I found this puzzle interesting.

- Code: Select all
`. 8 . | 7 . 5 | . 4 .`

. 9 2 | 8 . 1 | 6 3 .

7 . . | . . . | . . 1

-------+-------+------

. . 6 | . 8 . | 2 . .

. . . | . . . | . . .

. . 9 | . 5 . | 4 . .

-------+-------+------

3 . . | . . . | . . 9

. 1 7 | 5 . 3 | 8 2 .

. 6 . | 4 . 9 | . 1 .

I got to the position below *very* quickly -- maybe 5 minutes -- without using any pencil marks:

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

5 9 2 | 8 4 1 | 6 3 7

7 . . | . . . | 5 8 1

-------+-------+------

. . 6 | . 8 . | 2 . 3

. . . | . . . | 1 . .

. . 9 | . 5 . | 4 7 .

-------+-------+------

3 . . | 2 1 8 | 7 6 9

9 1 7 | 5 6 3 | 8 2 4

2 6 8 | 4 7 9 | 3 1 5

It took a couple minutes longer to see that at this point I would have to enter some pencil marks. As is my custom, I filled in only those cells that had exactly 2 candidates, shown below. Only if I cannot solve a puzzle with 2 candidates per cell do I fill the cells that need 3, etc. It's too much work to fill them in and is often overkill. Plus it makes it harder for me to see the connections. Other cells are left blank here for clarity:

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`. . . | . . . | . . .`

. . . | . . . | . . .

. 34 34 |69 29 26 | . . .

----------+-----------+----------

14 57 . |19 . 47 | . 59 .

48 . 35 | . 29 47 | . 59 68

18 23 . | . . 26 | . . 68

----------+-----------+----------

. 45 45 | . . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

No other pencil marks or any other kind of notations were required to finish the puzzle.

At this point, I identified the groups of cells that were connected in such a way that they could only be in one of two states, for example, the eight cells r3c12468 + r4c168 + r5c1 could only be in one of these two states:

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`. . . | . . . | . . .`

. . . | . . . | . . .

. . . | . . . | . . .

-------+-------+------

1 7 . | 9 . 4 | . 5 .

4 . . | . . 7 | . 9 .

8 . . | . . . | . . .

-------+-------+------

. . . | . . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

or

. . . | . . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

-------+-------+------

4 5 . | 1 . 7 | . 9 .

8 . . | . . 4 | . 5 .

1 . . | . . . | . . .

-------+-------+------

. . . | . . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

I find four such groups plus one lone cell:

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`. . . | . . . | . . .`

. . . | . . . | . . .

. b b | c c c | . . .

-------+-------+------

a a . | a . a | . a .

a . b | . c a | . a d

a e . | . . c | . . d

-------+-------+------

. b b | . . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

A proof, either by contradiction or forcing chains, must form a loop connecting at least two groups. (I haven't proved this, but since I'm defining a group as having two possible internally consistent states, it must be so.)

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`. . . | . . . | . . .`

. . . | . . . | . . .

. 34 34 |69 29 26 | . . .

----------+-----------+----------

14 57 . |19 . 47 | . 59 .

48 . 35 | . 29 47 | . 59 68

18 23 . | . . 26 | . . 68

----------+-----------+----------

. 45 45 | . . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

Looking only for connections between groups, I find one using groups A, C and D:

1) r3c6=2 => r6c6=6 => r6c9=8 => r6c1=1

2) r3c6=6 => r3c4=9 => r4c4=1 => r4c1=4 => r6c1=1

3) Therefore, r6c1=1 and the rest solves quickly.

Just the cells involved in the solution with the rest removed for clarity:

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`. . . | . . . | . . .`

. . . | . . . | . . .

. . . |69 . (26)| . . .

----------+-----------+----------

14 . . |19 . . | . . .

. . . | . . . | . . .

(18) . . | . . 26 | . . 68

----------+-----------+----------

. . . | . . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

Ok, my point: Other than filling in the candidate pairs, my total solving time was *maybe* 10 minutes, yet this puzzle was rated TOUGH.

(The puzzle is from today's LA Times.)

Paul's Pages (http://www.paulspages.co.uk/sudoku/) says: "WARNING - This puzzle has an 'outlaw' rating. During verification, it was not solved using the five solving rules described in 'How to solve Sudoku'. It may be necessary to use guesswork to solve it. Try it if you're tough enough!"

Argusj's Simple Sudoku is unable to solve it.

The solver at http://act365.com/sudoku/ says it requires a "guess".

It seems like the consensus on ratings for puzzles that require forcing chains and similar tactics is out of whack. Some of the puzzles in the Times and the UK Telegraph have even simpler forcing chains at the end and still get these ratings. This one stands out for me because the first part of the puzzle was soooo easy and solved so quickly. Puzzles that require swordfish or multiple x-wings, hidden trips, etc. are subjectively *much* more difficult for me, by a level of magnitude. You have to decide what to look for, where to look, and even then, it can be tedious. Plus, it is often *objectively* more difficult because it *requires* that you fill in *all* the candidates, even in cells that might have 4 or more -- and do so in a way that you can read your own writing. Even if I KNOW there's a swordfish and I'm taking the easy way out by using Argusj's Simple Sudoku to filter for it -- it's still easy for me to overlook it.

I know that difficulty is partly subjective, but maybe puzzles that include this type of structure shouldn't automatically be considered especially difficult. This was really only medium difficulty -- and if rated exclusively by solving time, it was the very easy.

Also, I don't claim the way I solved this is the only way or even the best way, only *a* way. It's what is easiest for me given my personal preferences -- I prefer to use less pencil marks if possible and solve as much in my head. I've posted several solutions using forcing chains previously, but this is the first time that I've tried show that I actually have a method to find them rather than just blindly searching. (Explaining is much harder than the doing.)

Is the identifying of connected groups like this of any interest to anyone else?

What do you see are the pros and cons of using different tactics to complete this puzzle from the sticking point? Would you have used colors? Assuming you solved this on paper, would you use additional notations?

Does anyone see a way that this puzzle could have been finished without using *any* pencil marks or notations of any kind?