The New Sudoku Players' Forum

[SpAce]

Your mission (syctai) is to come up with a single step, preferably a valid AIC, that eliminates the target candidates.

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.---------------------.--------------------.----------------------.
| 1      678    567   | 489    2     789   | 568-9  48-9   3      |
| 789    378    4     | 5      37-9  6     | 2      189    19     |
| 569    2      356   | 13489  349   1389  | 5689   7      4569   |
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| 3      1467   12567 | 129    569   129   | 1679   1249   8      |
| 268    9      1268  | 7      36    4     | 136    5      126    |
| 24567  1467   12567 | 12389  3569  12389 | 13679  12349  124679 |
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| 267    167    9     | 236    8     2357  | 4      23     1257   |
| 2478   5      12378 | 2349   3479  2379  | 1789   6      1279   |
| 24678  34678  23678 | 23469  1     23579 | 35789  2389   2579   |
'---------------------'--------------------'----------------------'

? => -9 r1c78,r2c5

[SpAce]

24 hours and no answers? Too easy? Too hard? Too boring?

Well, here's a huge hint. This probably makes it too easy, but the puzzle is the first example here. The necessary logic is explained there, especially in David's post. However, David's solution uses two separate logic steps, and I'm looking for a single self-contained chain (without any weird "quantum" logic). Shouldn't be particularly hard once you know the necessary ingredients.

[SCLT]

24 hours and no answers? Too easy? Too hard? Too boring?

I can't speak for others, but from my point of view, this sort of challenge is a bit too rigid to be enjoyable for me - a little too close to "guess what I'm thinking".

[SCLT]

I also think that trying to express the logic here as a single chain is a bit pointless and confusing. Why not say something like:

r23456c5 is an ALS so must contain either 4 or 7 --> Therefore r1c46 cannot be 4 and 7 --> Thus either 8 or 9 must appear in r1c46 --> So David's chain is then allowed

Seems like that could be expressed as one (branched) AIC but it would be less clear.

Note that you can actually get even more eliminations than you suggest (using the (9=8)r1c46 inference) via the following chain: (9=8)r1c46 - r1c2 = r2c12 - (8=19)r2c89 - 9r1c78 = 9r1c46 => -9 r1c78, r2c5, r3c456

[SpAce]

Hi SCLT,

I can't speak for others, but from my point of view, this sort of challenge is a bit too rigid to be enjoyable for me - a little too close to "guess what I'm thinking".

Fair enough. Thank you for honest feedback. Personally I would probably enjoy these kinds of exercises, but I've never seen them, so I made one up. How is it different from, say, a chess problem with a specific scenario and goal? The artificial rigidity forces one to think in new ways. Our normal "requirement" of a single step solution is a similar, though possibly less rigid, constraint. If that were removed, there would be countless more ways to solve these puzzles and most of them would be trivial.

I also think that trying to express the logic here as a single chain is a bit pointless and confusing. Why not say something like:

r23456c5 is an ALS so must contain either 4 or 7 --> Therefore r1c46 cannot be 4 and 7 --> Thus either 8 or 9 must appear in r1c46 --> So David's chain is then allowed

You can certainly say something like that, and it's a perfectly understandable solution -- just like David's original. It's just not what I asked, nor what I would prefer. To me the exercise of expressing a piece of logic as a single AIC is not pointless and confusing either, but I don't blame anyone for disagreeing (Note: before I posted the hint, I would have accepted that. Now that the logic is out in the open, the only question remaining is expressing it as a self-contained AIC. That's not found in the original thread.)

Seems like that could be expressed as one (branched) AIC but it would be less clear.

A matter of taste. I happen to prefer self-contained chains almost always. They may not always be the clearest possible solutions, but writing them is fun (for me), especially when it's not quite obvious how to do it. That was the point of this exercise (especially post-hint). In this particular case I don't see any clarity problems either.

The bottom line: I still haven't seen what I asked. I have one solution in mind, of course, but I'd be interested in seeing if someone comes up with something better. Mine uses a nested chain, but otherwise it's very simple. You use those a lot so at least that version should be a piece of cake for you. If someone can do it without a nested chain, even better (though I'm pretty sure it wouldn't be clearer).

Note that you can actually get even more eliminations than you suggest (using the (9=8)r1c46 inference) via the following chain: (9=8)r1c46 - r1c2 = r2c12 - (8=19)r2c89 - 9r1c78 = 9r1c46 => -9 r1c78, r2c5, r3c456

Of course, but it just lengthens the chain without adding any real value when claiming takes care of the extras right after. I'd see the point if it were the difference between stte and btte. Otherwise not.

[SCLT]

How is it different from, say, a chess problem with a specific scenario and goal?

Well, I'm not the world's biggest fan of those either!

Our normal "requirement" of a single step solution is a similar, though possibly less rigid, constraint. If that were removed, there would be countless more ways to solve these puzzles and most of them would be trivial.

You're right of course, but within the requirement of "present your solution as a single step" there's still scope for enormous variety - as evidenced by the eclectic mix of solutions we see to the daily puzzles. And not all challenges presented like the one in this thread will be quite as constrained, but with these particular target eliminations, the endpoints of the chain are pretty much fixed!

To me the exercise of expressing a piece of logic as a single AIC is not pointless and confusing either, but I don't blame anyone for disagreeing

I happen to prefer self-contained chains almost always. They may not always be the clearest possible solutions, but writing them is fun (for me), especially when it's not quite obvious how to do it.

I think my criterion is that if expressing the logic as a single chain obscures something about the configuration, or artificially makes the step harder to follow, then I'd prefer not to bother. Although I can understand the attraction to some in being creative with chain presentations, it's not my cup of tea! In this case, since the conclusion "r1c46 cannot contain both 4 and 7" stands on its own without needing to fit into a chain, including it (or something like it) in the middle of a chain is (to me) obscuring something about the structure. And considering how dense the candidates are in this grid, I'll be very surprised if there's anything fundamentally different!

The bottom line: I still haven't seen what I asked (the "preferably" part).

I look forward to seeing what others come up with (or your intended solution). I just hope you aren't offended if I don't try it myself Although here's a tongue-in-cheek answer using the fact that (anything) = (contradiction) implies the truth of (anything):

[(9=8)r1c46 - r1c2 = r2c12 - (8=19)r2c89] = (4r1c4 & 7r1c6) - (4r3c5 | 7r2c5) = (4&7)r8c5

[blue]

Not that I like this sort of thing, but ...

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(9=18)r2c89 - 8r2c12 = 8r1c2 - 8r1c46
    = ((4r1c4 & 7r1c6) | 9r1c46)
    - ((4r3c5 | 7r2c5) & 9r1c78)
    = ((4r8c5 & 7r8c5) | 9r1c46)   [ stop here ? ... if not, ... ]
    - ((7r8c5 | 4r8c5) & 9r1c78)
    = ((7r2c4 & 4r3c5) | 9r1c46)
    - ((7r1c6 | 4r1c4) & 9r1c78)
    = (89r1c46 | 9r1c46)

[SpAce]

And not all challenges presented like the one in this thread will be quite as constrained, but with these particular target eliminations, the endpoints of the chain are pretty much fixed!

Pretty much, though the obvious choice can sometimes be a deception. In this case it's not. The fun part is that knowing the end points doesn't make it trivial to come up with what's between them.

I think my criterion is that if expressing the logic as a single chain obscures something about the configuration, or artificially makes the step harder to follow, then I'd prefer not to bother. Although I can understand the attraction to some in being creative with chain presentations, it's not my cup of tea!

I understand, and I'm glad if you understand the other perspective. My preferred idea of a step is an expression that could be plugged into a parser and validated without further explanations. In theory, I'd like a software solver that wouldn't just accept manual eliminations or placements but would require one to enter the logic used! That would be kind of hard with natural language. (Personally I prefer to read formal expressions as well. It's often harder for me to follow someone's logic if it's explained in natural language, even if the corresponding chain is quite complex.)

In this case, since the conclusion "r1c46 cannot contain both 4 and 7" stands on its own without needing to fit into a chain, including it (or something like it) in the middle of a chain is (to me) obscuring something about the structure.

That's one way to look at it. I see it as formalizing the logic. If anything, it seems to me that this exercise has demonstrated that even otherwise obvious things are not necessarily trivial to express formally.

And considering how dense the candidates are in this grid, I'll be very surprised if there's anything fundamentally different!

Me too! Yet there's usually more than one way to see and express even the same fundamentals.

I look forward to seeing what others come up with (or your intended solution). I just hope you aren't offended if I don't try it myself

Of course not This is supposed to be fun.

Although here's a tongue-in-cheek answer using the fact that (anything) = (contradiction) implies the truth of (anything):

[(9=8)r1c46 - r1c2 = r2c12 - (8=19)r2c89] = (4r1c4 & 7r1c6) - (4r3c5 | 7r2c5) = (4&7)r8c5

Well, that's a valid AIC so it qualifies Of course using a contradiction feels a bit like cheating, but it wasn't specifically forbidden. As far as I'm concerned, "false=something" is a valid strong link, even though only one side can be true (and is).

[SpAce]

Not that I like this sort of thing, but ...

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(9=18)r2c89 - 8r2c12 = 8r1c2 - 8r1c46
    = ((4r1c4 & 7r1c6) | 9r1c46)
    - ((4r3c5 | 7r2c5) & 9r1c78)
    = ((4r8c5 & 7r8c5) | 9r1c46)   [ stop here ? ... if not, ... ]
    - ((7r8c5 | 4r8c5) & 9r1c78)
    = ((7r2c4 & 4r3c5) | 9r1c46)
    - ((7r1c6 | 4r1c4) & 9r1c78)
    = (89r1c46 | 9r1c46)

That qualifies too Seems that you ran into the same problem as I when I tried to do it without nesting: an embedded contradiction. I'd probably do what you suggested and stop there (it still qualifies), though points for completing the chain!

Makes me wonder if there's really no way to write it so that it has neither nesting nor an obvious contradiction.

[eleven]

ok, i repeat myself ...
this is one of many samples (starting with remote pairs), where i find the obsession to express everything with an aic ridiculous.
an almost hidden pair gives a strong link, and thus the elimination, that's all.

[SpAce]

Seems like no more entries are coming, so here's what I had.

With nesting (my preference):

(91=8)r2c89 - r3c7 = r3c46 - r1c46 = [(9=4)r1c4 - r3c5 = (4-7)r8c5 = r2c5 - (7=9)r1c6] => -9 r1c78,r2c5

Without nesting (but including an internal contradiction in r8c5):

(98=4|7)r1c46 - (47)r23c5 = (4|7-47)r8c5 = (4|7)r23c5 - (47=8|9)r1c46 - (89)b2p5789 = (8,19)b3p756|(9)r1c46 => -9 r1c78,r2c5

Added. What bothers me with that, just like the corresponding part in blue's solution, is the weak link (4|7-47)r8c5 which seems unavoidable with the non-nested approach. It contradicts not only itself but what I recently said about deadly patterns being incapable of weak links, which still makes sense to me. (47)r8c5 is clearly an impossible pattern (thus false) so how could it have a meaningful weak link with anything? The hypothetical logic (to the left of the contradiction) still works, but it just seems weird and superfluous. It would seem simpler and more efficient to use the contradiction (or DP, with externals 4r3c5 and 7r2c5) directly:

(47)r8c5[!] = (4|7)r23c5 - (47=8|9)r1c46 - (89)b2p5789 = (8,19)b3p756|(9)r1c46 => -9 r1c78,r2c5

Now we know that all the left-hand sides of the strong links are false and the right-hand sides are true. Not that elegant, perhaps, but maybe it's pretty close to how most people would see the logic. Even closer with an embedded end point (@), (added:) but that's not an AIC so it wouldn't qualify:

(47)r8c5[!] = (4|7)r23c5 - (47=9@|8)r1c46 - (8)r3c46 = r3c7 - (8=19)r2c89 => -9 r1c78,r2c5

In other words, I can't find a way to write this so that I could avoid both nesting and a contradiction. Personally I think the nested version is the more elegant one, but the contradiction version could be simpler to understand.

Added. If clarity is the number one priority, I would probably prefer a matrix:

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 91r2c89 8r2c8
         8r3c7 8r3c46
 9r1c4         8r1c4  4r1c4
                      4r3c5 4r8c5
                            7r8c5 7r2c5
 9r1c6         8r1c6              7r1c6
---------------------------------------
-9r1c78,r2c5

Any other expression can be easily extracted from that. Including this spaceship:

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                           @(9)r1c4
                            ||
                            (4)r1c4 - r3c5 = (4)r8c5
                            ||                |
                            (8)r1c4           |
                           /                  |
(91=8)r2c89 - r3c7 = r3c46                    |
                           \                  |
                            (8)r1c6           |
                            ||                |
                            (7)r1c6 - r2c5 = (7)r8c5
                            ||
                           @(9)r1c6


Edits: 1,2,3 (additions), 4 (correction of a typo)

[SpAce]

[deleted]

[eleven]

i want to apologize for the word ridiculous.
but be aware, that with the corset of aic thinking Steve K. would not have made his great discoveries.

[blue]

Wishing I had remained silent ...

[SpAce]

FWIW, I think this would do the trick without obvious DP nodes (contradictions), embedded end points, or nesting:

(9=4|8)r1c4 - (48)r3c456 = (47)r82c5|(819)b3p756 - (7)r1c6&(9)r1c78 = (8|9)r1c46 - (89)b2p5789 = (8,19)b3p756|(9)r1c46 => -9 r1c78,r2c5

Or minus one direct elimination (but the same end result) without the comma:

... = (819)b3p756|(9)r1c46 => -9 r1c78

Obviously that horror story is a ridiculous way to see and to present the logic, but just as obviously that's not anywhere near the point.