The New Sudoku Players' Forum

[ArkieTech]

Code: Select all

 *-----------*
 |.1.|.8.|...|
 |2..|9..|3..|
 |...|..3|.7.|
 |---+---+---|
 |.2.|.9.|7..|
 |9..|2.7|..5|
 |..4|.5.|.3.|
 |---+---+---|
 |.6.|1..|...|
 |..7|..6|..1|
 |...|.4.|.8.|
 *-----------*

.1..8....2..9..3.......3.7..2..9.7..9..2.7..5..4.5..3..6.1.......7..6..1....4..8.


Play/Print this puzzle online

[SpAce]

Code: Select all

.-----------------------.-------------------------.-------------------.
| 34567    1      3569  |   4567   8         245  | 259   2569  2469  |
| 2        4578   568   |   9      167       145  | 3     156   468   |
| 4568     4589   5689  |   456    126       3    | 18    7     24689 |
:-----------------------+-------------------------+-------------------:
| 1358     2      1358  |   348    9         148  | 7     16    68    |
| 9      a[83]    1368  |   2    a[3](16)    7    | 18    4     5     |
| 167-8    7-8    4     | b(68)    5       b(18)  | 29    3     29    |
:-----------------------+-------------------------+-------------------:
| 3458     6      23589 |   1      237       2589 | 2459  259   37    |
| 3458     34589  7     |   358    23        6    | 2459  259   1     |
| 135      359    12359 |   357    4         259  | 6     8     37    |
'-----------------------'-------------------------'-------------------'

WXYZ-Wing

(83)r5c25 = (168)b5p579 => -8 r6c12; btte (one locked pair)

stte: Show

(723)r278 = (7,8)r26c2 - (8=163)b5p795 - (38,7)r562c2 = (723)r278c5 => +723r278c5; stte

[Cenoman]

Code: Select all

 +--------------------------+----------------------+------------------------+
 |  34567   1       3569    |  4567   8     245    |  259    2569   2469    |
 |  2      f4578    568     |  9     g67-1  145    |  3     c156    468     |
 |  4568    4589    5689    |  456   a16-2  3      | b18     7      24689   |
 +--------------------------+----------------------+------------------------+
 | e1358    2      e1358    |  348    9     148    |  7     d16     68      |
 |  9      e38      1368    |  2      136   7      |  18     4      5       |
 |  1678   e78      4       |  68     5     18     |  29     3      29      |
 +--------------------------+----------------------+------------------------+
 |  3458    6       23589   |  1     h237   2589   |  2459   259    37      |
 |  3458    34589   7       |  358   h23    6      |  2459   259    1       |
 |  135     359     12359   |  357    4     259    |  6      8      37      |
 +--------------------------+----------------------+------------------------+

(1)r3c5 = r3c7 - r2c8 = r4c8 - (1=3587)b4p1358 - r2c2 = (7)r2c5^ - (7=32)r78c5 => -1 r2c5^, -2 r3c5; ste

[Ajò Dimonios]

Hi everyone

Code: Select all

+-------------------+---------------+-----------------+
| 34567 1     3569  | 4567 8   245  | 259  2569 2469  |
| 2     4578  568   | 9    167 145  | 3    156  468   |
| 4568  4589  5689  | 456  126 3    | 18   7    24689 |
+-------------------+---------------+-----------------+
| 1358  2     1358  | 348  9   148  | 7    16   68    |
| 9     38    1368  | 2    136 7    | 18   4    5     |
| 1678  78    4     | 68   5   18   | 29   3    29    |
+-------------------+---------------+-----------------+
| 3458  6     23589 | 1    237 2589 | 2459 259  37    |
| 3458  34589 7     | 358  23  6    | 2459 259  1     |
| 135   359   12359 | 357  4   259  | 6    8    37    |
+-------------------+---------------+-----------------+


7R7C5-R2C5=R1C4-R1C1=R6C1-(7=8)R6C2-(8=3)R5C2;
7R7C5-R2C5=R1C4-R1C1=R6C1-(7=8)R6C2-8R6C46=16R6C46-16R5C5=3R5C5=>contradiction two 3s in R5=>-7R7C5=>Stte.

Ciao a Tutti
Paolo

[eleven]

Hi Paolo,

nice demonstration, how manual solvers find a solution. In most cases it is not the shortest.

Code: Select all

7R7C5-R2C5=R1C4-R1C1=R6C1-(7=8)R6C2-(8=3)R5C2;
7R7C5-R2C5=R1C4-R1C1=R6C1-(7=8)R6C2-8R6C46=16R6C46-16R5C5=3R5C5=>contradiction two 3s in R5=>-7R7C5=>Stte.


If you look at it, you can leave out, what is equal in the 2 lines up to 8R6C2, the last one. Then you get the shorter contradiction for 8r6c2:
8R6C2-(8=3)R5C2;
8R6C2-8R6C46=16R6C46-16R5C5=3R5C5=>contradiction two 3s in R5=>-8R6C2=>Stte.
If 8 cannot be in r6c2, then (reading back your double chain), 7 must be there, and in r1c1, r2c5, so it can't be in r7c5 too. You just can leave that out.

Personally i don't mind, but contradictions are not beloved here. No problem, everybody can reformulate it by putting together the 2 lines, e.g. take the second line leaving out the double term 8R6C2, and continue with the first line backwards:
8R6C46=16R6C46-16R5C5=3R5C5-(3=8)R5C2
So you have either 8 in r6c46 or in r5c2, i.e. no 8 in r6c2 (and in r6c1)
You can also write the same this way:
(8=3)r5c2 - (3=8)b5p579 => -8r6c12

If you look at the first post, you can find the same there.

[eleven]

(1)r3c5 = r3c7 - r2c8 = r4c8 - (1=3587)b4p1358 - r2c2 = (7)r2c5^ - (7=32)r78c5 => -1 r2c5^, -2 r3c5; ste

Wow, nice. DIdn't find anything.

[Ajò Dimonios]

Hi Eleven


Yes, you're absolutely right, you can get a shorter chain. In fact I think that in fairly simple schemes like this, it seems to me that there are more than twenty backdoors, many resolutions may contain the same contradiction. Frankly, I think that any resolving technique always seeks a contradiction.

Ciao a Tutti

[SpAce]

Hi Paolo,

First of all, you're very welcome to participate here! Nice to see some new blood.

7R7C5-R2C5=R1C4-R1C1=R6C1-(7=8)R6C2-(8=3)R5C2;
7R7C5-R2C5=R1C4-R1C1=R6C1-(7=8)R6C2-8R6C46=16R6C46-16R5C5=3R5C5=>contradiction two 3s in R5=>-7R7C5=>Stte.

As SteveC said (in a now deleted post), a solution expressed as a contradiction is called Nishio. Well, I guess it's only Nishio if it can be expressed with relatively simple chains like here. Otherwise it's just T&E, though I don't really know where the line is drawn. Both are valid logic but not especially interesting to look at, so contradiction solutions are not very popular here (like eleven mentioned).

However, Nishio solutions are easy to convert into AICs or krakens by reading backwards from the contradiction and building a verity (as eleven also demonstrated). Verities are much nicer to look at than contradictions, so I would recommend learning that even if you find your solutions as contradictions. In fact, my (equivalent) solution was found as a contradiction too (by coloring the 7s), but I converted it into something prettier. (Most of my solutions are found as direct verities, but not today.)

PS. Like mine, your solution is actually btte (basics to the end), not stte (singles to the end). For an stte solution, see Cenoman's or my hidden one.

PPS. Please use lower-case letters in your coordinates (like r2c5) and some whitespace (like 7r7c5 - r2c5 = r1c4 ...). It's the convention here, and much easier on the eyes anyway Thanks!

[SpAce]

Frankly, I think that any resolving technique always seeks a contradiction.

Wrong.

[Ajò Dimonios]

Hi Space

Space Wrote

WXYZ-Wing

This is a contradiction. If R6C2 or R6C1 is equal to 8 then R6C4 = 6 and R6C6 = 1 and therefore R5C5 = 3, but also R5C2 = 3. This is certainly a contradiction.

Ciao a Tutti
Paolo

[SpAce]

This is a contradiction. If R6C2 or R6C1 is equal to 8 then R6C4 = 6 and R6C6 = 1 and therefore R5C5 = 3, but also R5C2 = 3. This is certainly a contradiction.

Yes, it is, because it's assumptive Nishio logic. My WXYZ-Wing is not (and I didn't assume those 8s to find it). It's a verity pattern (like any AIC) and valid as such even if there's nothing to eliminate. And when it eliminates, it's a verity elimination, not a contradiction.

Assumptions and eliminations are not part of such patterns but they're certainly part of any contradiction (because they all start with assuming the eventual elimination). If 8r6c12 are eliminated, the WXYZ-Wing pattern is still there (though without anything to do). Your Nishio isn't.

Every elimination can be both found and expressed as either a contradiction or a verity, and they can be converted into each other at will, but they're still very different concepts both in theory and practice. I suggest you learn the difference (and use lower-case coordinates )

A couple of different verity expressions for the same four cells:

1) Death Blossom (136)r5c5:

(1)r5c5 - (1=8)r6c6
(3)r5c5 - (3=8)r5c2
(6)r5c5 - (6=8)r6c4
=> -8 r6c12

This is actually very close to how I found my solution, because the contradiction I spotted first made r5c5 empty. After that it was obviously very easy to see the above Death Blossom, and then turn it into the WXYZ-Wing that I chose to present. So, like I said before, in this case I didn't find a verity directly. The coloring process I use can produce both verities and contradictions, so I take whatever comes first and works, but I prefer direct verities if available. It's very different from a process that can find contradictions only. Also notice that I found the contradiction for the 7s (which I used as the obvious coloring seed) but didn't actually use it to eliminate any 7s. Rather it prompted me to spot the above pattern and its verity eliminations, in that order.

2) ADDS (136[8]): {5N25 6N46 \ 3r5 16b5 [8r6 8b4]} => -8 r6c12

3) eleven's style (same as 2 but in words): four digits in four cells, only 8 can be twice (so at least one must be true) => -8 r6c12

Number 2 is also an almost-MSLS, which you were interested in elsewhere. (Probably not what you were looking for, lol.)

[Ajò Dimonios]

Hi Space.

Space wrote:

Yes, it is, because it's assumptive Nishio logic. My WXYZ-Wing is not (and I didn't assume those 8s to find it). It's a verity pattern (like any AIC) and valid as such even if there's nothing to eliminate. And when it eliminates, it's a verity elimination, not a contradiction.

What is "it's a verity elimination"? For me it is clearly a contradiction, that is, a failure to verify a truth.

Any elimination performed with an AIC can be translated with a Nisho. The difference is that the AIC starts with a strong inference and finds the lacked truth or contradiction at the end of the chain where the element that can be eliminated is at the same time true and false and therefore false. The Nisho chain starts from the eliminable element and finds a lacked truth or contradiction in a different area of the scheme, an empty cell or a double insertion in a row or column or block.

Assumptions and eliminations are not part of such patterns but they're certainly part of any contradiction (because they all start with assuming the eventual elimination). If 8r6c12 are eliminated, the WXYZ-Wing pattern is still there (though without anything to do). Your Nishio isn't.

I agree with you that it is probably more elegant to start from a chain, apparently not linked to elimination, but in the end the lack of truth or contradiction is the same. There are no series A or Series B logics there are correct or incorrect logics. The Nishio, after the elimination no longer exists because the initial hypothesis proves to verity elimination or a contradiction.

Ciao a Tutti
Paolo

[StrmCkr]

Wxyz - wing technique {upgraded by me years ago} directly posted as its parent configuration technique that it derives from:

Code: Select all

Almost Locked Set XZ-Rule: A=r5c2 {38}, B=r5c5,r6c46 {1368}, X=3, Z=8 => r6c12<>8

N cells with N+1 digits linked to N cells with N+1 digit where digit 3 is a restricted common between both sets thus set A or B is a locked set of n digits with n cells when 3 is limited to being in a or b exclusively directly implying all non restricted commons of the set cannot appear out side the set ie z = 8 => thus any cells that are visible to the 8 cells cannot contain 8. which is R6C12

this is not an assumptive technique as it is based on set mathematics.

it is both forward and backwards compatible:
the elimination placed proves the set contains an error of placement {n cells with less then n digits ie a construction contradiction}
and the logic of the sets prove the elimination must be true.

The Nisho chain starts from the eliminable element and finds a lacked truth or contradiction in a different area of the scheme, an empty cell or a double insertion in a row or column or block.

Assumptions and eliminations are not part of such patterns but they're certainly part of any contradiction (because they all start with assuming the eventual elimination). If 8r6c12 are eliminated, the WXYZ-Wing pattern is still there (though without anything to do). Your Nishio isn't.




I agree with you that it is probably more elegant to start from a chain, apparently not linked to elimination, but in the end the lack of truth or contradiction is the same. There are no series A or Series B logics there are correct or incorrect logics. The Nishio, after the elimination no longer exists because the initial hypothesis proves to verity elimination or a contradiction.

my last comment is the problem with nisho chains most often it cannot prove the elimination in the reverse logic:
when it can it is usually mimicking : a forward & reversible logic chain of some sort.

which is why brute force/t&e techniques are often not used on these forums.

really if i wanted to i could simply list the back door cells of a puzzle and solve it with (1-3) key placements based on the grids required backdoor size and the singles trigger.
for no rhyme or reason but a blind guess at those placements. and that ain't fun

[Ajò Dimonios]

Hi StrmCkr

Code: Select all

  StrmCkr wrote:


my last comment is the problem with nisho chains most often it cannot prove the elimination in the reverse logic:
when it can it is usually mimicking : a forward & reversible logic chain of some sort.

which is why brute force/t&e techniques are often not used on these forums.

really if i wanted to i could simply list the back door cells of a puzzle and solve it with (1-3) key placements based on the grids required backdoor size and the singles trigger.
for no rhyme or reason but a blind guess at those placements. and that ain't fun

What you say is true the NIsho chain is not always reversible (it always starts from a weak inference), but this is of no use when we have obtained a contradiction, while it is fundamental when we get a backdoor. The contradiction is always valid even if it is not easy to build a reversible chain or network with a simple logic. In this scheme we have many examples. The backdoor 7r7c9 is reversible as are the backdoors 8r3c7,6r4c8,8r4c9,1r5c7 (even if the reverse network is very complex and profound), while the backdoors 3r8c5, 2r8c8, 6r1c9,6r2c3 and others are not reversible with a simple logic.

Ciao a Tutti
Paolo

[eleven]

We have to distinguish two very different things:

The one is to eliminate a digit by assuming it and showing with a chain (or maybe 2 or 3 ... chains), that it leads to a contradiction.
This is always reversable and can be expressed without contradiction. Logically you have a chain
A => B => ... => X => contradiction
and you can reverse it with
NO contradiction => not X => ... => not B => not A
Note, that in such a chain each step/implication is independant from the others.

The other is to GUESS a number, which solves the puzzle (without proving that the other candidates do not lead to a solution).
Here you show many candidates to be true, the next one depending on possibly all of the former ones, which means, that the solution is build from a complex net - and the reverse would be a complex net too.

Guessing is, what it is. You maybe lucky or not (in which case you have to erase everything and restart).
As SpAce mentioned, AIC chains may be useless, but they are always true - so you don't have to erase anything.