Almost-binary puzzle

Advanced methods and approaches for solving Sudoku puzzles

Almost-binary puzzle

Postby qiuyanzhe » Sat Dec 14, 2019 5:50 am

A puzzle rated SE10.2, by ssxsssxs.
Code: Select all
123..4.567......3.8......2....3.8675.........5..9........6...914915..3..3..2..5..

after basic techniques:
Code: Select all
+-------------------+-------------------+-------------------+
| 1     2     3     | 78    9     4     | 78    5     6     |
| 7     456   569   | 18    56    2     | 149   3     489   |
| 8     456   569   | 17    3     56    | 149   2     479   |
+-------------------+-------------------+-------------------+
| 9     1     4     | 3     2     8     | 6     7     5     |
| 6     378   278   | 4     57    15    | 29    18    389   |
| 5     378   278   | 9     67    16    | 248   148   348   |
+-------------------+-------------------+-------------------+
| 2     578   578   | 6     4     3     | 78    9     1     |
| 4     9     1     | 5     8     7     | 3     6     2     |
| 3     678   678   | 2     1     9     | 5     48    478   |
+-------------------+-------------------+-------------------+

It gives me a feeling that the rest of the grid is 'binary' except r9c8, so I just guessed r9c8=8, which solves the puzzle.
The pattern after placing r9c8=4 seems quite interesting, something similar to MUG or something else. Is there any way to explain it?
*I wonder if this has already been discussed before, if so please attach a link here, thx.
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Re: Almost-binary puzzle

Postby totuan » Sat Dec 14, 2019 5:09 pm

As reference.

Hi qiuyanzhe,
qiuyanzhe wrote:A puzzle rated SE10.2, by ssxsssxs.
Code: Select all
123..4.567......3.8......2....3.8675.........5..9........6...914915..3..3..2..5..

after basic techniques:
Code: Select all
+-------------------+-------------------+-------------------+
| 1     2     3     | 78    9     4     | 78    5     6     |
| 7     456   569   | 18    56    2     | 149   3     489   |
| 8     456   569   | 17    3     56    | 149   2     479   |
+-------------------+-------------------+-------------------+
| 9     1     4     | 3     2     8     | 6     7     5     |
| 6     378   278   | 4     57    15    | 29    18    389   |
| 5     378   278   | 9     67    16    | 248   148   348   |
+-------------------+-------------------+-------------------+
| 2     578   578   | 6     4     3     | 78    9     1     |
| 4     9     1     | 5     8     7     | 3     6     2     |
| 3     678   678   | 2     1     9     | 5     48    478   |
+-------------------+-------------------+-------------------+

Thanks for the puzzle - a weird puzzle, it is not too difficult to eliminate r7c2 <> 7 by a quite complex move then the puzzle is downgraded to ER9.0 and still hard to finish. It is fairly for the puzzle with ER10.2 but in this case - the puzzle has so many bivalue cell, T&E is the best way to solve this one :D.

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Re: Almost-binary puzzle

Postby eleven » Sat Dec 14, 2019 8:54 pm

qiuyanzhe wrote:The pattern after placing r9c8=4 seems quite interesting, something similar to MUG or something else. Is there any way to explain it?

Interesting puzzle.
A MUG has 2 or more solutions, so it seems to be just a deadly pattern (or meta BUG?).
You can arrive at a BUG e.g. by all combinations of 34r6c9 and 78r9c9 (and it seems all pairs of binary cells with 4 digits).

[Added:]On second glance i would call it an extended BUG. All digits are in patterns with 2n possible resolutions (2 for a normal BUG), and probably it can be proved similar, that it cannot have a single solution. But never heard, that someone had mentioned that.

If it's true, this puzzle is very easy to solve:
Code: Select all
+----------------+----------------+----------------+
| 1    2    3    | 78   9    4    | 78   5    6    |
| 7    456  569  | 18   56   2    | 149  3    489  |
| 8    456  569  | 17   3    56   | 149  2    479  |
+----------------+----------------+----------------+
| 9    1    4    | 3    2    8    | 6    7    5    |
| 6    378  278  | 4    57   15   | 29   18   39+8 |
| 5    378  278  | 9    67   16   | 24   18+4 34+8 |
+----------------+----------------+----------------+
| 2    578  578  | 6    4    3    | 78   9    1    |
| 4    9    1    | 5    8    7    | 3    6    2    |
| 3    678  678  | 2    1    9    | 5    48   478  |
+----------------+----------------+----------------+

4r6c8 == 8r56c9 - (8=14)r56c8 => 4r6c8
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Re: Almost-binary puzzle

Postby SpAce » Sun Dec 15, 2019 1:06 am

Hi eleven,

eleven wrote:A MUG has 2 or more solutions.

I'm not sure if that's entirely accurate. BUGs and BUG-Lites don't have that requirement in their definitions, and in fact many such patterns have zero solutions instead of two (can they even have more?). It should thus logically follow that MUGs didn't have a multi-solution requirement either. I recently had an interesting discussion with blue about that, and he agreed.

There's also more than one way a pattern can be a DP without having multiple solutions. The one we normally call a "no-solution" case is when the isolated pattern itself has no solutions, but it's also possible that a pattern with one or more isolated solutions can't exist in a valid grid (i.e. it would cause an external contradiction). The latter case might be more common with MUGs. A long time ago Red Ed suggested calling those types of DPs Unreal or Totally Unreal, which I think is a good idea to distinguish them from the internally 0-solution patterns.

That said, I'm terrible at spotting or even verifying MUGs in practice, so this is just theoretical on my part. I just think that all uniqueness DPs should be governed by the same rules, i.e. multi-solutions, zero-solutions, and unreal patterns are counted as long as the pattern definition is otherwise fulfilled. BUGs and BUG-Lites are easy because they have clear definitions without the multi-solution requirement, but MUGs are obviously more complicated to recognize as such (requiring footprint analysis).

A practical benefit of distinguishing multi-solution and no-solution/unreal DPs is of course that the latter can be used without assuming uniqueness (even if it's a "uniqueness pattern"). Most players probably never check the actual solution counts for BUGs and BUG-Lites and just assume they have two, especially because sources like Hodoku falsely claim so. They might be surprised how often that's not true.
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Re: Almost-binary puzzle

Postby eleven » Sun Dec 15, 2019 1:50 am

I am not firm with the definitions, but what is an example of a 0 solution MUG, which is not a deadly pattern without needing uniqueness ?
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Re: Almost-binary puzzle

Postby totuan » Sun Dec 15, 2019 4:02 am

Hi eleven,
I'm vague about BUG :oops: , so can you explain more for me how (4)r6c8==(8)r56c9? Many thanks.

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Re: Almost-binary puzzle

Postby SpAce » Sun Dec 15, 2019 6:16 am

eleven wrote:I am not firm with the definitions

I'm not sure if anyone is regarding MUGs. In fact, I think Luke nailed it here:

Luke wrote:An actual MUG definition? How revolutionary :twisted:

That said, the definition you suggested there did include the "at least two internal solutions" requirement. What Myth Jellies had in mind might have been even more restricted, because apparently he didn't consider "unreal" patterns, such as this or this, MUGs even though they have multiple internal solutions (which just can't be extended to the full grid). Is it necessarily a useful restriction for practical purposes, though?

blue's footprint-based definition seems to accept even patterns without any internal solutions as MUGs, including the one in this puzzle. If 4r6c8 and 8r56c8 are removed (or any solution candidate, for that matter), his program states:

blue_mug_prog wrote:35 cells
0 solutions
0 footprints
valid MUG

Is that too inclusive? I don't know.

but what is an example of a 0 solution MUG, which is not a deadly pattern without needing uniqueness ?

I'm not sure if I interpret your question correctly. Anyway, I'm probably the wrong person to even try to answer it. My understanding of MUGs is far from solid. I'm just wondering if its definition should or should not include the multi-solution requirement, and if so, should it be restricted to the internal solutions only (accepting "unreal" patterns as MUGs).

If the (internal) multi-solution requirement is kept then we should accept that they're fundamentally different from BUGs and BUG-Lites. I guess there might be an argument for that because otherwise every no-solution-DP (with more than two candidates in at least one cell) could be called MUG, as far as I understand. That wouldn't be extremely useful either.
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Re: Almost-binary puzzle

Postby qiuyanzhe » Sun Dec 15, 2019 8:33 am

After placing r9c8=4:
Code: Select all
+-------------------+-------------------+-------------------+
| 1     2     3     | 78    9     4     | 78    5     6     |
| 7     456   569   | 18    56    2     | 149   3     489   |
| 8     456   569   | 17    3     56    | 149   2     479   |
+-------------------+-------------------+-------------------+
| 9     1     4     | 3     2     8     | 6     7     5     |
| 6     378   278   | 4     57    15    | 29    18    39    |
| 5     378   278   | 9     67    16    | 24    18    34    |
+-------------------+-------------------+-------------------+
| 2     578   578   | 6     4     3     | 78    9     1     |
| 4     9     1     | 5     8     7     | 3     6     2     |
| 3     678   678   | 2     1     9     | 5     4     78    |
+-------------------+-------------------+-------------------+

I have been thinking of this pattern in such a way:
From the two 56s in Box2 and 49 in Columns23, we know in Box 1, 56 are in different rows, 49 are in different columns.
So each pair is placed diagonally in Box 1.
Similar for 23-78 in Box 4, 56-78 in Box7, etc.
We can declare many variables, 1 for true and 0 for false, with xor addition:
A-4 in box 1 is in row 2
B-5 in box 1 is in row 2
C-4 in box 1 is in column 2
D-5 in box 1 is in column 2
E-2 in box 4 is in row 5
F-7 in box 4 is in row 5
G-2 in box 4 is in column 2
H-7 in box 4 is in column 2
I-5 in box 7 is in row 7
J-7 in box 7 is in row 7
K-5 in box 7 is in column 2
L-7 in box 7 is in column 2
.....
Then we'd would get many variables and equations(like C=1, D+K=1, A+B+C+D=1..), when added together, there is a parity problem.
2(A+B+C+D+....)=1.
Similar parity problems also occur in some BUGs(especially ones involving remote pairs or regular X-Chains. Also the one hidden below). But there are 4 more variables than equations in this(because of boxes 1347 with 3 candidates in each cell)(may call it 'degree of freedom'?), which makes it even difficult to get contradiction by T&E..
Hidden Text: Show
(Found in an 'ancient' post. There's also a new topic about it)
Code: Select all
| ab cd .  | ac bd .  | .  .  .  |
| ab cd .  | .  .  .  | ad bc .  |
| .  .  .  | ac bd .  | ad bc .  |

This is why I said the grid seems binary and I wonder if and how can we make it more straightforward.
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Re: Almost-binary puzzle

Postby SpAce » Sun Dec 15, 2019 3:54 pm

qiuyanzhe wrote:A puzzle rated SE10.2, by ssxsssxs.
Code: Select all
123..4.567......3.8......2....3.8675.........5..9........6...914915..3..3..2..5..

I don't know about MUGs but here's a solution with BUG-Lites only:

Step 1: Show
Code: Select all
.---------------.--------------.------------------.
| 1  2     3    | 78  9    4   | 78    5     6    |
| 7  45+6  69+5 | 18  56+  2   | 49+1  3     489  |
| 8  45+6  69+5 | 17  3    56+ | 149   2     49+7 |
:---------------+--------------+------------------:
| 9  1     4    | 3   2    8   | 6     7     5    |
| 6  38+7  27+8 | 4   57+  15+ | 29+   18+   39+8 |
| 5  38+7  27+8 | 9   67+  16+ | 24+   18+4  34+8 |
:---------------+--------------+------------------:
| 2  58-7  578  | 6   4    3   | 78    9     1    |
| 4  9     1    | 5   8    7   | 3     6     2    |
| 3  678   678  | 2   1    9   | 5     48    478  |
'---------------'--------------'------------------'

BUG-Lite+13

-- pairwise mirrored in step 2:

    (6)r23c2 - (6=95)r23c3 - r7c3 = (5)r7c2
    (5)r23c3 - r7c3 = (5)r7c2
    (7)r56c2
    (8)r56c3 - (8=37)r56c2
-- common with step 2:

    (1)r2c7 - (1=8)r2c4 - r1c4 = r1c7 - (8=7)r7c7
    (4)r6c8 - (4=8)r9c8 - (8=7)r7c7
    (7)r3c9 - r9c9 = (7)r7c7
    (8)r56c9 - r56c8 = r9c8 - (8=7)r7c7
=> -7 r7c2

Step 2: Show
Code: Select all
.---------------.--------------.------------------.
| 1  2     3    | 78  9    4   | 78    5    6     |
| 7  46+5  59+6 | 18  56+  2   | 49+1  3    489   |
| 8  46+5  59+6 | 17  3    56+ | 149   2    49+7  |
:---------------+--------------+------------------:
| 9  1     4    | 3   2    8   | 6     7     5    |
| 6  37+8  28+7 | 4   57+  15+ | 29    18    39+8 |
| 5  37+8  28+7 | 9   67+  16+ | 24    18+4  34+8 |
:---------------+--------------+------------------:
| 2  58    58-7 | 6   4    3   | 78    9     1    |
| 4  9     1    | 5   8    7   | 3     6     2    |
| 3  678   678  | 2   1    9   | 5     48    478  |
'---------------'--------------'------------------'

BUG-Lite+13

-- pairwise mirrored in step 1:

    (5)r23c2 - r23c3 = (5)r7c3
    (6)r23c3 - (6=45)r23c2 - r7c2 = (5)r7c3
    (8)r56c2 - (8=27)r56c3
    (7)r56c3
-- common with step 1:

    (1)r2c7 - (1=8)r2c4 - r1c4 = r1c7 - (8=7)r7c7
    (4)r6c8 - (4=8)r9c8 - (8=7)r7c7
    (7)r3c9 - r9c9 = (7)r7c7
    (8)r56c9 - r56c8 = r9c8 - (8=7)r7c7
=> -7 r7c3 (7 placements)

Step 3: Show
Code: Select all
.---------------.-------------.----------------.
| 1  2     3    | 7  9    4   | 8   5     6    |
| 7  456   56+9 | 8  56+  2   | 1   3     49   |
| 8  56+4  569  | 1  3    56+ | 49  2     7    |
:---------------+-------------+----------------:
| 9  1     4    | 3  2    8   | 6   7     5    |
| 6  78+3  278  | 4  57+  15+ | 29  18+   389  |
| 5  378   78+2 | 9  67+  16+ | 24  18+4  38-4 |
:---------------+-------------+----------------:
| 2  58+   58+  | 6  4    3   | 7   9     1    |
| 4  9     1    | 5  8    7   | 3   6     2    |
| 3  67+   67+  | 2  1    9   | 5   48    48   |
'---------------'-------------'----------------'

BUG-Lite+5

-- diagonally mirrored in step 4:

    (9)r2c3 - (9=4)r2c9
    (4)r3c2 - r3c7 = (4)r2c9
    (3)r5c2 - r5c9 = (3)r6c9
    (2)r6c3 - (2=4)r6c7
-- common guardian in all:

    (4)r6c8
=> -4 r6c9

Step 4: Show
Code: Select all
.---------------.-------------.---------------.
| 1  2     3    | 7  9    4   | 8   5     6   |
| 7  56+4  569  | 8  56+  2   | 1   3     9-4 |
| 8  456   56+9 | 1  3    56+ | 49  2     7   |
:---------------+-------------+---------------:
| 9  1     4    | 3  2    8   | 6   7     5   |
| 6  378   78+2 | 4  57+  15+ | 29  18    389 |
| 5  78+3  278  | 9  67+  16+ | 24  18+4  38  |
:---------------+-------------+---------------:
| 2  58+   58+  | 6  4    3   | 7   9     1   |
| 4  9     1    | 5  8    7   | 3   6     2   |
| 3  67+   67+  | 2  1    9   | 5   48    48  |
'---------------'-------------'---------------'

BUG-Lite+5

-- diagonally mirrored in step 3:

    (4)r2c2
    (9)r3c3 - (9=4)r3c7
    (2)r5c3 - (2=94)r53c7
    (3)r6c2 - (3=84)r69c9
-- common guardian in all:

    (4)r6c8 - r9c8 = (4)r9c9
=> -4 r2c9; stte

So yeah, I would agree that it's a pretty BUGgy puzzle. (Please inform if you find unintended bugs in my solution.)

Note that steps 1 and 2 have four common branches, and the others are mirrored. Steps 3 and 4 are mirrored as well. The one common guardian in all is 4r6c8 (also a backdoor). What can we conclude from this? Can we simplify it more?

--
Edit. Simplified the solution (6 -> 4 steps).
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Re: Almost-binary puzzle

Postby eleven » Sun Dec 15, 2019 8:03 pm

SpAce wrote:blue's footprint-based definition seems to accept even patterns without any internal solutions as MUGs, including the one in this puzzle.

In practice it is not relevant, if a uniqueness pattern like a MUG or BUG lite has internal solutions or not. Purists would not look for them, or they would find an impossible pattern with other methods.
But i do not want any 0 solution patterns to be called a MUG, e.g. an oddagon is definitely no MUG for me. Maybe we should postulate the potential of multiple solutions for a MUG or BUG-Lite.
What we are looking for are either known patterns (all having multiple solutions) or candidate links, which could lead to multiple or zero solutions of non unique puzzles - and we don't need to find out, what it has in a special case.
For patterns, which could be MUG's, it is often hard to verify them, but since the common ones all have multiple solutions, i tended to only call new patterns a MUG, if they have multiple solutions too. But i am open for other definitions.

But back to the question, if the grid above is a uniqueness pattern and if so, how can we verify that, hopefully manually.
qiuyanzhe wrote:[Then we'd would get many variables and equations(like C=1, D+K=1, A+B+C+D=1..), when added together, there is a parity problem.
2(A+B+C+D+....)=1.
Similar parity problems also occur in some BUGs(especially ones involving remote pairs or regular X-Chains
....
This is why I said the grid seems binary and I wonder if and how can we make it more straightforward.

As i said, my assumption is, that it is sufficient that all (connected) one digit patterns have an even number of (internal) solutions.
[Edit:] As usual for my sudoku conjectures also this one is wrong.
qiuyanzhe's explanation seems to be a good approach, how it could be proved. But it is still too vague for me. It does not convince me, that if the pattern would have a solution, it must have at least a second one.
This topic has many pitfalls, so we have to be very careful.
Last edited by eleven on Mon Dec 23, 2019 11:24 pm, edited 1 time in total.
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Re: Almost-binary puzzle

Postby Mauriès Robert » Sun Dec 15, 2019 9:44 pm

Hi,
This puzzle is very special indeed, because no traditional method works if we want to avoid the use of deadly patterns.
In TDP I define the notion of resolution tree which always allows to build the solution of a puzzle, and thus determines its level of difficulty (see TDP part 5)
Here is the resolution tree of this puzzle that shows the uniqueness of the solution and sets the TDP level of difficulty at 7.
Robert

Code: Select all
C = contradiction, S = solution

                  2r6c3 - - > C   
                   |
           4r2c2 - -
            |      |
            |     2r5c3 - - > S
6r9c2 - - - -
            |     2r6c3 - - > C   
            |      |
           4r2c2 - -
                   |
                  2r5c3 - - > C
 
-----------------------------

                  2r6c3 - - > C   
                   |
           4r2c2 - -
            |      |
            |     2r5c3 - - > C
6r9c3 - - - -
            |     2r6c3 - - > C   
            |      |
           4r2c2 - -
                   |
                  2r5c3 - - > C
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Re: Almost-binary puzzle

Postby totuan » Mon Dec 16, 2019 3:10 am

Mauriès Robert wrote:This puzzle is very special indeed, because no traditional method works if we want to avoid the use of deadly patterns.

I don’t think so. In my opinion, uniqueness pattern & deadly patterns are different and for this puzzle as I said:
totuan wrote:it is not too difficult to eliminate r7c2 <> 7 by a quite complex move then the puzzle is downgraded to ER9.0 and still hard to finish. It is fairly for the puzzle with ER10.2 but in this case - the puzzle has so many bivalue cell, T&E is the best way to solve this one :D.

I have not used uniqueness pattern to eliminate r7c2<>7.

BTW, I don't think zero or no solution is included in MUGs. My favorites when solving puzzles is that I always try to find stronglinks based on avoiding empty cell (deadly patterns? See example here) - IMO, it is not related to uniqueness pattern.

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Re: Almost-binary puzzle

Postby SpAce » Mon Dec 16, 2019 4:25 am

Hi Robert,

Mauriès Robert wrote:This puzzle is very special indeed, because no traditional method works if we want to avoid the use of deadly patterns.

Why would we want to avoid deadly patterns in general? They're fun and effective. Many deadly patterns have nothing to do with uniqueness in the first place, and even "uniqueness patterns" can often be used without assuming uniqueness (requires extra checks, though). For example, in this case my resolution proved uniqueness just as well as yours even though I used BUG-Lites all the way. I just didn't originally check that it did because it's not generally relevant to me.

Here is the resolution tree of this puzzle that shows the uniqueness of the solution and sets the TDP level of difficulty at 7.

Here's my comparable resolution tree that also shows the uniqueness of the solution without requiring nested T&E:

Code: Select all
T&E (using basics up to Naked Pair), 35 unsolved cells to start with

7r7c2 -> basics  -> BUG (no solution) 22 cells => -7 r7c2
7r7c3 -> basics  -> BUG (no solution) 22 cells => -7 r7c3 (7 placements)
4r6c9 -> singles -> BUG (no solution) 16 cells => -4 r6c9
4r2c9 -> singles -> BUG (no solution) 16 cells => -4 r2c9; stte (unique solution)

Such relatively small BUGs are trivial to recognize and verify as such, even more so than regular contradictions. It's also trivial to check that they're no-solution variants if one doesn't want to assume uniqueness. Also, since there are only 35 or 28 unsolved cells and max 3 candidates per cell to start with, they're quickly found manually using coloring or otherwise. So, pretty easy.

Here's another similar path:

Code: Select all
T&E (using singles only)

6r3c2 -> singles -> BUG (no solution) 20 cells => -6 r3c2
6r3c3 -> singles -> BUG (no solution) 20 cells => -6 r3c3 (7 placements)
4r6c9 -> singles -> BUG (no solution) 18 cells => -4 r6c9
4r9c8 -> singles -> BUG (no solution) 18 cells => -4 r9c8; stte (unique solution)

as a verity solution (like the original): Show
Step 1.

Code: Select all
.---------------.------------.------------------.
| 1  2     3    | 78+  9   4  | 78+   5    6    |
| 7  456   569  | 18+  56  2  | 19+4  3    89+4 |
| 8  45-6  569  | 17+  3   56 | 14+9  2    47+9 |
:---------------+-------------+-----------------:
| 9  1     4    | 3    2   8  | 6     7    5    |
| 6  37+8  27+8 | 4    57  15 | 29+   18   39+8 |
| 5  38+7  28+7 | 9    67  16 | 24+   148  34+8 |
:---------------+-------------+-----------------:
| 2  578   78+5 | 6    4   3  | 78+   9    1    |
| 4  9     1    | 5    8   7  | 3     6    2    |
| 3  78+6  678  | 2    1   9  | 5     48   78+4 |
'---------------'-------------'-----------------'

    BUG-Lite+13

    (4)r2c79 - r3c79 = (4)r3c2
    (9)r3c79 - (9=56)r3c36
    (6)r9c2
    (5)r7c3 - (5=96)r23c3
    --
    (8)r5c23,r56c9 - (8=15)r5c86 - (5=6)r3c6
    (7)r6c23 - (7=6)r6c5 - r6c6 = (6)r3c6
    (4)r9c9 - r9c8 = (41)r65c8 - (1=56)r53c6

    => -6 r3c2
Step 2.

Code: Select all
.---------------.-------------.-----------------.
| 1  2     3    | 78+  9   4  | 78+   5    6    |
| 7  456   569  | 18+  56  2  | 14+9  3    48+9 |
| 8  45    59-6 | 17+  3   56 | 19+4  2    79+4 |
:---------------+-------------+-----------------:
| 9  1     4    | 3    2   8  | 6     7    5    |
| 6  37+8  27+8 | 4    57  15 | 29+   18   39+8 |
| 5  38+7  28+7 | 9    67  16 | 24+   148  34+8 |
:---------------+-------------+-----------------:
| 2  78+5  578  | 6    4   3  | 78+   9    1    |
| 4  9     1    | 5    8   7  | 3     6    2    |
| 3  678   78+6 | 2    1   9  | 5     48   78+4 |
'---------------'-------------'-----------------'

    BUG-Lite+13

    (9)r2c79 - r3c79 = (9)r3c3
    (4)r3c79 - (4=56)r3c26
    (5)r7c2 - (5=46)r32c2
    (6)r9c3
    --
    (8)r5c23,r56c9 - (8=15)r5c86 - (5=6)r3c6
    (7)r6c23 - (7=6)r6c5 - r6c6 = (6)r3c6
    (4)r9c9 - r9c8 = (41)r65c8 - (1=56)r53c6

    => -6 r3c3
Step 3.

Code: Select all
.---------------.-----------.----------------.
| 1  2     3    | 78+  9  4 | 78+   5   6    |
| 7  46+   69+  | 18+  5  2 | 14+9  3   89+4 |
| 8  45+   59+  | 17+  3  6 | 14+9  2   79+4 |
:---------------+-----------+----------------:
| 9  1     4    | 3    2  8 | 6     7   5    |
| 6  38    28   | 4    7  5 | 29    1   389  |
| 5  378   278  | 9    6  1 | 24    48  38-4 |
:---------------+-----------+----------------:
| 2  57+8  58+7 | 6    4  3 | 78+   9   1    |
| 4  9     1    | 5    8  7 | 3     6   2    |
| 3  67+8  68+7 | 2    1  9 | 5     48  78+4 |
'---------------'-----------'----------------'

    BUG-Lite+9

    (8)r79c2 - (8=3)r5c2 - r5c9 = (3)r6c9
    (7)r79c3 - r6c3 = (73)r6c29
    (9)r23c7 - (9=24)r56c7
    (4)r239c9

    => -4 r6c9
Step 4.

Code: Select all
.---------------.-----------.-----------------.
| 1  2     3    | 78+  9  4 | 78+   5    6    |
| 7  46+   69+  | 18+  5  2 | 19+4  3    48+9 |
| 8  45+   59+  | 17+  3  6 | 19+4  2    47+9 |
:---------------+-----------+-----------------:
| 9  1     4    | 3    2  8 | 6     7    5    |
| 6  38    28   | 4    7  5 | 29    1    389  |
| 5  378   278  | 9    6  1 | 24    48   38   |
:---------------+-----------+-----------------:
| 2  58+7  57+8 | 6    4  3 | 78+   9    1    |
| 4  9     1    | 5    8  7 | 3     6    2    |
| 3  68+7  67+8 | 2    1  9 | 5     8-4  78+4 |
'---------------'-----------'-----------------'

    BUG-Lite+9

    (7)r79c2 - r6c2 = (724)r6c378
    (8)r79c2 - (8=2)r5c3 - r5c7 = (24)r6c78
    (4)r23c7 - r6c7 = (4)r6c8
    (9)r23c9 - (9=384)b6p698
    (4)r9c9

    => -4 r9c8; stte
--
Seems that this variant might be slightly simpler than my original.

There are probably other such combos as well but these seemed like the most obvious ones. Thus, it doesn't even require a lot of luck or mundane testing to find the solution this way. Poorly chosen or unlucky trials might result in a bit longer path, like my original 6-step solution, but it's still easy. Therefore, I think the actual difficulty level of this puzzle is pretty low, if T&E and BUGs are accepted. Even the uniqueness of the solution got proved as a side effect, even though it wasn't a goal.

I'll be surprised if anyone finds an even simpler way to solve this, at least using proven methods that are easily understandable and manually applicable.

--
Edit 1. Added the verity solution for the second variant.
Edit 2. Fixed two typos.
Last edited by SpAce on Mon Dec 16, 2019 1:48 pm, edited 2 times in total.
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Re: Almost-binary puzzle

Postby SpAce » Mon Dec 16, 2019 8:27 am

Hi totuan,

totuan wrote:In my opinion, uniqueness pattern & deadly patterns are different

I'm sure you're not alone with that opinion. However, I personally think uniqueness patterns are a subtype of deadly patterns, not something totally separate. In other words, to me a deadly pattern is any pattern that can't exist in a puzzle according to the agreed rules of engagement. All of them depend on some kind of assumption. Uniqueness techniques depend on the assumption that the puzzle has at most (or exactly) one solution, while the others (and normal solving techniques) depend on the assumption that the puzzle has at least one solution. And, as I've now said many times, using uniqueness patterns doesn't necessarily require assuming uniqueness of the solution.

I have not used uniqueness pattern to eliminate r7c2<>7.

I'd be interested in seeing how you did it!

My favorites when solving puzzles is that I always try to find stronglinks based on avoiding empty cell (deadly patterns? See example here)

Thanks for that link! I've sometimes (rarely) used that technique myself, for example here. In fact, I almost thought that I invented it myself because I couldn't remember seeing anyone else use it :D Of course I knew that couldn't be true, and now you showed me where I must have seen it (not really sure if I internalized it at the time, though). Nevertheless, I'd forgotten all about it but started thinking about it after reading Allan Barker's dark logic stuff and having a discussion about headless fishes (which are DPs if all fins are removed).

- IMO, it is not related to uniqueness pattern.

Of course not. It's in the same DP category as Oddagons and Headless Fishes which have nothing to do with uniqueness. However, no-solution BUGs and BUG-Lites work exactly the same way even though they're uniqueness patterns! Even URs can be used without the assumption of uniqueness, though it must be proved externally (as the UR pattern is too simple to have intrinsic no-solution variants).
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Re: Almost-binary puzzle

Postby totuan » Mon Dec 16, 2019 12:51 pm

As reference.

Hi SpAce,
SpAce wrote:
I have not used uniqueness pattern to eliminate r7c2<>7.

I'd be interested in seeing how you did it!

Code: Select all
 *--------------------------------------------------*
 | 1    2    3    | 78   9    4    | 78   5    6    |
 | 7    456  569  | 18   56   2    | 149  3    489  |
 | 8    456  569  | 17   3    56   | 149  2    479  |
 |----------------+----------------+----------------|
 | 9    1    4    | 3    2    8    | 6    7    5    |
 | 6    378  278  | 4    57   15   | 29   18   389  |
 | 5    378  278  | 9    67   16   | 24   148  348  |
 |----------------+----------------+----------------|
 | 2    578  578  | 6    4    3    | 78   9    1    |
 | 4    9    1    | 5    8    7    | 3    6    2    |
 | 3    678  678  | 2    1    9    | 5    48   478  |
 *--------------------------------------------------*

It is unfair for this puzzle with many bivalue cell :D.
Present as diagram: => r7c2<>7
Code: Select all
(7)r3c9-r1c7=r7c7*
 ||
 ||      -(4=19)r23c7-(9=2)r5c7-r5c3=r6c3-------(7)r6c3
 ||     |                                        || 
(4)r3c9--r3c2=r2c2-(5)r2c2                      (7)r6c2* 
 ||                 ||                           ||
 ||                (5)r3c2-r3c6=r2c5-(5=7)r5c5--(7)r6c5
 ||                 ||
 ||                (5)r7c2*     
 ||
 ||                         (5)r2c2-r2c5=r5c5---(7)r5c5
 ||                          ||                  ||       
 ||       (4)r2c7-r2c2=r3c2—(5)r3c2             (7)r5c2*     
 ||        ||                ||                  ||
 ||     --(9)r2c7           (5)r7c2*            (7)r5c3
 ||    |   ||                                    |
 ||    |  (1)r2c7----------------------------    |
 ||    |                                     |   |
 ||    |                                     |   |
(9)r3c9--(7)r3c9=(7-8)r1c7=(8)r2c9-(8=1)r2c4-    |     
       |                                         |   
        -(9=14)r23c7-(4=2)r6c7-r6c3=r5c3---------


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