Yes, I may be a few decades late to this realisation.
I was doing a sudoku on a flight and got horribly stuck towards the end. All of the remaining cells only had one of two values that they could be, but there didn’t seem to be any logical basis to rule out either value for any cell.
After staring at it for about half an hour, I eventually realised that both sets of values made complete and correct sudoku grids.
I had always assumed sudokus were always solvable to one unique solution.
A Sudoku with more than one solution is considered “invalid” and/or “not well formed.”
But still possible, apparently.
Years ago I got massively stuck and handed the puzzle to somebody else in the house, who couldn’t exactly get unstuck but who decided it was a valid puzzle because they were able, by taking a guess, to come out with a successful answer. I’ve used that technique in desperation occasionally since and sometimes had it work; and have always assumed that what was going on was that I (and that one time the other person) had missed some bit of information. Maybe we hadn’t; maybe it was a two-solutions puzzle.
In the other direction I have a sudoku puzzle book I’ve been very slowly occasionally working through during slow spells at farmers’ market. A number of the puzzles I’ve gotten massively stuck on; I assumed that the problem was that at market I can’t fully concentrate, due to keeping an eye out for customers and repeatedly shifting gears to marketing. Then one day I was halfway through a puzzle when I suddenly realized it was totally impossible – because the pre-filled numbers included two of the same number on one line. The puzzlemakers had simply gotten it wrong, and included at least one puzzle that was un-doable from the beginning. I now wonder whether at least some of the ones I got stuck on were also set up wrong from the start, just in a less immediately obvious fashion.
I’ll throw a related question in here:
How is “taking a guess and seeing if it works out” different than the other types of logical solving steps? Isn’t it just more layers of deduction before you hit the contradiction (or don’t) to determine what the first number should be? Perhaps more layers than you can hold in your head, and thus it feels more like a guess than a logical solution. Is there some limit to the number of steps that “should” be needed before the contradiction/solution appears?
I understand that in a “well formed” puzzle one of the two possible answer should lead to a contradiction, and thus reveal the one true solution.
You should never have to guess. If you watch Cracking The Cryptic, they have all kinds of ways to eliminate possibilities that are not obvious – X-wings, Swordfish, other weirder stuff.
Of course a Sudoku grid might have multiple solutions – just put the number 5 in the middle – how many solutions would that have? However, any puzzle offered up as a solvable Sudoku should only have one unique solution. So, if you find one with two, then either the puzzle maker made a mistake or you have a mistake somewhere.
A well-formed sudoku has exactly one solution. Whether you can find it without guessing is a function of how hard it is, and your skills, but does not affect whether or not it’s a well-formed sudoku.
I have a friend who was the world sudoku champion for several years, and now helps run the championship competitions. He said that under time pressure, he often guessed, and thinks that’s a fine strategy for solving hard sudokus. Dunno if there’s some computationally harder things you can do without “guessing” that would always work, but i don’t think that’s relevant to the quality of the puzzle.
A few years back bought a book of sudokus that was reasonable. After I finished it I ordered it’s sequel.
Right away I realized that a good percentage of the puzzles were invalid due to having multiple solutions. I soon threw it out. Just not at all fun.
If you think such invalid puzzles are reasonable, consider the worst case scenario. A completely blank puzzle. (Much more flexible than a puzzle with a single “5”.) Is this going to be interesting to solve at all?
Guessing (also called bifurcation, because you meander down one path until something is wrong and then you come back and try the other one) is definitely useful when you’re under time pressure, but you shouldn’t have to do it for a good puzzle.
That would be one terrible book. It’s possible that you were missing some logical deductions that would have obviated the need for guessing.
The puzzle maker.
When I say it was on a flight, I mean it was on the airplane entertainment system, and they are notoriously basic. I played a chess game once where the CPU castled out of check.
Having said that I was reluctant to post up the image of the point I got stuck but screw it, let’s hear where I went wrong 
Yeah, that’s a crappy book. Each puzzle should have exactly one valid solution.
I’ve done Sudoku for years, in fact I do at least one puzzle a day (usually more).
I’m aware of those techniques but once I got into puzzles that were hard enough to require learning them I pumped the brakes. I don’t like puzzles that much.
I like Sudoku because I can solve one methodically without having to do too much calculation, like a game of Solitaire. Once it gets to be like solving a complicated physics problem, it stops being a leisure activity.
May I recommend the Android app, Sudoku, by Genina. There’s not much difficulty between their “easy”, “medium”, “hard”, and “extreme” puzzles, but I think you will find them to be well-formed puzzles that are at the level you enjoy.
I now use the one by Mobilityware (on my iPhone). It has “challenges” beyond simply finishing a puzzle; like solve all even numbers, place a 6 then a 1 consecutively 3 times, complete 2 rows without completing a column, etc. It’s really fun because it takes the game to a different level and has challenges you wouldn’t normally have in Sudoku.
I don’t do sudokus, but I do do Ken Kens and they are also supposed to have unique solutions and I often use that fact to help solve them. If a 4 and 5 go here and also there, they can be exchanged. No good. So at least one of them is not 4 5.
The convention that a Sudoku puzzle has only one solution is just that: a convention. One could create a similar puzzle (maybe called a Suduoku) that has exactly two solutions, and you must find both to solve it.
That doesn’t prove that it was valid. A valid sudoku has exactly one solution. If a puzzle has zero solutions, that’s obviously an invalid sudoku, and your housemate was able to prove that that wasn’t the case there. But finding one solution doesn’t prove there aren’t others, so it’s still possible that the puzzle you had was invalid in the other direction.
Proper solving technique, I think, shouldn’t rely upon the assumption that a puzzle is valid, because if someone does hand you an invalid puzzle, you want to be able to discover that fact. But even if you don’t rely upon that assumption, you can still use it. The key to the guess-and-check method (which is a perfectly valid method, and is in every sense a “logical” method) is that you make a guess which you think is wrong, so that, when you eventually prove it wrong, you can definitively show that that square wasn’t that value. And you can still use things like “this puzzle ought to only have one solution” to guide you in making wrong guesses.
I stumbled on a sudoku website a few years ago that would solve puzzles. It had an option to go step-by-step through the puzzle, explaining at each step which technique it was using to draw the next inferrence. Some of them were quite complicated, both to recognize, and to explain why they worked.
At some point it passes from “interesting way to kill a few minutes”, to “exercise in frustrating theoretical logic”. It’s interesting that such a simple type of puzzle gives rise to such intricate solving techniques, but it’s not something I need to bash my brain against.
You shouldn’t have to do it, but guessing and backtracking may be a quicker way to make progress than scanning the whole puzzle for some of the more esoteric solving steps.
There’s a difference between solving with pencil-and-paper and online. There’s a site I go to for logic puzzles, and all the different puzzle types have an undo button to back up. On paper (or sites without undo), I can never remember which steps to back up.
I’ve tried doing sudokus on Delta’s in-flight system. I get stuck on their difficult puzzles; they either have multiple solutions, or use solving techniques that I don’t recognize. Either way, feels like they’re more difficult that they ought to be.
As I said above in my post. We were not at the time considering the possibility of a puzzle being published that was invalid in having two solutions; but I was aware of that when I posted in this thread – which is why, in the post that you quoted, I said:
On the Cracking the Cryptic Youtube channel, they refuse to use uniqueness to help solve a puzzle.
For example, if four cells would work either way with one number but not another, they won’t eliminate the number that would guarantee uniqueness, even though they know that all the puzzles they solve have a unique solution.
Simon refuses to use uniqueness and he also refuses to guess. He said once that Mark had told him he won’t get anywhere in competitive Sudoku if he won’t use bifurcation. He then acknowledges that he indeed didn’t get anywhere in competitive Sudoku.