The New Sudoku Players' Forum

[urhegyi]

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1......76..5.4.........9.........8.2.4..5......73.6.1.8...3.5..3.92......6...7...

[Cenoman]

Four steps, the first one only being not so simple:

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 +------------------------+-----------------------+------------------------+
 |  1     B2389   2348    |  58      28    2358   | v2349   7       6      |
 |  67    B2389   5       |  67      4     238    |  1239   2389    1389   |
 | x2467  B238    23468   |  167     167   9      | v234   w23458  w3458   |
 +------------------------+-----------------------+------------------------+
 |  569 rAa135    136     |  1479    179   14     |  8     s3569    2      |
 |  269    4     D2368-1  |  189     5     28     |  67     369     379    |
 |  259   C258    7       |  3       289   6      | u49     1      t459    |
 +------------------------+-----------------------+------------------------+
 |  8      127   z124     |  1469    3     14     |  5      269     179    |
 |  3      157    9       |  2       168   58     |  67     468     1478   |
 | y245    6     z124     |  14589   189   7      |  1239   2389    1389   |
 +------------------------+-----------------------+------------------------+

1. Kraken cell (135)r4c2
(1)r4c2
(3)r4c2 - (3=298)r123c2 - r6c2 = (8)r5c3
(5)r4c2 - r4c8 = (5-4)r6c9 = r6c7 - r13c7 = r3c89 - r3c1 = r9c1 - (4=21)r79c3
=> -1 r5c3; 16 placements & lcls

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 +-----------------------+-------------------+-----------------------+
 |  1      2389   34-8   |  58   28   2358   |  2349   7      6      |
 |  67     2389   5      |  67   4    238    |  1239   289    138    |
 |  67+24  238    3468   |  67   1    9      |  234    2458   3458   |
 +-----------------------+-------------------+-----------------------+
 |  56     135    136    |  9    7    4      |  8      35     2      |
 |  29     4      38     |  1    5    28     |  6      39     7      |
 |  259    258    7      |  3    28   6      |  49     1      45     |
 +-----------------------+-------------------+-----------------------+
 |  8      7      2      |  4    3    1      |  5      6      9      |
 |  3      15     9      |  2    6    58     |  7      48     148    |
 |  45     6      14     |  58   9    7      |  123    28     138    |
 +-----------------------+-------------------+-----------------------+

2. UR(67)r23c14 =>+24r3c1
3. Kite (8)r1c5 = r6c5 - r5c6 = r5c3 => -8 r1c3; 20 placements & lcls

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 +------------------+-----------------+-----------------+
 |  1    9     34   |  58   2    58   |  34   7    6    |
 |  7    28    5    |  6    4    3    |  12   9    18   |
 |  24   38+2  6    |  7    1    9    |  34   28   5    |
 +------------------+-----------------+-----------------+
 |  6    13    13   |  9    7    4    |  8    5    2    |
 |  9    4     8    |  1    5    2    |  6    3    7    |
 |  25   25    7    |  3    8    6    |  9    1    4    |
 +------------------+-----------------+-----------------+
 |  8    7     2    |  4    3    1    |  5    6    9    |
 |  3    15    9    |  2    6    58   |  7    4    18   |
 |  45   6     14   |  58   9    7    |  12   28   3    |
 +------------------+-----------------+-----------------+

4. BUG+1 => +2 r3c2; ste

[denis_berthier]

.

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Resolution state after Singles and whips[1]:
   +-------------------+-------------------+-------------------+
   ! 1     2389  2348  ! 58    28    2358  ! 2349  7     6     !
   ! 2679  2389  5     ! 1678  4     1238  ! 1239  2389  1389  !
   ! 2467  238   23468 ! 1678  12678 9     ! 1234  23458 13458 !
   +-------------------+-------------------+-------------------+
   ! 569   1359  136   ! 1479  179   14    ! 8     3569  2     !
   ! 269   4     12368 ! 189   5     128   ! 3679  369   379   !
   ! 259   2589  7     ! 3     289   6     ! 49    1     459   !
   +-------------------+-------------------+-------------------+
   ! 8     127   124   ! 1469  3     14    ! 5     2469  1479  !
   ! 3     157   9     ! 2     168   1458  ! 1467  468   1478  !
   ! 245   6     124   ! 14589 189   7     ! 12349 23489 13489 !
   +-------------------+-------------------+-------------------+
215 candidates

A very interesting puzzle, illustrating the balance one has to make between simplicity of steps and number of steps.
I give only 3 of all the possible choices.


1) Simplest-first solution, in S+Z4 (using only very simple, reversible chains)
Notice the long series of Singles before the last two non-W1 eliminations.

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naked-pairs-in-a-column: c6{r4 r7}{n1 n4} ==> r8c6≠4, r8c6≠1, r5c6≠1, r2c6≠1
whip[1]: r8n4{c9 .} ==> r7c8≠4, r7c9≠4, r9c7≠4, r9c8≠4, r9c9≠4
hidden-pairs-in-a-column: c7{n6 n7}{r5 r8} ==> r8c7≠4, r8c7≠1, r5c7≠9, r5c7≠3
hidden-pairs-in-a-row: r2{n6 n7}{c1 c4} ==> r2c4≠8, r2c4≠1, r2c1≠9, r2c1≠2
whip[1]: c1n9{r6 .} ==> r4c2≠9, r6c2≠9
whip[1]: r2n1{c9 .} ==> r3c7≠1, r3c9≠1
finned-x-wing-in-columns: n1{c6 c2}{r4 r7} ==> r7c3≠1
hidden-triplets-in-a-block: b2{n1 n6 n7}{r3c5 r3c4 r2c4} ==> r3c5≠8, r3c5≠2, r3c4≠8
finned-swordfish-in-rows: n6{r7 r2 r4}{c8 c4 c1} ==> r5c1≠6
z-chain[3]: r9n5{c1 c4} - r9n4{c4 c3} - r7c3{n4 .} ==> r9c1≠2
z-chain[3]: c1n2{r6 r3} - c1n4{r3 r9} - r7c3{n4 .} ==> r5c3≠2
biv-chain[4]: c7n6{r5 r8} - c5n6{r8 r3} - b2n1{r3c5 r3c4} - r5n1{c4 c3} ==> r5c3≠6
whip[1]: r5n6{c8 .} ==> r4c8≠6
biv-chain[4]: c8n4{r8 r3} - c1n4{r3 r9} - b7n5{r9c1 r8c2} - r8c6{n5 n8} ==> r8c8≠8
z-chain[4]: c9n5{r3 r6} - c9n4{r6 r8} - c8n4{r8 r3} - r3n5{c8 .} ==> r3c9≠3
z-chain[4]: c9n5{r3 r6} - c9n4{r6 r8} - c8n4{r8 r3} - r3n5{c8 .} ==> r3c9≠8
biv-chain[4]: b7n5{r8c2 r9c1} - c1n4{r9 r3} - r3c9{n4 n5} - b6n5{r6c9 r4c8} ==> r4c2≠5
biv-chain[3]: r6n8{c5 c2} - c2n5{r6 r8} - r8c6{n5 n8} ==> r5c6≠8, r8c5≠8, r9c5≠8
36 singles ==> r5c6=2, r5c1=9, r1c5=2, r6c5=8, r5c4=1, r4c6=4, r7c6=1, r8c5=6, r8c7=7, r5c7=6, r5c8=3, r5c3=8, r5c9=7, r7c9=9, r7c4=4, r7c3=2, r7c2=7, r7c8=6, r8c8=4, r9c5=9, r4c5=7, r3c5=1, r4c4=9, r4c8=5, r4c1=6, r2c1=7, r2c4=6, r3c4=7, r6c9=4, r3c9=5, r6c7=9, r2c8=9, r1c2=9, r3c3=6
whip[1]: r1n8{c6 .} ==> r2c6≠8
singles ==> r2c6=3, r9c9=3
naked-pairs-in-a-column: c7{r2 r9}{n1 n2} ==> r3c7≠2
biv-chain[4]: c1n4{r3 r9} - r9n5{c1 c4} - r9n8{c4 c8} - c8n2{r9 r3} ==> r3c1≠2
stte

This is 16 non-W1 steps.

2) By allowing slightly longer chains (max length 6), the fewer step method found a solution in 9 non-W1 steps:

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=====> STEP #1
naked-pairs-in-a-column: c6{r4 r7}{n1 n4} ==> r8c6≠4, r8c6≠1, r5c6≠1, r2c6≠1
whip[1]: r8n4{c9 .} ==> r7c8≠4, r7c9≠4, r9c7≠4, r9c8≠4, r9c9≠4

=====> STEP #2
hidden-pairs-in-a-row: r2{n6 n7}{c1 c4} ==> r2c1≠9, r2c4≠8, r2c4≠1, r2c1≠2
whip[1]: r2n1{c9 .} ==> r3c7≠1, r3c9≠1
whip[1]: c1n9{r6 .} ==> r4c2≠9, r6c2≠9

=====> STEP #3
hidden-pairs-in-a-column: c7{n6 n7}{r5 r8} ==> r5c7≠9, r8c7≠4, r8c7≠1, r5c7≠3

=====> STEP #4
biv-chain[6]: r8c7{n7 n6} - b8n6{r8c5 r7c4} - r2n6{c4 c1} - b1n7{r2c1 r3c1} - c1n4{r3 r9} - b7n5{r9c1 r8c2} ==> r8c2≠7
hidden-single-in-a-block ==> r7c2=7

=====> STEP #5
whip[6]: r3n1{c5 c4} - r5n1{c4 c3} - c2n1{r4 r8} - b7n5{r8c2 r9c1} - c1n4{r9 r3} - r3n7{c1 .} ==> r3c5≠6
singles ==> r8c5=6, r8c7=7, r5c7=6, r7c8=6, r7c3=2, r5c9=7
whip[1]: b6n3{r5c8 .} ==> r2c8≠3, r3c8≠3, r9c8≠3
whip[1]: r7n4{c6 .} ==> r9c4≠4

=====> STEP #6
biv-chain[4]: r8c6{n8 n5} - b7n5{r8c2 r9c1} - c1n4{r9 r3} - c8n4{r3 r8} ==> r8c8≠8
naked-single ==> r8c8=4

====> STEP #7
whip[6]: b7n1{r9c3 r8c2} - b7n5{r8c2 r9c1} - c1n4{r9 r3} - c9n4{r3 r6} - r6n5{c9 c2} - b4n8{r6c2 .} ==> r5c3≠1
singles ==> r5c4=1, r4c6=4, r7c6=1, r7c9=9, r7c4=4, r3c5=1, r4c5=7, r4c4=9, r9c5=9

=====> STEP #8
z-chain[6]: c5n8{r6 r1} - c3n8{r1 r3} - c3n6{r3 r4} - c3n1{r4 r9} - r8n1{c2 c9} - r8n8{c9 .} ==> r5c6≠8
singles ==> r5c6=2, r5c1=9, r5c8=3, r4c8=5, r4c1=6, r2c1=7, r2c4=6, r6c9=4, r6c7=9, r5c3=8, r6c5=8, r1c5=2, r2c8=9, r1c2=9, r3c3=6, r3c4=7, r3c9=5, r2c6≠8, r2c6=3, r9c9=3

=====> STEP #9
biv-chain[4]: r3c1{n4 n2} - c8n2{r3 r9} - r9n8{c8 c4} - r9n5{c4 c1} ==> r9c1≠4
stte

Sure, the number of steps could be reduced, by doing more tries or by allowing slightly longer chains.


3) 1-step solution with chains of unrestricted length, for those who like the nukes.
Mathematically speaking, it's reasoning by cases on cell r9c1.
Call it Kraken if you want to make it look more respectable.

Code: Select all

FORCING[3]-T&E(W1) applied to trivalue candidates n2r9c1, n4r9c1 and n5r9c1 :
===> 29 values decided in the three cases: n9r6c7 n4r6c9 n5r4c8 n5r3c9 n2r1c5 n8r6c5 n8r5c3 n4r4c6 n1r7c6 n9r9c5 n7r4c5 n9r4c4 n1r5c4 n6r4c1 n9r5c1 n7r2c1 n6r2c4 n4r7c4 n1r3c5 n7r3c4 n6r8c5 n4r8c8 n7r7c2 n9r7c9 n2r5c6 n3r2c6 n6r7c8 n3r5c8 n3r9c9
===> 129 candidates eliminated in the three cases: n2r1c2 n3r1c2 n8r1c2 n2r1c3 n8r1c3 n8r1c5 n2r1c6 n3r1c6 n2r1c7 n9r1c7 n2r2c1 n6r2c1 n9r2c1 n2r2c2 n3r2c2 n1r2c4 n7r2c4 n8r2c4 n1r2c6 n2r2c6 n8r2c6 n3r2c7 n9r2c7 n3r2c8 n8r2c8 n3r2c9 n9r2c9 n6r3c1 n7r3c1 n3r3c2 n3r3c3 n4r3c3 n8r3c3 n1r3c4 n6r3c4 n8r3c4 n2r3c5 n6r3c5 n7r3c5 n8r3c5 n1r3c7 n2r3c7 n2r3c8 n3r3c8 n4r3c8 n5r3c8 n1r3c9 n3r3c9 n4r3c9 n8r3c9 n5r4c1 n9r4c1 n5r4c2 n9r4c2 n6r4c3 n1r4c4 n4r4c4 n7r4c4 n1r4c5 n9r4c5 n1r4c6 n3r4c8 n6r4c8 n9r4c8 n2r5c1 n6r5c1 n1r5c3 n2r5c3 n3r5c3 n6r5c3 n8r5c4 n9r5c4 n1r5c6 n8r5c6 n3r5c7 n9r5c7 n6r5c8 n9r5c8 n3r5c9 n9r5c9 n9r6c1 n8r6c2 n9r6c2 n2r6c5 n9r6c5 n4r6c7 n5r6c9 n9r6c9 n1r7c2 n2r7c2 n1r7c3 n4r7c3 n1r7c4 n6r7c4 n9r7c4 n4r7c6 n2r7c8 n4r7c8 n9r7c8 n1r7c9 n4r7c9 n7r7c9 n7r8c2 n1r8c5 n8r8c5 n1r8c6 n4r8c6 n1r8c7 n4r8c7 n6r8c7 n6r8c8 n8r8c8 n4r8c9 n2r9c3 n1r9c4 n4r9c4 n9r9c4 n1r9c5 n8r9c5 n3r9c7 n4r9c7 n9r9c7 n3r9c8 n4r9c8 n9r9c8 n1r9c9 n4r9c9 n8r9c9 n9r9c9

stte