The New Sudoku Players' Forum

[Wecoc]

I made this sudoku variant I'd like to share

The puzzle has the basic sudoku rules but it would have more than one solution without an extra information.
The new 'layer' of givens are colors based on some of the basic sudoku tips (naked pairs, X-wing...) but with a twist: it tells you where you will "forcely" find some of them.

PUZZLE 1



Code:
....531.2.......57...9......9......4.17..5.3..2......8...7..6..9...4...17.3......

PUZZLE 2



Code:
3.2.9....4.....8.57...1...9.....63.7...74....5.......8..7.....3956......8....5.2.

[creint]

Can you post a step by step process on how to use those extra tips?
Have tried many things but still can't figure it out.

[Wecoc]

There are different ways to approach this, what works best for me is trying to solve the sudoku as always (ignoring the extra tips) until you get stuck, then write all candidates on the marked squares to find for something useful to use. In terms of algorithm-like systems, getting all candidates in each cell and using the extra tips as soon as possible to remove some candidates is also valid, but those tips aren't very useful in the very start of the solving. These puzzles are hard, using entire notation may be quite helpful.

I'm afraid this ruins a bit the challenge but I'll explain how to approach some of the cases.

Explanation: Show

Naked pairs: If there are candidates only available in one of the squares, we can automatically remove them because they must share all the candidates.
Once we reduced the candidates to only 2 we can make use of the naked pair to remove those candidates in other squares on that row, column or region. In special cases we can use this to deduce none of the marked squares will be a naked pair with another square of the row/column/region other than the other marked one.

Hidden subset: You may think same logic can be applied, but since it's a subset of unknown number we can't use it to remove non-shared candidates directly (for example if only 2 squares are marked, a third one could be part of the subset) so other steps are required to make use of it. It's complicated and very case-specific so I won't say anything else in this one, my recommendation is don't use it till the end

X-Wing: Without removing anything, we search for the candidates of the X-wing shared in the 4 squares. If there's one single candidate with that condition, we can use the X-wing to remove that candidate on other squares of the 2 rows and 2 columns outside the wing. If there's more than one possible candidate, we can use trial and error to deduce which one is the correct one.

Single's Chain: If there's a candidate in one square not included in the adjacent squares of the chain, we can remove it directly. To make use of the chain first we should know where it starts/ends. We know the first 2 squares of the start of the chain must have exactly the same 2 candidates. Using that we should be able to deduce the start, and remove other candidates. Once there are only a few candidates in the first squares we can go through the chain using its logic to remove some of the others. Trial and error may be required in the last steps.

[SpAce]

Can you post a step by step process on how to use those extra tips?
Have tried many things but still can't figure it out.

I agree. Kind of a neat idea, but neither example seems to work in any way that would make sense to me. The first example is close, as the X-Wing(1) and the Naked Pair (28) tips work, but the "hidden subset" part seems broken. If used the way that seems the most logical (12)r23c1, it leads to no solution. I can find one solution by adding r3c2 to the subset (and using digits 258), which gives a naked single (2)r2c1, but the presented puzzle doesn't really hint that. The second example makes even less sense, so I haven't put much effort into it. What is that "single's chain" anyway?? If it means X-Chain, it wouldn't eliminate anything, so it's not a real (or at least useful) pattern.

Such pattern-based tips are ambiguous anyway if it's not clearly stated whether degenerate patterns are acceptable or not. If all available (even very advanced) moves are exhausted before turning to the tips, it's quite possible that those patterns become degenerate. On the other hand, if degenerate patterns are accepted, then most of those tips become useless as clues.

[Wecoc]

I can find one solution by adding r3c2 to the subset (and using digits 258), which gives a naked single (2)r2c1, but the presented puzzle doesn't really hint that.

That's the correct solution indeed. Maybe I should have made it a bit more obvious adding the hint on r3c2, but because it was the only way to crack the puzzle anyway I decided to make it a bit more difficult.
Sorry for that, I still consider you successfully cracked it

The second one may seem broken at first glance but it also has one solution, and as an extra clue I'll say in that case the subset is a hidden pair.

What is that "single's chain" anyway?

Usually I call it 'logic chain' but here is defined as Single's Chain so I used that terminology.

Such pattern-based tips are ambiguous anyway if it's not clearly stated whether degenerate patterns are acceptable or not. If all available (even very advanced) moves are exhausted before turning to the tips, it's quite possible that those patterns become degenerate.

That's a very good point. I tried making those hints "unskippable" so they can't become degenerate, and as far as I tried I always have to use them in one step or another to get the unique solution, but I can't guarantee this will always be the case. The chain is easily the most delicate since it involves more squares.

Thank you for trying it

[SpAce]

Wecoc, thanks for replying! That explains a lot. I see now that you'd already replied to creint's original question when I wrote mine, but it wasn't visible then (probably because new users' posts require approval). Just wanted to mention that in case someone wonders why I questioned things you'd already explained (now that your reply appears before mine).

I can find one solution by adding r3c2 to the subset (and using digits 258), which gives a naked single (2)r2c1, but the presented puzzle doesn't really hint that.

That's the correct solution indeed. Maybe I should have made it a bit more obvious adding the hint on r3c2, but because it was the only way to crack the puzzle anyway I decided to make it a bit more difficult. Sorry for that, I still consider you successfully cracked it

So I accidentally stumbled on the correct solution, and even the correct way to find it? Good to know. I had no way of knowing until now that I saw this:

Hidden subset: You may think same logic can be applied, but since it's a subset of unknown number we can't use it to remove non-shared candidates directly (for example if only 2 squares are marked, a third one could be part of the subset) so other steps are required to make use of it. It's complicated and very case-specific so I won't say anything else in this one, my recommendation is don't use it till the end

That does explain what you were after, and why my solution worked, but the naming is very very confusing. You should definitely call it something other than "Hidden subset" because the latter has a specific and well-known sudoku meaning and it's very different from yours. If you color two cells and call it "Hidden subset" the only logical interpretation is a "Hidden Pair" and not a "subset of unknown number" where extra cells may need to be added. I kind of suspected you meant something like that, though, because you didn't call it "Hidden Pair" but it seemed too weird to be true. If you want to use that kind of a hint, I'd suggest something like "Unknown subset" instead, accompanied with a clear explanation that the colored cells could be just part of it. Edit: that's a pretty poor name too as it's not a sudoku pattern and doesn't really hint that more cells need to be considered. How about A*LS (=(Almost...) Locked Set)? That would be a valid pattern name for a cell group with (possibly) more digits than cells.

Btw, I really don't think you need or should make these puzzles harder by purposely adding ambiguity. A bit related to that:

If there's more than one possible candidate, we can use trial and error to deduce which one is the correct one.
...
Trial and error may be required in the last steps.

Stating it like that will put off many manual solvers, and even more if it's actually true. If a good player needs actual T&E, the puzzle is probably too difficult for normal consumption. I'd suggest you kept your puzzles a bit easier at least until any glitches in your rules are worked out. Testing them is pretty tedious otherwise.

The second one may seem broken at first glance but it also has one solution, and as an extra clue I'll say in that case the subset is a hidden pair.

I might take a closer look at that puzzle later. However, I'm still confused by this:

What is that "single's chain" anyway?

Usually I call it 'logic chain' but here is defined as Single's Chain so I used that terminology.

I think the more common and easily understood name would be "Simple Coloring", if that's indeed what you mean ("logic chain" would be really bad because it could mean anything). Still, your colored cells don't represent a normal Simple Coloring pattern, at least not one used by your SudokuWiki reference. Some definitions do use grouped nodes, though, and I guess that would sort of work here, but either way it wouldn't be a chain capable of eliminating anything -- which is why I can't really see it as a pattern. At most it's a chain fragment.

Such pattern-based tips are ambiguous anyway if it's not clearly stated whether degenerate patterns are acceptable or not. If all available (even very advanced) moves are exhausted before turning to the tips, it's quite possible that those patterns become degenerate.

That's a very good point. I tried making those hints "unskippable" so they can't become degenerate, and as far as I tried I always have to use them in one step or another to get the unique solution, but I can't guarantee this will always be the case. The chain is easily the most delicate since it involves more squares.

I think you should guarantee that or it might get very frustrating to try to rely on those hints. Here's the second puzzle after using all available SudokuWiki moves and the "Hidden subset" hint (you said it was a pair so I assumed 14 because they were the only digits with both left in those cells):

Code: Select all

+---------------+-----------------+-----------------+
| 3   168 2     | 5    9    478   |  1467 1467 14   |
| 4   169 19    | 236  236  237   |  8    1367 5    |
| 7   68  5     | 3468 1    348   |  2    346  9    |
+---------------+-----------------+-----------------+
| 12  24  1489  | 89   5    6     |  3    149  7    |
| 6   19  38    | 7    4    38    |  15   159  2    |
| 5   7   349   | 1239 23  *1239  | *46   469  8    |
+---------------+-----------------+-----------------+
| 12  24  7     | 1469 68  *19    | *145  1458 3    |
| 9   5   6     | 1234 238  1234  |  47   1478 14   |
| 8   3   14    | 14   7    5     |  9    2    6    |
+---------------+-----------------+-----------------+

I don't see anything resembling an X-Wing in the indicated cells (at most a Sashimi one with 1s, but which way would it be oriented anyway?). Has it degenerated, or has something else been messed up? Also, I still don't see much of a "Single's chain" (as an elimination capable pattern), though the 8s seem to be the most likely candidate for what you could possibly mean (if so, does it mean we could eliminate (8)r3c6 or what?).

Thank you for trying it

No problem. Like I said, I kind of like the idea, but I think you should work on the implementation details to avoid unnecessary confusion and resulting frustration. The "Naked Pair" and "X-Wing" hints were pretty unambiguous and worked as expected (for the first puzzle), but the "Hidden subset" and "Single's Chain" need new names (at least the first one) and better definitions.

Also, like I said, I think you should use easier puzzles to hone the details so there's no more ambiguity left. I don't think anyone is willing to solve puzzles as difficult as these manually unless the rules are crystal clear and can be trusted to work as expected. Otherwise there's too much risk of wasting time by making mistakes and never knowing if it was the player's or the puzzle designer's fault. If that happens, it's unlikely that the player will bother to try again.

[Wecoc]

That does explain what you were after, and why my solution worked, but the naming is very very confusing.

Yes, I see the problem now, probably the best option is using 'Naked pair' and 'Naked triple' and mark all cells of the subset to avoid that ambiguity.

Trial and error may be required in the last steps.

It may be needed but I'm talking about a few steps/candidates to check when it's almost finished, not something over complicated for someone who is solving by hand or in a spritesheet.

Single's Chain = Simple Colouring, it's exactly the same

Anyway, here's the solution of the examples.
Extreme strategies aren't required, but may be used as shortcuts.

Code: Select all

PUZZLE 1
Default solving until you get stuck

*--------------------------------------------------------*
| 468    7    4689   | 468   5      3      | 1   4689 2  |
| 12468  3    124689 | 12468 1268   12468  | 489 5    7  |
| 124568 4568 124568 | 9     12678  124678 | 48  468  3  |
|--------------------------------------------------------|
| 3568   9    568    | 12368 123678 12678  | 57  12   4  |
| 48     1    7      | 248   289    5      | 29  3    6  |
| 3456   2    456    | 1346  13679  14679  | 57  19   8  |
|--------------------------------------------------------|
| 12458  458  12458  | 7     3      289    | 6   248  59 |
| 9      568  2568   | 2568  4      268    | 3   7    1  |
| 7      4568 3      | 12568 125689 12689  | 248 248  59 |
*--------------------------------------------------------*

Naked pair (HINT) must be 28 (repeated candidates)
X-Wing (HINT): Remove 1 (repeated) from r2c4, r9c4, r4c5, r4c6, r6c5, r6c6
Default solving until you get stuck again

*-------------------------------------------*
| 468  7  4689 | 468  5     3    | 1   89 2 |
| 126  3  1469 | 2468 268   1246 | 489 5  7 |
| 258  58 125  | 9    278   1247 | 48  6  3 |
|-------------------------------------------|
| 3568 9  568  | 13   23678 678  | 57  12 4 |
| 48   1  7    | 248  289   5    | 29  3  6 |
| 3456 2  456  | 13   69    4679 | 57  19 8 |
|-------------------------------------------|
| 12   4  128  | 7    3     9    | 6   28 5 |
| 9    58 258  | 268  4     268  | 3   7  1 |
| 7    6  3    | 5    1     28   | 28  4  9 |
*-------------------------------------------*

Naked triple (HINT) between r2c1/r3c1/r7c1 removes 6 from r2c1 & 8 from r3c1
Default solving again

*-----------------------*
| 6 7 4 | 8 5 3 | 1 9 2 |
| 1 3 9 | 2 6 4 | 8 5 7 |
| 5 8 2 | 9 7 1 | 4 6 3 |
|-----------------------|
| 3 9 5 | 1 8 6 | 7 2 4 |
| 8 1 7 | 4 2 5 | 9 3 6 |
| 4 2 6 | 3 9 7 | 5 1 8 |
|-----------------------|
| 2 4 1 | 7 3 9 | 6 8 5 |
| 9 5 8 | 6 4 2 | 3 7 1 |
| 7 6 3 | 5 1 8 | 2 4 9 |
*-----------------------*

And here the puzzle 2. You can use the chain or the naked pair first. It doesn't matter which one you choose, the result is the same. In my example I start with the chain (simple coloring). Avoiding to use the resolved X-wing hint for too long can lead to a dead end.

Code: Select all

PUZZLE 2
Default solving until you get stuck

*------------------------------------------------*
| 3  168 2    | 5    9   478  | 1467  1467   146 |
| 4  169 19   | 236  236 237  | 8     1367   5   |
| 7  68  5    | 3468 1   348  | 2     346    9   |
|------------------------------------------------|
| 12 249 1489 | 189  5   6    | 3     149    7   |
| 6  139 1389 | 7    4   1389 | 159   159    2   |
| 5  7   1349 | 1239 23  1239 | 1469  1469   8   |
|------------------------------------------------|
| 12 24  7    | 1469 68  149  | 14569 145689 3   |
| 9  5   6    | 1234 238 1234 | 47    1478   14  |
| 8  134 134  | 1469 7   5    | 1469  2      146 |
*------------------------------------------------*

Simple Coloring (HINT) we know it can't start with 19-19 on the bottom so it can only start on the top
Simple Coloring (HINT) removes 1 from r1c2 (starts with 68-68)
Naked pair (HINT) sets candidates 14 on r9c3, r9c4 and therefore r9c7 must be 9
X-Wing (HINT) the 1/9 disambiguation has been solved because of r9c7, therefore it must be 1
Default solving again

*---------------------------------*
| 3 68 2  | 5   9  478 | 46 67  1 |
| 4 1  9  | 23  6  237 | 8  37  5 |
| 7 68 5  | 38  1  48  | 2  346 9 |
|---------------------------------|
| 1 2  48 | 89  5  6   | 3  49  7 |
| 6 9  38 | 7   4  38  | 5  1   2 |
| 5 7  34 | 239 23 1   | 46 469 8 |
|---------------------------------|
| 2 4  7  | 6   8  9   | 1  5   3 |
| 9 5  6  | 1   23 23  | 7  8   4 |
| 8 3  1  | 4   7  5   | 9  2   6 |
*---------------------------------*

Simple Coloring (HINT) must be 6-8-3-8-9 or it would get stuck halfways

*-----------------------*
| 3 6 2 | 5 9 8 | 4 7 1 |
| 4 1 9 | 2 6 7 | 8 3 5 |
| 7 8 5 | 3 1 4 | 2 6 9 |
|-----------------------|
| 1 2 4 | 8 5 6 | 3 9 7 |
| 6 9 8 | 7 4 3 | 5 1 2 |
| 5 7 3 | 9 2 1 | 6 4 8 |
|-----------------------|
| 2 4 7 | 6 8 9 | 1 5 3 |
| 9 5 6 | 1 3 2 | 7 8 4 |
| 8 3 1 | 4 7 5 | 9 2 6 |
*-----------------------*

For other puzzles other strategies could be used as possible hints (Pointing Pairs, Y-Wing, X-Cycles...)
I think it's a nice challenge to think the logic required to solve that "inverse step" even if it would be hard to apply in a real puzzle in some cases

[SpAce]

Yes, I see the problem now, probably the best option is using 'Naked pair' and 'Naked triple' and mark all boxes cells of the subset to avoid that ambiguity.

That's a good idea. (Btw, box is a synonym for a 9x9 nonet/block. The common term for your meaning is cell.)

Single's Chain = Simple Colouring, it's exactly the same

Yes, though I still don't see what your indicated cells and explanations actually have to do with it (more about that below).

X-Wing (HINT): Remove 1 (repeated) from B4, J4, D5, D6, F5, F6

FYI: on this site it's preferable to use the rncn cell references.

Simple Coloring (HINT) we know it can't start with 19-19 on the bottom so it can only start on the top
Simple Coloring (HINT) removes 1 from A2 (starts with 68-68)
Simple Coloring (HINT) must be 6-8-3-8-9 or it would get stuck halfways

I have no idea how any of this is related to Simple Coloring. It's a single-digit technique, but you're listing multiple digits here. What exactly do you mean? Also, it doesn't really have any starting point -- it's a coloring pattern of conjugate pairs (i.e. bilocation candidates of a single digit) that can be "started" anywhere within the pattern. I'm still confused. (Yours looks more like a fragment of an XY-Chain if anything.)

For other puzzles other strategies could be used as possible hints (Pointing Pairs, Y-Wing, X-Cycles...)

Looks like you've taken your pattern names from SudokuWiki. It's actually not a great reference for that, or anything related to chaining really. Almost all other sources call it XY-Wing instead of Y-Wing. Also, these days almost no one else talks about X-Cycles (a single-digit Nice Loop) or any other kinds of Nice Loops because AICs are used instead. The AIC version of X-Cycle is X-Chain -- but they're not the same pattern; X-Chain is better because AICs are better than Nice Loops in general. Unfortunately SudokuWiki confuses Nice Loops and AICs with each other, which makes its chaining stuff a mess. I'd say Hodoku is your best source for accurate naming of patterns, or at least they're the ones used on this site.

[Wecoc]

Oops, I know... I meant to say cell, sorry.
I also edited the post to use the rncn format.

You are right, looks like I messed up badly with the chain, what a disaster.

Let's better forget about the chain, I'll try explain what the mistake was in all this, but what I'll do is make a third puzzle a bit easier and without chains, after all this is about the idea and not about those examples. I think using chains would still be possible but it's complicated to see how exactly.

Explanation: Show

Mistake: The chain in the example is meant to be an X-Chain of 8s, but the "last" cell (r6c4) doesn't have an 8 and it should. That's because the chain ends 1 step earlier: The last one is a 9 simply because that's the only cell a 9 can go after you solve the chain. Besides that the puzzle can be solved the same way, at least I think so. This is important anyway because the way it is now you can't see directly it's a chain of 8s.

The chain itself doesn't have a start nor end, but you always start solving it from one cell or another and going through the chain to solve each other cell, and that's what I was referring to by "start" and "end", basically. We know it's a X-Chain of 8s because it's the only shared candidate but we need to set a start where the first 2 cells share candidates and reduce them along the chain to 2 options (or colors) and finally by elimination pick the only option that goes through the entire chain. That was the process there and it's how I got the 6-8-3-8-9.

I'd say Hodoku is your best source for accurate naming of patterns, or at least they're the ones used on this site.

Thank you so much, I'll be using these from now.
And thanks again for trying to solve the second puzzle, you helped me a lot to make progress on this.

[SpAce]

Oops, I know... I meant to say cell, sorry.
I also edited the post to use the rncn format.

Good! It helps those of us who're much more used to reading that.

Let's better forget about the chain

That's a good idea for now. I think you should first get the basic patterns to work consistently and then add more complicated stuff. Now that I looked at your solutions more closely, there still seems to be problems with Puzzle #1 as well regarding the Naked Triple hint (more about that below).

I'd say Hodoku is your best source for accurate naming of patterns, or at least they're the ones used on this site.

Thank you so much, I'll be using these from now.

Good idea. For this kind of puzzle to work you need to have a consistent naming system, so there's no confusion about what the pattern names actually mean.

And thanks again for trying to solve the second puzzle, you helped me a lot to make progress on this.

No problem. This is kind of interesting. Seems that it's not an easy puzzle variant to get to work perfectly, though. That's why I think it's a good idea to start with (much) easier puzzles where the normal solving steps are obvious and lead to the same stuck states and intended hint patterns consistently. Once that works without any ambiguity, then it's possible to increase both the puzzle difficulty and the hint complexity.

Anyway, here's the solution of the examples.
Extreme strategies aren't required, but may be used as shortcuts.

What do you count as extreme strategies tactics? Looks like even your own solutions must have used forcing chains for the "default solving" steps, and though not extreme, I'd still count them as pretty difficult for most manual players. Anyway, here's my take on your solution:

Comments on Puzzle #1 solution: Show

Code: Select all

PUZZLE 1
Default solving until you get stuck

*--------------------------------------------------------*
| 468    7    4689   | 468   5      3      | 1   4689 2  |
| 12468  3    124689 | 12468 1268   12468  | 489 5    7  |
| 124568 4568 124568 | 9     12678  124678 | 48  468  3  |
|--------------------------------------------------------|
| 3568   9    568    | 12368 123678 12678  | 57  12   4  |
| 48     1    7      | 248   289    5      | 29  3    6  |
| 3456   2    456    | 1346  13679  14679  | 57  19   8  |
|--------------------------------------------------------|
| 12458  458  12458  | 7     3      289    | 6   248  59 |
| 9      568  2568   | 2568  4      268    | 3   7    1  |
| 7      4568 3      | 12568 125689 12689  | 248 248  59 |
*--------------------------------------------------------*

I agree so far (except that you've forgotten to remove two 3s in c5). Max techniques required: simple AICs.

Naked pair (HINT) must be 28 (repeated candidates)
X-Wing (HINT): Remove 1 (repeated) from r2c4, r9c4, r4c5, r4c6, r6c5, r6c6

I agree with this, too.

Code: Select all

Default solving until you get stuck again

*-------------------------------------------*
| 468  7  4689 | 468  5     3    | 1   89 2 |
| 126  3  1469 | 2468 268   1246 | 489 5  7 |
| 258  58 125  | 9    278   1247 | 48  6  3 |
|-------------------------------------------|
| 3568 9  568  | 13   23678 678  | 57  12 4 |
| 48   1  7    | 248  289   5    | 29  3  6 |
| 3456 2  456  | 13   69    4679 | 57  19 8 |
|-------------------------------------------|
| 12   4  128  | 7    3     9    | 6   28 5 |
| 9    58 258  | 268  4     268  | 3   7  1 |
| 7    6  3    | 5    1     28   | 28  4  9 |
*-------------------------------------------*

At this point I must ask what you count as "default solving". Some of your eliminations have obviously required more than normal AICs. Yet you haven't applied every possible technique, which means that there's variability in the puzzle state when the hint is applied. If the cut-off point is at AICs, I get here:

Code: Select all

.------------------.-------------------.------------.
| 468   7    4689  | 468   5     3     | 1    89  2 |
| 1268  3    1469  | 2468  268   12468 | 489  5   7 |
| 1258  58   12458 | 9     278   1247  | 48   6   3 |
:------------------+-------------------+------------:
| 3568  9    568   | 13    2678  678   | 57   12  4 |
| 48    1    7     | 248   289   5     | 29   3   6 |
| 3456  2    456   | 13    679   467   | 57   19  8 |
:------------------+-------------------+------------:
| 128   4    128   | 7     3     9     | 6    28  5 |
| 9     568  258   | 2568  4     268   | 3    7   1 |
| 7     56   3     | 56    1     28    | 28   4   9 |
'------------------'-------------------'------------'

Notice that there are still quite a few candidates in your hint cells (r237c1). At that point the most logical subset would seem to be (128) since those are the only digits present in all hint cells. Believing that would lead to a deadly pattern with two solutions, however:

Code: Select all

+--------------+--------------+--------------+
| 6   7   4    | 8   5   3    | 1   9   2    |
| 12  3   9    | 4   26  16   | 8   5   7    |
| 8   5   12   | 9   27  17   | 4   6   3    |
+--------------+--------------+--------------+
| 3   9   8    | 1   67  67   | 5   2   4    |
| 4   1   7    | 2   8   5    | 9   3   6    |
| 5   2   6    | 3   9   4    | 7   1   8    |
+--------------+--------------+--------------+
| 12  4   12   | 7   3   9    | 6   8   5    |
| 9   8   5    | 6   4   2    | 3   7   1    |
| 7   6   3    | 5   1   8    | 2   4   9    |
+--------------+--------------+--------------+

So, let's accept that the hint can't be applied without some further solving (but how should we know that?). That, however, requires forcing chains. If all of them are exhausted, we get here:

Code: Select all

.----------------.----------------.-----------.
| 46    7    469 | 68   5     3   | 1   89  2 |
| 126   3    19  | 24   268   146 | 89  5   7 |
| 258   58   125 | 9    27    17  | 4   6   3 |
:----------------+----------------+-----------:
| 35    9    568 | 13   2678  678 | 57  12  4 |
| 48    1    7   | 24   289   5   | 29  3   6 |
| 3456  2    456 | 13   69    467 | 57  19  8 |
:----------------+----------------+-----------:
| 12    4    128 | 7    3     9   | 6   28  5 |
| 9     568  258 | 568  4     26  | 3   7   1 |
| 7     56   3   | 56   1     28  | 28  4   9 |
'----------------'----------------'-----------'

I think that's as far as one can get without the hints. Notice that it now has fewer candidates than your puzzle state. The hint candidates are the same, though, so let's look at how you used the hint:

Naked triple (HINT) between r2c1/r3c1/r7c1 removes 6 from r2c1 & 8 from r3c1

In other words you're saying that the naked triple has the digits (125), right? Assuming they were correct (which I don't agree with), how exactly should we know that? The only digit present in all three cells is 2 so that seems pretty fixed, and 1 is there twice so perhaps we could keep that as well (already getting more ambiguous, though), but how should we choose between 5, 6, and 8? Is this the part where you suggest trial and error? Doesn't sound like a lot of fun to me. Furthermore, your own choice of removing 8 actually results in multiple solutions, so as far as I see, it can't even be correct. Or, can you explain how you got this as the only possible solution:

Code: Select all

Default solving again

*-----------------------*
| 6 7 4 | 8 5 3 | 1 9 2 |
| 1 3 9 | 2 6 4 | 8 5 7 |
| 5 8 2 | 9 7 1 | 4 6 3 |
|-----------------------|
| 3 9 5 | 1 8 6 | 7 2 4 |
| 8 1 7 | 4 2 5 | 9 3 6 |
| 4 2 6 | 3 9 7 | 5 1 8 |
|-----------------------|
| 2 4 1 | 7 3 9 | 6 8 5 |
| 9 5 8 | 6 4 2 | 3 7 1 |
| 7 6 3 | 5 1 8 | 2 4 9 |
*-----------------------*

?? Myself, I only get this far after removing 6 and 8 from the triple:

Code: Select all

+--------------+--------------+--------------+
| 46  7   46   | 8   5   3    | 1   9   2    |
| 1   3   9    | 2   6   4    | 8   5   7    |
| 5   8   2    | 9   7   1    | 4   6   3    |
+--------------+--------------+--------------+
| 3   9   56   | 1   8   67   | 57  2   4    |
| 8   1   7    | 4   2   5    | 9   3   6    |
| 46  2   456  | 3   9   67   | 57  1   8    |
+--------------+--------------+--------------+
| 2   4   1    | 7   3   9    | 6   8   5    |
| 9   56  8    | 56  4   2    | 3   7   1    |
| 7   56  3    | 56  1   8    | 2   4   9    |
+--------------+--------------+--------------+

Notice the obvious deadly rectangle in (56)r89c24, and the other deadly pattern (MUG) with the rest of the cells. Thus, there's no way to get a single solution if you choose (125) for the triple r237c1. In fact, I don't really think you can use those cells for the triple hint at all. Like I said in my first post, I got it to work by using the cells r23c1,r3c2 for the triple, which quite clearly hints at (258) so we can remove 1 and 6 from r2c1. That gives us a single solution:

Code: Select all

+--------------+--------------+--------------+
| 6   7   4    | 8   5   3    | 1   9   2    |
| 2   3   9    | 4   6   1    | 8   5   7    |
| 8   5   1    | 9   2   7    | 4   6   3    |
+--------------+--------------+--------------+
| 3   9   8    | 1   7   6    | 5   2   4    |
| 4   1   7    | 2   8   5    | 9   3   6    |
| 5   2   6    | 3   9   4    | 7   1   8    |
+--------------+--------------+--------------+
| 1   4   2    | 7   3   9    | 6   8   5    |
| 9   8   5    | 6   4   2    | 3   7   1    |
| 7   6   3    | 5   1   8    | 2   4   9    |
+--------------+--------------+--------------+

Do you agree about the problems I identified?