Pure guess-and-check is easy in the sense that you could easily write a computer program that would, eventually, solve the problem. But the total amount of programmer time plus computer time would be more than the lifetime of the Universe. I have a hard time calling something with a total solution time longer than the lifetime of the Universe “easy”.
Guess-and-check is always valid, but for it to be useful, it has to be used in combination with other methods. Precisely what other methods is flexible.
In the following sudoku, the green highlighted 1s eliminate the red highlighted 1s. What is the guess here? Specifically, which cell are you guessing to be which number and how are you checking if it is correct?
Anyway, I was originally referring to “pure brute force guess-check” and you answered your own question as to why I discounted it.
You could present that as “Suppose that top red square is a 1…”, and then eventually proceed to the conclusion that the bottom-left area doesn’t have any 1s in it, which is a contradiction. Since the assumption that the top red square was a 1 led to a contradiction, we conclude that the top red square isn’t a 1. And then repeat for the other two red squares.
You probably wouldn’t go through all of that in detail, because you can see a shortcut. But ultimately, that shortcut is based on “if it were any other way, there’d be a contradiction”.
