The New Sudoku Players' Forum

[denis_berthier]

.

Code: Select all

+-------+-------+-------+
! 1 2 . ! . . 6 ! 7 8 . !
! . 5 . ! . . . ! 2 . 6 !
! . . . ! . . 2 ! . 1 5 !
+-------+-------+-------+
! 2 . 5 ! . 7 . ! . . . !
! . 1 . ! . . . ! . . . !
! 8 7 . ! 2 1 . ! . 6 . !
+-------+-------+-------+
! 5 . . ! 9 4 8 ! . . . !
! 7 8 . ! 3 6 . ! . . . !
! . . . ! . 2 7 ! 6 . 8 !
+-------+-------+-------+
12...678..5....2.6.....2.152.5.7.....1.......87.21..6.5..948...78.36........276.8;10190;240260
SER = 10.4

Code: Select all

Resolution state after Singles and whips[1]:
   +----------------------+----------------------+----------------------+
   ! 1      2      349    ! 45     359    6      ! 7      8      349    !
   ! 349    5      34789  ! 1478   389    1349   ! 2      349    6      !
   ! 3469   3469   346789 ! 478    389    2      ! 349    1      5      !
   +----------------------+----------------------+----------------------+
   ! 2      3469   5      ! 468    7      349    ! 13489  349    1349   !
   ! 3469   1      3469   ! 4568   3589   3459   ! 34589  234579 23479  !
   ! 8      7      349    ! 2      1      3459   ! 3459   6      349    !
   +----------------------+----------------------+----------------------+
   ! 5      36     1236   ! 9      4      8      ! 13     237    1237   !
   ! 7      8      1249   ! 3      6      15     ! 1459   2459   1249   !
   ! 349    349    1349   ! 15     2      7      ! 6      3459   8      !
   +----------------------+----------------------+----------------------+
177 candidates

[pjb]

1) Type 3 unique rectangle of 27 at r57c89, eliminating 3 from r7c23, 1 from r7c3, 1 from r8c79, 3 from r9c8 =>

Code: Select all

 1       2      *349    | 45     359    6      | 7      8     *349   
*349     5       78     | 1478   389    1349   | 2     *349    6     
 6      *349     78     | 478    389    2      |*349    1      5     
------------------------+----------------------+---------------------
 2      *349     5      | 6      7      349    | 8     *349    1     
*349     1       6      | 458    3589   3459   |*5-349  27     27     
 8       7      *349    | 2      1      3459   | 3459   6     *349   
------------------------+----------------------+---------------------
 5       6       2      | 9      4      8      | 1      37     37     
 7       8       149    | 3      6      15     | 49     2459   249   
 349     349     1349   | 15     2      7      | 6      459    8     


2) Type 1 Tridagon => -349 from r5c7; stte

Phil

[denis_berthier]

.
Hi pjb
I don't see how your step 1 can be justified.
What's your starting PM?

[DEFISE]

After basics :

Code: Select all

|-----------------------------------------------------------|
| 1     2     349   | 45    359   6     | 7     8     349   |
| 349   5     78    | 1478  389   1349  | 2     349   6     |
| 3469  3469  78    | 478   389   2     | 349   1     5     |
|-----------------------------------------------------------|
| 2     3469  5     | 468   7     349   | 13489 349   1349  |
| 3469  1     3469  | 4568  3589  3459  | 34589 27    27    |
| 8     7     349   | 2     1     3459  | 3459  6     349   |
|-----------------------------------------------------------|
| 5     36    1236  | 9     4     8     | 13    237   1237  |
| 7     8     1249  | 3     6     15    | 149   2459  1249  |
| 349   349   1349  | 15    2     7     | 6     3459  8     |
|-----------------------------------------------------------|

Tridagon (3,4,9) in b1p348, b3p357, b4p249, b6p249
with 5 guardians: 6r3c2,6r4c2,6r5c1,5r5c7,8r5c7

Note that :
-6r3c2 & -6r4c2 -> 6r3c1 -> 6r4c4 -> 8r4c7
So : -6r3c2 & -6r4c2 => -6r5c1 & -8r5c7
So just consider the 3 guardians 6r3c2, 6r4c2, 5r5c7.

OR3-g-braid[6] : r8c6{n1 n5}- r6n5{c6 c7}- c8n5{r8 r9}- r9n3{c8 c123}- r7c2{n3 n6}- OR3{all guardians | .} => -1r2c6
Single(s): 1r2c4, 5r9c4, 4r1c4, 1r8c6, 5r1c5, 7r2c3, 8r3c3, 7r3c4, 8r2c5, 5r8c8, 1r9c3
Hidden pairs: 27c8r57 => -3r7c8
whip[4]: r1n9{c9 c3}- r2n9{c1 c6}- r6n9{c6 c9}- r8n9{c9 .} => -9r3c7
whip[5]: r1n3{c9 c3}- r2n3{c1 c6}- r6n3{c6 c9}- r4n3{c9 c2}- r7n3{c2 .} => -3r3c7
Single(s): 4r3c7, 9r8c7, 4r2c1
Xwing in columns: 4c28r49 => -4r4c6 -4r4c9
Naked pairs: 39c6r24 => -3r5c6 -9r5c6 -3r6c6 -9r6c6
Xwing in rows: 9r16c39 => -9r4c9 -9r5c3
Hidden pairs: 49b6p29 => -3r4c8 -3r6c9
Xwing in columns: 9c68r24 => -9r4c2
whip[4]: r2c8{n3 n9}- r1n9{c9 c3}- c2n9{r3 r9}- r9c1{n9 .} => -3r9c8
STTE

[denis_berthier]

Tridagon (3,4,9) in b1p348, b3p357, b4p249, b6p249
with 5 guardians: 6r3c2,6r4c2,6r5c1,5r5c7,8r5c7
Note that :
-6r3c2 & -6r4c2 -> 6r3c1 -> 6r4c4 -> 8r4c7
So : -6r3c2 & -6r4c2 => -6r5c1 & -8r5c7
So just consider the 3 guardians 6r3c2, 6r4c2, 5r5c7.

Good idea to reduce the number of guardians before using them. How to rate such steps?

[denis_berthier]

.
Thanks for your solutions.

Contrary to other puzzles that require much cleaning before an anti-tridagon pattern appears, here we get one almost immediately:

Code: Select all

hidden-pairs-in-a-row: r5{n2 n7}{c8 c9} ==> r5c9≠9, r5c9≠4, r5c9≠3, r5c8≠9, r5c8≠5, r5c8≠4, r5c8≠3
whip[1]: c8n5{r9 .} ==> r8c7≠5
hidden-pairs-in-a-column: c3{n7 n8}{r2 r3} ==> r3c3≠9, r3c3≠6, r3c3≠4, r3c3≠3, r2c3≠9, r2c3≠4, r2c3≠3
   +-------------------+-------------------+-------------------+
   ! 1     2     349   ! 45    359   6     ! 7     8     349   !
   ! 349   5     78    ! 1478  389   1349  ! 2     349   6     !
   ! 3469  3469  78    ! 478   389   2     ! 349   1     5     !
   +-------------------+-------------------+-------------------+
   ! 2     3469  5     ! 468   7     349   ! 13489 349   1349  !
   ! 3469  1     3469  ! 4568  3589  3459  ! 34589 27    27    !
   ! 8     7     349   ! 2     1     3459  ! 3459  6     349   !
   +-------------------+-------------------+-------------------+
   ! 5     36    1236  ! 9     4     8     ! 13    237   1237  !
   ! 7     8     1249  ! 3     6     15    ! 149   2459  1249  !
   ! 349   349   1349  ! 15    2     7     ! 6     3459  8     !
   +-------------------+-------------------+-------------------+

OR5-anti-tridagon[12] for digits 3, 4 and 9 in blocks:
        b1, with cells: r1c3, r2c1, r3c2
        b3, with cells: r1c9, r2c8, r3c7
        b4, with cells: r6c3, r5c1, r4c2
        b6, with cells: r6c9, r5c7, r4c8
with 5 guardians: n6r3c2 n6r4c2 n6r5c1 n5r5c7 n8r5c7

In the simplest-first approach, because there are no short ORk-whips at this point, other rules apply before the pattern is used:

Code: Select all

biv-chain[3]: r1c4{n4 n5} - b8n5{r9c4 r8c6} - c6n1{r8 r2} ==> r2c6≠4
whip[1]: c6n4{r6 .} ==> r4c4≠4, r5c4≠4
biv-chain[4]: r7c7{n1 n3} - r7c2{n3 n6} - r4n6{c2 c4} - r4n8{c4 c7} ==> r4c7≠1
hidden-single-in-a-block ==> r4c9=1
whip[7]: r3n6{c1 c2} - r7c2{n6 n3} - r9c1{n3 n4} - b1n4{r3c1 r1c3} - r1c4{n4 n5} - r9n5{c4 c8} - r9n3{c8 .} ==> r3c1≠9
whip[8]: r9n9{c3 c8} - c8n5{r9 r8} - b8n5{r8c6 r9c4} - r1c4{n5 n4} - r1c3{n4 n3} - r6c3{n3 n4} - c9n4{r6 r8} - r8n2{c9 .} ==> r8c3≠9
whip[1]: r8n9{c9 .} ==> r9c8≠9
whip[8]: r6n5{c7 c6} - b8n5{r8c6 r9c4} - b8n1{r9c4 r8c6} - r8c7{n1 n4} - r9c8{n4 n3} - r4c8{n3 n4} - b3n4{r2c8 r1c9} - r1c4{n4 .} ==> r6c7≠9

Its turn has now come:
Trid-OR5-whip[8]: r7c7{n1 n3} - r7c2{n3 n6} - c3n6{r7 r5} - r4n6{c2 c4} - r4n8{c4 c7} - OR5{{n8r5c7 n6r5c1 n6r4c2 n6r3c2 | n5r5c7}} - b5n5{r5c4 r6c6} - r8c6{n5 .} ==> r8c7≠1
hidden-single-in-a-block ==> r7c7=1
Trid-OR5-whip[7]: c6n5{r6 r8} - r8n1{c6 c3} - c3n2{r8 r7} - c3n6{r7 r5} - c1n6{r5 r3} - OR5{{n6r5c1 n5r5c7 n6r4c2 n6r3c2 | n8r5c7}} - r5c4{n8 .} ==> r5c5≠5
singles ==> r1c5=5, r1c4=4
naked-pairs-in-a-row: r3{c3 c4}{n7 n8} ==> r3c5≠8
z-chain[7]: r8c7{n9 n4} - b3n4{r3c7 r2c8} - c8n9{r2 r8} - c8n5{r8 r9} - c4n5{r9 r5} - c4n6{r5 r4} - r4n8{c4 .} ==> r4c7≠9
whip[7]: r3n6{c1 c2} - r7c2{n6 n3} - r9n3{c3 c8} - r9c1{n3 n9} - c2n9{r9 r4} - r4c8{n9 n4} - b3n4{r2c8 .} ==> r3c1≠4
Trid-OR5-whip[5]: r3n4{c7 c2} - r3n6{c2 c1} - OR5{{n6r5c1 n8r5c7 n5r5c7 n6r3c2 | n6r4c2}} - r4c4{n6 n8} - c7n8{r4 .} ==> r5c7≠4
whip[6]: r1c3{n9 n3} - r3c1{n3 n6} - b4n6{r5c1 r4c2} - b4n3{r4c2 r5c1} - r5c5{n3 n8} - r4c4{n8 .} ==> r5c3≠9
Trid-OR5-whip[6]: r8c7{n9 n4} - r3n4{c7 c2} - r3n6{c2 c1} - OR5{{n6r5c1 n8r5c7 n5r5c7 n6r3c2 | n6r4c2}} - r4c4{n6 n8} - c7n8{r4 .} ==> r5c7≠9
Back to ordinary rules:

Code: Select all

biv-chain[3]: b6n9{r4c8 r6c9} - c9n4{r6 r8} - r8c7{n4 n9} ==> r8c8≠9
z-chain[3]: c8n9{r4 r2} - r1c9{n9 n3} - b9n3{r7c9 .} ==> r4c8≠3
whip[6]: r4c8{n9 n4} - r2n4{c8 c1} - c2n4{r3 r9} - c2n9{r9 r3} - r1n9{c3 c9} - b6n9{r6c9 .} ==> r4c6≠9
whip[6]: c7n8{r4 r5} - c7n5{r5 r6} - c7n3{r6 r3} - r3c5{n3 n9} - r5c5{n9 n3} - r4c6{n3 .} ==> r4c7≠4
whip[5]: r1n9{c3 c9} - r1n3{c9 c3} - r6c3{n3 n4} - b6n4{r6c9 r4c8} - c8n9{r4 .} ==> r9c3≠9
whip[5]: r5n4{c3 c6} - r4n4{c6 c8} - c8n9{r4 r2} - c6n9{r2 r6} - b6n9{r6c9 .} ==> r6c3≠4
naked-pairs-in-a-column: c3{r1 r6}{n3 n9} ==> r9c3≠3, r7c3≠3, r5c3≠3
whip[7]: r3n4{c2 c7} - r2n4{c8 c1} - b1n9{r2c1 r1c3} - b3n9{r1c9 r2c8} - r4n9{c8 c2} - r9n9{c2 c1} - b7n3{r9c1 .} ==> r3c2≠3
whip[7]: r4c8{n9 n4} - r2n4{c8 c1} - c2n4{r3 r9} - r9c3{n4 n1} - c4n1{r9 r2} - r2c6{n1 n3} - r4c6{n3 .} ==> r2c8≠9
hidden-single-in-a-column ==> r4c8=9
whip[1]: b6n4{r6c9 .} ==> r6c6≠4
biv-chain[3]: c1n6{r3 r5} - b4n9{r5c1 r6c3} - c3n3{r6 r1} ==> r3c1≠3
naked-single ==> r3c1=6

At least one candidate of a previous Trid-OR5-relation has just been eliminated.
There remains a Trid-OR4-relation between candidates: n6r3c2 n6r4c2 n5r5c7 n8r5c7
   +----------------+----------------+----------------+
   ! 1    2    39   ! 4    5    6    ! 7    8    39   !
   ! 349  5    78   ! 178  389  139  ! 2    34   6    !
   ! 6    469  78   ! 78   39   2    ! 349  1    5    !
   +----------------+----------------+----------------+
   ! 2    346  5    ! 68   7    34   ! 38   9    1    !
   ! 349  1    46   ! 568  389  3459 ! 358  27   27   !
   ! 8    7    39   ! 2    1    359  ! 345  6    34   !
   +----------------+----------------+----------------+
   ! 5    36   26   ! 9    4    8    ! 1    237  237  !
   ! 7    8    124  ! 3    6    15   ! 49   245  249  !
   ! 349  349  14   ! 15   2    7    ! 6    345  8    !
   +----------------+----------------+----------------+

At least one candidate of a previous Trid-OR4-relation has just been eliminated.
There remains a Trid-OR3-relation between candidates: n6r4c2 n5r5c7 n8r5c7

   +----------------+----------------+----------------+
   ! 1    2    39   ! 4    5    6    ! 7    8    39   !
   ! 349  5    78   ! 178  389  139  ! 2    34   6    !
   ! 6    49   78   ! 78   39   2    ! 349  1    5    !
   +----------------+----------------+----------------+
   ! 2    346  5    ! 68   7    34   ! 38   9    1    !
   ! 349  1    46   ! 568  389  3459 ! 358  27   27   !
   ! 8    7    39   ! 2    1    359  ! 345  6    34   !
   +----------------+----------------+----------------+
   ! 5    36   26   ! 9    4    8    ! 1    237  237  !
   ! 7    8    124  ! 3    6    15   ! 49   245  249  !
   ! 349  349  14   ! 15   2    7    ! 6    345  8    !
   +----------------+----------------+----------------+

And a final ORk-whip with only 3 guardians:
Trid-OR3-whip[3]: OR3{{n8r5c7 n5r5c7 | n6r4c2}} - r4n3{c2 c6} - r4n4{c6 .} ==> r5c7≠3
end in S2Fin+BC3

[Leren]

1. Phil's UR Type 3 : After basics:

Code: Select all

*--------------------------------------------------------*
| 1    2    349   | 45   359  6    | 7      8      349   |
| 349  5    78    | 1478 389  1349 | 2      349    6     |
| 3469 3469 78    | 478  389  2    | 349    1      5     |
|-----------------+----------------+---------------------|
| 2    3469 5     | 468  7    349  | 13489  349    1349  |
| 3469 1    3469  | 4568 3589 3459 | 34589 *27    *27    |
| 8    7    349   | 2    1    3459 | 3459   6      349   |
|-----------------+----------------+---------------------|
| 5    6-3  26-13 | 9    4    8    |*13    *27+3  *27+13 |
| 7    8    1249  | 3    6    15   | 49-1   2459   249-1 |
| 349  349  1349  | 15   2    7    | 6      459-3  8     |
*--------------------------------------------------------*

UR Type 3 => Quantum Naked Pair (13) r7c789 => - 3 r7c2, -13 r7c3, - 1 r8c79, - 3 r9c8; 7 placements

2. Trigadon as shown in Phil's post with 1 guardian => r5c7 = 5; stte

Leren

[denis_berthier]

OK, thanks, Leren. I had forgotten this UR case

[DEFISE]

Tridagon (3,4,9) in b1p348, b3p357, b4p249, b6p249
with 5 guardians: 6r3c2,6r4c2,6r5c1,5r5c7,8r5c7
Note that :
-6r3c2 & -6r4c2 -> 6r3c1 -> 6r4c4 -> 8r4c7
So : -6r3c2 & -6r4c2 => -6r5c1 & -8r5c7
So just consider the 3 guardians 6r3c2, 6r4c2, 5r5c7.

Good idea to reduce the number of guardians before using them. How to rate such steps?

This sequence of implications (-6r3c2 & -6r4c2 -> 6r3c1 -> 6r4c4 -> 8r4c7)
correspond to this partial-braid structure:
-6r3c2 & -6r4c2 => r3n6{c2 c1}- r4n6{c2 c4}- r4n8{c4 c7} => -6r5c1 & -8r5c7
I think its total length is 4, to be consistent with the length defined for a braid.
We can also use this partial-whip structure (of length 5) for the same conclusion:
-6r3c2 & -6r4c2 => r3n6{c2 c1}- b4n6{c1 c3}- c4n6{r5 r4}- r4n8{c4 c7} => -6r5c1 & -8r5c7

Another remark: My OR3-g-braid[6] can be replaced by this one:
OR3-g-braid[5] : r6n5{c6 c7}- c8n5{r8 r9}- r9n3{c8 c123}- r7c2{n3 n6}- OR3{all guardians | .} => -5r8c6
without changing the rest of the path.
N.B: without the preliminary implications, it is possible to directly eliminate this 5r8c6 with an OR5-g-braid[8]. Obviously these preliminary implications would have been more interesting if I had to use several ORk-whips, like you.