Is it possible to solve this sudoku puzzle without guesswork?

I’m sure many of the Teeming Millions are familiar with Sudoku. (Rules in the link)

The puzzle in the 11 January 2008 edition of the Los Angeles Times is classified as “Diabolical”. Here it is:

``````

_ _ _   _ _ _   _ _ _
_ 4 1   6 _ 7   _ 5 _
_ _ 9   2 _ 8   7 _ _

_ 8 3   _ 7 9   6 _ _
_ _ _   _ _ _   _ _ _
_ _ 2   5 4 _   1 7 _

_ _ 7   4 _ 2   8 _ _
_ 3 _   7 _ 5   2 6 _
_ _ _   _ _ _   _ _ _

``````

(The puzzle is credited to “The Mepham Group”)

To solve the “Diabolical” puzzles of the LA Times, I use my habitual methods and then at some point I resort to trial and error - make a guess in one cell, follow the guess through until I encounter a contradiction, and then backtrack and make another guess.

Someone told me that you should never have to guess and that it’s always possible to find a logical way to solve the puzzle without guessing, using advanced methods such as the ones you find here.

I am curious to know if anyone here could solve the puzzle in my post without doing the guessing and backtracking. Needless to say, I couldn’t. In fact, I must have chosen poorly the squares in which to guess, because I guessed in one square, followed the path from my guess, had to backtrack; and then had to guess again because I only got so far after my second guess; and even after the second guess and backtracking after finding a contradiction, I only got so far and had to make a third guess!

I am by no means a Sudoku fiend, but can usually solve the regular 9x9 ones. I never “guess”: if I can’t solve it by deduction, I’ll stop (and perhaps come back to it later, perhaps not). I usually only do Sudokus while riding the bus or train, and if it’s not finished by the time I reach my destination, I just stop there.

I’ve not done an online Sudoku before, but thought it might be fun to take up the challenge, Arnold, so loaded up the LA Times’ Jan 11 puzzle [available here ].

The puzzle starts with 28 of the 81 squares filled in. I managed to fill in a further 22 squares (by deduction using my undoubtedly-limited skill set, not “guessing”), ending up with:

2 7 _ _ _ 4 _ 8 _
8 4 1 6 _ 7 _ 5 2
3 _ 9 2 _ 8 7 4 _

_ 8 3 1 7 9 6 2 _
7 1 _ 8 2 6 _ _ _
_ _ 2 5 4 3 1 7 8

_ _ 7 4 _ 2 8 1 _
1 3 _ 7 _ 5 2 6 _
_ 2 _ _ _ 1 _ _ 7

[It doesn’t look as though it should be too hard from here: all of the unfilled squares in the first six rows are “paired up” (i.e. each square can only have one of two values, and has a counterpart within the same 3x3 square), but the bottom three rows are more complex.]

I have to admit that I, personally, can’t see how to go beyond this without guessing (and potentially backtracking) – or at least leaving it and coming back later. Does this help, Arnold? Is this about where you are? If needed, I can explain the various steps to get to this point. I’d also be interested in hearing how to go beyond this for anyone who has solved it.

Sorry, not a definitive GQ answer, but at least another (weak) vote for “unsolvable without guesswork”, and a free bump…

You’ve done the heavy lifting on this one Antonius. The answer is below, and that’s just looking at rows and columns from where you left off, no complicated methods. There was one occasion where a column had three vacancies, xy, xy and xyz. So the third vacancy had to be z, I’d imagine that this would be a basic technique for a diabolical sudoku. Is diabolical the hardest level in the LA Times? - The difficulty level varies with different newspapers - super fiendish in the Times is the most difficult over here. Never guess a Sudoku! There is always a logical solution; although it’s been pointed out on this board before that the difference between guessing, trialling out solutions and using more structured methods can be vague.

2 7 6 9 5 4 3 8 1
8 4 1 6 3 7 9 5 2
3 5 9 2 1 8 7 4 6

4 8 3 1 7 9 6 2 5
7 1 5 8 2 6 4 3 9
9 6 2 5 4 3 1 7 8

5 9 7 4 6 2 8 1 3
1 3 8 7 9 5 2 6 4
6 2 4 3 8 1 5 9 7

Antonius Block, I can’t remember now, but I thought I got about as far as you did without “guesswork”. I am going to have to print out your answers and then try again from scratch. Thanks guys!
Diabolical is the most difficult level in the L.A. Times. But the L.A. Times knows that it’s dealing with Southern Californians, and we are prized more for our good looks than our brains.

I tried it, and got as far as

``````

___ __4 _8_
841 6_7 _52
3_9 2_8 7__

_83 179 62_
___ 826 ___
__2 543 178

__7 4_2 8__
_3_ 7_5 26_
___ __1 __7

``````

I plugged it into Simple Sudoku, and got a couple of hints for eliminating candidate numbers from some of the squares, but then it got stuck too.

Antonius Block, how did you get from where I am to where you are?

I agree that a well-made sudoku puzzle should be solvable without guesswork, but guesswork is really no less logical than the normal tools for solving. Guesswork just comes in where the logic is too complicated to fit in your brain all at once, but fundamentally it’s the same as the normal rules.

[/silly nitpick]

<ignore>

I didn’t quite have to use guesswork, but I had to look for “chains” of two value cells (“if this is 1, that has to be 5, if that’s 5 then that has to be 4”…) until I found a contradiction. In other words, I did the trial and error in my head. It definitely wasn’t an " a-ha! There’s the clue" deal.