrete answers anyway, I'm not gonna waste time fighting it. $\endgroup$ – NPS Jan 16 '17 at 14:54
  • $\begingroup$ (After correction of a typo) "But what is guessing?  How do we define it?" $\endgroup$ – Peregrine Rook Jan 16 '17 at 21:25
  • $\begingroup$ Yeah, how do we define the term in the context of puzzle solving, not what does the word mean in English or any other language. Same as we define many other words to mean one precise thing in the context of some subject, e.g. neighborhood in mathematics. The word has a broader meaning (or multiple meanings) in the language but we talk about a specific one in a specific context. $\endgroup$ – NPS Jan 17 '17 at 7:11
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A guess is a supposition where the chain of deductions to reach a conclusion on the validity of that supposition exceeds the mental capacity of the solver to process without the assistance of visual cues. i.e. writing hypothetical numbers onto the grid.

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  • $\begingroup$ That would mean the definition of 'guessing' depends on the mental capacity of the solver (which varies widely), making the definition rather shaky. $\endgroup$ – Ankoganit Jan 15 '17 at 14:59
  • $\begingroup$ That is absolutely not the definition of guessing. Based on that computers would be unable to guess im most scenarios (because they a lot of capacity and processing power) and brute force wouldn't be guessing. No, we might assume that a solver has a limitless capacity. That's why I wrote that solving on paper and in mind is the same thing. $\endgroup$ – NPS Jan 15 '17 at 15:41
  • $\begingroup$ @Ankoganit I'd say it's more the case that the point at which a solver is reduced to guessing depends on the solver. $\endgroup$ – Neil W Jan 15 '17 at 16:15
  • $\begingroup$ @NPS I don't think anything a sudoku solving program does really can be called a guess, nor is brute force search guessing. It's a logical, deterministic algorithm guaranteed to find the solution. $\endgroup$ – Neil W Jan 15 '17 at 16:23

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