There is a difference from *knowing* that a puzzle CAN be solved without guessing/trial and error/backtracking etc and *demonstrating* this in a particular case. I'm saying that each and every one of the grazillion possible 9x9 sudoku's has a logical path from start to finish -- regardless of whether or not you, I or any human or that has ever lived -- can find it.
Let's set that point aside for the moment since so few seem to agree with me about it and there are really no other arguments I can make to convince anyone. Instead, lets stipulate that some puzzles DO require G/T&E/B, etc. These puzzles CANNOT be solved, by flesh or silicone short of a brute force search. Sudoku's can now be put into two groups -- those solveable without brute force search, and those NOT. Remember, those in the second group cannot be solved without brute force regardless of what other tactics and patterns are discovered. If they can, then they were mis-labled to begin with and belong to the first group.
Here's the problem -- given Sudoku "X" (not the diagonal kind -- the unknown kind), how do we decide which group to put it in? Well, we can try to solve it by non-brute force methods. If we are successful, it goes in the first group. If not -- well, we can't be sure *where* it goes, can we? In fact, deciding which group it goes in is more difficult and will take more time than the most protracted brute force search. That second group MUST stay empty and instead, we fill a third group -- the "unknown".
I cannot prove that there is or isn't at least one Sudoku that cannot be solved without a brute force search. Maybe that question is flawed. Maybe the question is whether or not a particular puzzle can be solved
with more or less computation time (computer computation, not human) than a brute force search.As puzzles get more complex, the number of tactics to apply and patterns to look for goes up and the number of times per puzzle any particular tactic or pattern comes up goes down. Taken to the ridiculous extreme would be to name each and every one of the 6 grazillion possible starting grids as a "pattern". All we have to do is identify the pattern and the solution is known -- but searching through this enormous database of "patterns" will take zillions of times longer than a simple brute force search of the puzzle itself. The location of this line, the line between puzzles that can be solved more quickly by looking through a limited database of patterns and applying them and the puzzles that will always be faster to solve by brute force may be a more interesting discussion.
In human terms, it gets more muddled, as we all have solved puzzles by protracted but logical means when we knew that at some point a brute force search would have been quicker.
There are puzzles that I have been unable to solve for years that I eventualy figured out (or tossed). For the longest time, I though solitaire-peg puzzle ...
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Remove one peg to start. Then jump pegs orthoganally, removing the peg jumped over. Goal: Given a specific peg to remove to start and specific hole, end with exactly one peg -- in the chosen hole.
... was beyond the ability of humans, that the only way to find any specific solution was trial and error. Turns out I was wrong. The Conway and Guy book "Winning Ways for your Mathematical Plays" shows in just a few pages a way to break down any peg jumping puzzle and make the impossible only mildly difficult.
I don't think there is a qualitative difference between working on a puzzle that cannot be solved without BFS and one which is merely beyond the solvers ability to solve without BFS. I think the question "Can this puzzle be solved without guessing?" has no meaning. Asking "Is there a puzzle that cannot be solved without guessing" is like asking "Is there a weight that cannot be lifted?" By who and what means? Which puzzles require guessing depends on which person is solving and what tactics are at her disposal -- again, the extreme case being that she has all 6 grazillion starting patterns memorized.