The New Sudoku Players' Forum

[daj95376]

While examining this PM, my solver found a chain that loops back through the originating cell (*) before continuing on and encountering a contradiction (@).

Code: Select all

 +-----------------------------------------------------------------------------------------+
 |  124589   149      1589     |  2479     14789    12789    |  6        3        12589    |
 |  12489    7        13689    |  5        13489    123689   |  289      1289     1289     |
 |  12589    1369     135689   |  2369     1389     123689   |  4        7        12589    |
 |-----------------------------+-----------------------------+-----------------------------|
 |  6        58       4        |  379      3579     3579     |  1        289      2389     |
 |  179      19       179      |  8        2        36       |  5        46       346      |
 |  3        58       2        |  469      1459     1569     |  7        689      89       |
 |-----------------------------+-----------------------------+-----------------------------|
 |  4589     349      3589     |  1        6        23589    |  2389     2489     7        |
 |  14       2        16       |  379      3789     3789     |  389      5        146      |
 |  15789    1369     1356789  |  239      3589     4        |  2389     12689    12689    |
 +-----------------------------------------------------------------------------------------+


Code: Select all

********                                            ********
[r2c6]=6=[r2c3]-6-[r8c3]-1-[r8c1]-4-[r2c1]=4=[r2c5]=3=[r2c6]-3-[r5c6]=3=[r5c9]=4=[r8c9]
                           @@@@@@@@                                            @@@@@@@@
_______________________________________________________________________________________


I could stop the chain when it returns to the originating cell ...

Code: Select all

[r2c6]=6=[r2c3]-6-[r8c3]-1-[r8c1]-4-[r2c1]=4=[r2c5]=3=[r2c6] => [r2c6]<>1289


... but the contradiction lets me conclude that [r2c6]=6, which cracks the puzzle.

Is my first chain (and contradiction) considered acceptable

(Yes, I know there are other chains that crack this puzzle.)

[ronk]

I could stop the chain when it returns to the originating cell ...

Code: Select all

[r2c6]=6=[r2c3]-6-[r8c3]-1-[r8c1]-4-[r2c1]=4=[r2c5]=3=[r2c6] => [r2c6]<>1289


When you add the necessary strong inference between r2c3 and r2c6, you've actually got a net.

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              +=3=====================================+
[r2c6]=6=[r2c3]-6-[r8c3]-1-[r8c1]-4-[r2c1]=4=[r2c5]=3=[r2c6] => [r2c6]<>1289

Reading left-to-right ... when r2c5<>3 then r2c6=3 because r2c3<>3 already occurred. IOW you've got an almost-nice-loop (ANL) (aka a triple implication chain, a kraken row, an AAIC) deduction.

Is my first chain (and contradiction) considered acceptable

With the above already a net, I don't wish to go there.

[daj95376]

ronk:

If I follow your logic, you are saying that =4=[r2c5]=3=[r2c6] isn't true because it relies on an elimination in 3 that I performed during [r2c3]-6- . This made it a net instead of a chain? I'm confused!!! I'll need time to review (again) the differences between a chain and a network.

[Pat]

This made it a net instead of a chain?

yes, exactly

they call it a "chain" only where each step stands on its own as a valid implication

anything more complicated ( when a step depends on something which happened upstream ) is called a "net"

Code: Select all

 124589   149   1589     |  2479     14789    12789    |  6     3     12589
 12489    7     13689    |  5        13489    123689   |  289   1289  1289 
 12589    1369  135689   |  2369     1389     123689   |  4     7     12589
-------------------------+-----------------------------+--------------------
 6        58    4        |  379      3579     3579     |  1     289   2389 
 179      19    179      |  8        2        36       |  5     46    346   
 3        58    2        |  469      1459     1569     |  7     689   89   
-------------------------+-----------------------------+--------------------
 4589     349   3589     |  1        6        23589    |  2389  2489  7     
 14       2     16       |  379      3789     3789     |  389   5     146   
 15789    1369  1356789  |  239      3589     4        |  2389  12689 12689



each arrow here is a valid implication on its own
( the left-hand side directly implies the right-hand side )
except the one step marked
( where the implication also depends on r2 having no 3 at r2c3 )
making this a "net"

r2c3 = 6 --> r8c3 = 1 --> r8c1 = 4 --> r2c5 = 4 --> r2c6 = 3 --> r5c9 = 3 --> r8c9 = 4

conflict in r8

therefore, r2c3 not 6

[ronk]

If I follow your logic, you are saying that =4=[r2c5]=3=[r2c6] isn't true because it relies on an elimination in 3 that I performed during [r2c3]-6- . This made it a net instead of a chain? I'm confused!!!

"Chain" is arguably the most misused term in sudoku, so IMO there are many others confused as well. I use Jeff's definitions here.

[daj95376]

Thanks ronk and Pat for the clarification!

[Pat]

meanwhile i notice i had neglected to show one arrow -- at the final step !!
here's my amended example of a "net" --
each arrow is a valid implication on its own,
except the 2 steps marked --
• one where the implication also depends on r2 having no 3 at r2c3
• one ( the final step ) also depends on r8c1 = 4 upstream
r2c3 = 6 --> r8c3 = 1 --> r8c1 = 4 --> r2c5 = 4 --> r2c6 = 3 --> r5c9 = 3 --> r8c9 = 4 --> 2 4s in r8; therefore, r2c3 not 6


-- or as long as i'm revising,
might as well shorten it one step --
r2c3 = 6 --> r8c3 = 1 --> r8c1 = 4 --> r2c5 = 4 --> r2c6 = 3 --> r5c9 = 3 --> no 4 in c9; therefore, r2c3 not 6

[Carcul]

This one looks like fun.

Code: Select all

 *--------------------------------------------------------------------------*
 | 579     569     579    | 589     2       34569  | 34689   3569    1      |
 | 8       1569    4      | 1579    3569    135679 | 369     2       35679  |
 | 3       1569    2      | 15789   45689   145679 | 4689    5679    456789 |
 |------------------------+------------------------+------------------------|
 | 1579    359     13579  | 6       579     8      | 1239    4       2359   |
 | 459     2       368    | 159     459     1459   | 7       36      368    |
 | 14579   48      156789 | 3       4579    2      | 1689    1569    5689   |
 |------------------------+------------------------+------------------------|
 | 6       359     1359   | 4       359     3579   | 1239    8       2379   |
 | 249     7       389    | 289     1       369    | 5       369     3469   |
 | 12459   48      13589  | 25789   35689   35679  | 13469   13679   34679  |
 *--------------------------------------------------------------------------*


1) [r6c7]-8-[r5c9]=8=[r5c3]-8-[r8c3]=8=[r8c4]-8-[r1c4]=8=[r1c7]-8-
-[r6c7], => r6c7<>8.

2) [r8c1]=2=[r8c4]=8=[r8c3]-8-[r9c2]-4-[r8c1], => r8c1<>4.

3) [r8c6]=6=[r8c8]-6-[r5c8]-3-[r1c8](=3=[r1c6]-3-[r8c6])=3|8=[r1c4]-8-
-[r8c14]-9-[r8c6], => r8c6<>3,9.

4) [r8c8]=3=[r8c3]=8=[r8c4]-8-[r1c4]=8|3=[r1c8]-3-[r8c8], => r8c3<>9; r3c4, r9c4<>8; r5c8, r9c8<>3.

5) [r9c7]=6=[r2c7]-6-[r2c5]=6=[r3c5]=8=[r9c5]-8-[r8c4]=8=[r8c3]=3=
=[r8c8](-3-[r9c7])-3-[r1c8]=3=[r1c6]-3-[r2c5], => r9c7, r2c5<>3.

6) [r9c5]=3=[r7c5]-3-[r7c2]=3=[r8c3]=8=[r8c4]-8-[r9c5], => r9c5<>8.

7) [r7c2]=5=[r3c2]-5-[r3c46]=5=[r5c46]-5-[r5c1]=5=[r4c3]-5-[r9c3]=5=
=[r7c2], => r7c2=5 and the puzzle is solved.

[daj95376]

Solves easily with a net.

Code: Select all

 ..1....452...8................5.1...3.....8.....64.....541......6.....7.......2.. #44753
_________________________________________________________________________________________

 +--------------------------------------------------------------------------------+
 |  6789    3789    1       |  2379    2679    23679   |  3679    4       5       |
 |  2       3479    35679   |  3479    8       345679  |  13679   1369    13679   |
 |  45679   3479    35679   |  3479    1       345679  |  3679    28      28      |
 |--------------------------+--------------------------+--------------------------|
 |  46789   24789   6789    |  5       3       1       |  4679    269     24679   |
 |  3       14      56      |  279     279     279     |  8       156     146     |
 |  1579    1279    579     |  6       4       8       |  13579   12359   12379   |
 |--------------------------+--------------------------+--------------------------|
 |  789     5       4       |  1       2679    23679   |  369     3689    3689    |
 |  189     6       2       |  3489    59      349     |  13459   7       13489   |
 |  1789    13789   3789    |  4789    5679    4679    |  2       15689   14689   |
 +--------------------------------------------------------------------------------+


[daj95376]

Code: Select all

 ..6..3....1.....4.....5....2.....3...9.1.....5.........8....1.93...2.......4...5. #46801

 +-----------------------------------------------------------------------------------------+
 |  4789     2457     6        |  2789     1        3        |  25789    2789     2578     |
 |  789      1        23578    |  26789    6789     26789    |  256789   4        235678   |
 |  789      237      278      |  26789    5        4        |  26789    1236789  12678    |
 |-----------------------------+-----------------------------+-----------------------------|
 |  2        467      1478     |  56789    46789    56789    |  3        16789    145678   |
 |  4678     9        3478     |  1        34678    25678    |  245678   2678     245678   |
 |  5        3467     1478     |  236789   46789    26789    |  246789   126789   124678   |
 |-----------------------------+-----------------------------+-----------------------------|
 |  467      8        2457     |  3567     67       567      |  1        2367     9        |
 |  3        567      579      |  56789    2        1        |  4678     678      4678     |
 |  1        267      279      |  4        36789    6789     |  2678     5        23678    |
 +-----------------------------------------------------------------------------------------+


[Carcul]

Solves easily with a net.

Code: Select all

 *------------------------------------------------------------------------*
 | 6789    3789    1     | 2379    2679    23679  | 3679    4       5     |
 | 2       3479    35679 | 3479    8       345679 | 13679   1369    13679 |
 | 45679   3479    35679 | 3479    1       345679 | 3679    28      28    |
 |-----------------------+------------------------+-----------------------|
 | 46789   24789   6789  | 5       3       1      | 4679    269     24679 |
 | 3       14      56    | 279     279     279    | 8       156     146   |
 | 1579    1279    579   | 6       4       8      | 13579   12359   12379 |
 |-----------------------+------------------------+-----------------------|
 | 789     5       4     | 1       2679    23679  | 369     3689    3689  |
 | 189     6       2     | 3489    59      349    | 13459   7       13489 |
 | 1789    13789   3789  | 4789    5679    4679   | 2       15689   14689 |
 *------------------------------------------------------------------------*


1) [r8c7]=5=[r6c7]-5-[r6c1]=5=[r3c1]=4=[r4c1](-4-[r5c2]-1-[r6c1])-4-
-[r4c7]=4=[r8c7], => r8c7<>1,3,9; r6c3, r6c8<>5; r3c1<>6,7,9; r6c1<>1; r4c2, r4c9<>4.

2) [r4c2]=2=[r6c2]=1=[r5c2]=4=[r4c1](=6=[r1c1]=8=[r1c2]-8-[r4c2])-
-4-[r3c1]-5-[r6c13]-7,9-[r46c2], => r4c2, r6c2<>7,9; r4c2<>8, and the puzzle is solved.

[Carcul]

For the last puzzle posted by Daj95376,

Code: Select all

 *------------------------------------------------------------------------------*
 | 4789     2457     6     | 2789     1        3     | 25789    2789     2578   |
 | 789      1        23578 | 26789    6789     26789 | 256789   4        235678 |
 | 789      237      2378  | 26789    5        4     | 26789    1236789  123678 |
 |-------------------------+-------------------------+--------------------------|
 | 2        467      1478  | 56789    46789    56789 | 3        16789    145678 |
 | 4678     9        3478  | 1        34678    25678 | 245678   2678     245678 |
 | 5        3467     13478 | 236789   346789   26789 | 246789   126789   124678 |
 |-------------------------+-------------------------+--------------------------|
 | 467      8        2457  | 3567     367      567   | 1        2367     9      |
 | 3        567      579   | 56789    2        1     | 4678     678      4678   |
 | 1        267      279   | 4        36789    6789  | 2678     5        23678  |
 *------------------------------------------------------------------------------*


1) [r7c3]=4=[r7c1]-4-[r1c1]=4=[r1c2]=5=[r2c3]-5-[r7c3], => r7c3<>5.

2) [r7c8]=3=[r3c8]-3-[r2c9]=3=[r2c3]=5=[r1c2]=4=[r1c1]-4-[r7c1]=4=[r7c3]=2=[r7c8], => r7c8<>6,7; r3c9<>3; r2c3<>2,7,8; r1c2<>2,7; r5c1<>4; r7c3<>7.

3) [r3c2]=2=[r3c3]-2-[r7c3]-4-[r7c1]=4=[r1c1]-4-[r1c2]-5-[r2c3]-3-[r3c2], => r3c2<>3, which solves the puzzle.

[eleven]

This one looks like fun.

Code: Select all

 *--------------------------------------------------------------------------*
 | 579     569     579    | 589     2       34569  | 34689   3569    1      |
 | 8       1569    4      | 1579    3569    135679 | 369     2       35679  |
 | 3       1569    2      | 15789   45689   145679 | 4689    5679    456789 |
 |------------------------+------------------------+------------------------|
 | 1579    359     13579  | 6       579     8      | 1239    4       2359   |
 | 459     2       368    | 159     459     1459   | 7       36      368    |
 | 14579   48      156789 | 3       4579    2      | 1689    1569    5689   |
 |------------------------+------------------------+------------------------|
 | 6       359     1359   | 4       359     3579   | 1239    8       2379   |
 | 249     7       389    | 289     1       369    | 5       369     3469   |
 | 12459   48      13589  | 25789   35689   35679  | 13469   13679   34679  |
 *--------------------------------------------------------------------------*


1) [r6c7]-8-[r5c9]=8=[r5c3]-8-[r8c3]=8=[r8c4]-8-[r1c4]=8=[r1c7]-8-
-[r6c7], => r6c7<>8.

2) [r8c1]=2=[r8c4]=8=[r8c3]-8-[r9c2]-4-[r8c1], => r8c1<>4.

3) [r8c6]=6=[r8c8]-6-[r5c8]-3-[r1c8](=3=[r1c6]-3-[r8c6])=3|8=[r1c4]-8-
-[r8c14]-9-[r8c6], => r8c6<>3,9.

4) [r8c8]=3=[r8c3]=8=[r8c4]-8-[r1c4]=8|3=[r1c8]-3-[r8c8], => r8c3<>9; r3c4, r9c4<>8; r5c8, r9c8<>3.

5) [r9c7]=6=[r2c7]-6-[r2c5]=6=[r3c5]=8=[r9c5]-8-[r8c4]=8=[r8c3]=3=
=[r8c8](-3-[r9c7])-3-[r1c8]=3=[r1c6]-3-[r2c5], => r9c7, r2c5<>3.

6) [r9c5]=3=[r7c5]-3-[r7c2]=3=[r8c3]=8=[r8c4]-8-[r9c5], => r9c5<>8.

7) [r7c2]=5=[r3c2]-5-[r3c46]=5=[r5c46]-5-[r5c1]=5=[r4c3]-5-[r9c3]=5=
=[r7c2], => r7c2=5 and the puzzle is solved.

Note the similarity to my solution posted here: http://forum.enjoysudoku.com/viewtopic.php?p=55314#p55314. Steps 2 to 4 are almost the same as my steps 1 to 3. So this puzzle does not seem to offer a great variety of effective solution paths.

[hobiwan]

#44753 (without net, a bit more complicated):
Continuous Nice Loop [r4c1]=4=[r3c1]=5=[r6c1]-5-[r6c7]=5=[r8c7]=4=[r4c7]-4-[r4c1] => [r8c7]<>139, [r4c29]<>4, [r6c38]<>5, [r3c1]<>679
Almost Locked Set XZ-Rule: A=[r5c3] - {56}, B=[r16789c1] - {156789}, X=5, Z=6 => [r23c3],[r4c1]<>6
SSTS
Discontinuous Nice Loop [r6c1]=5=[r3c1]=4=[r4c1]=8=[r4c3]=6=[r5c3]=5=[r6c1] => [r6c1]<>179

[hobiwan]

#46801:
Almost Locked Set XZ-Rule: A=[r7c14568] - {234567}, B=[r123c1],[r3c3] - {24789}, X=4, Z=2 => [r3c8],[r7c3]<>2
Puzzle solved