The New Sudoku Players' Forum

[denis_berthier]

.

Code: Select all

+-------+-------+-------+
! . 2 3 ! 4 . 6 ! . . . !
! . . . ! 1 8 . ! . . . !
! . . 9 ! . . 7 ! . . . !
+-------+-------+-------+
! 2 . . ! . . . ! 1 . 3 !
! 3 . 5 ! . 1 . ! 8 9 . !
! . . 1 ! . 3 . ! . 5 2 !
+-------+-------+-------+
! . . . ! . . . ! 3 . 8 !
! . 3 . ! 5 . 1 ! 9 2 . !
! 9 . 2 ! . . . ! . 1 5 !
+-------+-------+-------+
.234.6......18......9..7...2.....1.33.5.1.89...1.3..52......3.8.3.5.192.9.2....15;146;17266
SER = 11.7

Code: Select all

Resolution state after Singles and whips[1]:
   +-------------------+-------------------+-------------------+
   ! 1578  2     3     ! 4     59    6     ! 57    78    179   !
   ! 4567  4567  467   ! 1     8     2359  ! 24567 3467  4679  !
   ! 14568 14568 9     ! 23    25    7     ! 2456  3468  146   !
   +-------------------+-------------------+-------------------+
   ! 2     46789 4678  ! 6789  45679 4589  ! 1     467   3     !
   ! 3     467   5     ! 267   1     24    ! 8     9     467   !
   ! 4678  46789 1     ! 6789  3     489   ! 467   5     2     !
   +-------------------+-------------------+-------------------+
   ! 14567 14567 467   ! 2679  24679 249   ! 3     467   8     !
   ! 4678  3     4678  ! 5     467   1     ! 9     2     467   !
   ! 9     467   2     ! 3678  467   348   ! 467   1     5     !
   +-------------------+-------------------+-------------------+
179 candidates.

[totuan]

Code: Select all

 *--------------------------------------------------------------------*
 | 157-8  2      3      | 4      59     6      | 57     78     179    |
 | 4567   4567   467    | 1      8      2359   | 24567  3467   4679   |
 | 1456-8 14568  9      | 23     25     7      | 2456   3468   146    |
 |----------------------+----------------------+----------------------|
 | 2      46789 *467+8  | 6789   45679  4589   | 1     *467    3      |
 | 3     *467    5      | 267    1      24     | 8      9     *467    |
 |*467+8  46789  1      | 6789   3      489    |*467    5      2      |
 |----------------------+----------------------+----------------------|
 | 15     15    *467    | 2679   24679  249    | 3     *467    8      |
 |*467+8  3      4678   | 5      467    1      | 9      2     *467    |
 | 9     *467    2      | 38     467    38     |*467    1      5      |
 *--------------------------------------------------------------------*

My path for this one:
Tridagon (467) * marked cells => (8)r68c1=(8)r4c3
01: (8)r3c2=r13c1-r68c1==r4c3-r46c2=r3c2 => r3c2=8, some singles.

Code: Select all

 *--------------------------------------------------------------------*
 | 17     2      3      | 4      59     6      | 57     8      179    |
 | 467    5     *467    | 1      8      239    | 2467  *467+3  4679   |
 | 146    8      9      | 23     25     7      | 2456   346    146    |
 |----------------------+----------------------+----------------------|
 | 2      4679  *467+8  | 6789  A467+9  5      | 1     *467    3      |
 | 3      467    5      | 267    1      24     | 8      9      467    |
 | 467-8  4679   1      | 6789   3      489    | 467    5      2      |
 |----------------------+----------------------+----------------------|
 | 5      1     *467    | 2679   24679  249    | 3     *467    8      |
 |*467+8  3      4678   | 5     *467    1      | 9      2     *467    |
 | 9     *467    2      | 38    *467    38     |*467    1      5      |
 *--------------------------------------------------------------------*

Impossible pattern (467) * marked cells => (8)r4c3/r8c1=(3)r2c8=(9)r4c5
02: Present as diagram: => r6c1<>8, r4c3=8, r8c1=8.

Code: Select all

(8)r4c3/r8c1*
 ||
(9)r4c5-(59=2)r13c5-(2=3)r3c4--(3=8)r9c4-r4c4=r4c3*
 ||                           |
(3)r2c8-r2c6=r3c4-------------

Code: Select all

 *--------------------------------------------------------------------*
 | 17     2      3      | 4      59     6      | 57     8      179    |
 | 467    5      467    | 1      8      239    | 2467   3467   4679   |
 | 146    8      9      | 23     25     7      | 2456   346    146    |
 |----------------------+----------------------+----------------------|
 | 2     *4679   8      | 679   *467+9  5      | 1     *467    3      |
 | 3     *467    5      | 267    1      24     | 8      9     *467    |
 |*467   *4679   1      | 6789   3      489    |*467    5      2      |
 |----------------------+----------------------+----------------------|
 | 5      1      467    | 2679   24679  249    | 3     *467    8      |
 | 8      3     *467    | 5     *467    1      | 9      2     *467    |
 | 9     *467A   2      | 38    *467    38     |*467    1      5      |
 *--------------------------------------------------------------------*

03: Impossible pattern (467) * marked cells – like twin => r4c5=9, some singles then ER-6.6 and one simple move to finish.

Thanks for the puzzle!
totuan

[marek stefanik]

Bigger 3-digit pattern but less guardians.
After placing 8r3c2:

Code: Select all

.------------------.------------------.------------------.
| 17    2     3    | 4    c59     6   | 57    8     179  |
| 467   5    #467  | 1     8     d29–3|2467 a#3467  4679 |
| 146   8     9    | 23    25     7   | 2456  346   146  |
:------------------+------------------+------------------:
| 2    *4679 *4678 |6789 b#467+9  5   | 1    #467   3    |
| 3    *467   5    | 267   1      24  | 8     9    #467  |
|*4678 *4679  1    | 6789  3      489 |#467   5     2    |
:------------------+------------------+------------------:
| 5     1    *467  | 2679  24679  249 | 3    #467   8    |
|*4678  3    *4678 | 5    #467    1   | 9     2    #467  |
| 9    *467   2    | 38   #467    38  |#467   1     5    |
'------------------'------------------'------------------'

impossible pattern 467b47#, internals 3r2c8 9r4c5

Impossibility proof: Show

Code: Select all

+-------+-------+-------+
| . . . | . . . | . . . |
| . . # | . . . | . # . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . * * | . # . | . A . |
| . * . | . . . | . . B |
| * * . | . . . | C . . |
+-------+-------+-------+
| . . A | . . . | . A . |
| B . B | . # . | . . B |
| . C . | . # . | C . . |
+-------+-------+-------+

Note that r8c59 take b7p38 in some order. Therefore b7 has one of 467 in each of A-, B-, and C-marked cells.
The ABC permutations in b679 have the same parity, therefore either they form RTs or b67 have the same permutation.
The digit in r4c5 cannot appear in the A-marked cells (it then couldn't appear twice in r89), so the latter is the case.
Let a,b,c be the digits in A-, B-, and C-marked cells in b67, in this order.

Code: Select all

+-------+-------+-------+
| . . . | . . . | . . . |
| . . # | . . . | . # . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . * c | . # . | . a . |
| . * . | . . . | . . b |
| * * . | . . . | c . . |
+-------+-------+-------+
| . . a | . . . | . c . |
| # . # | . c . | . . # |
| . c . | . # . | # . . |
+-------+-------+-------+

Now c in b4 takes r4c3, in c5 r8c5, and in b9 r7c8, therefore r2c38 are both restricted to b, i.e. contra.

3r2c8 = 9r4c5 – 9r1c5 = 9r2c6 => –3r2c6

After basics (roughly the same as totuan's third step):

Code: Select all

.----------------.------------------.-----------------.
| 17   2     3   | 4     59     6   | 57    8    179  |
| 467  5     467 | 1     8      29  | 2467  3    4679 |
| 146  8     9   | 3     25     7   | 25    46   146  |
:----------------+------------------+-----------------:
| 2   *4679  8   | 679   9–467  5   | 1    A467  3    |
| 3   *467   5   | 267   1      24  | 8     9   B467  |
|*467 *4679  1   | 679   3      8   |C467   5    2    |
:----------------+------------------+-----------------:
| 5    1    A467 |*2679 *24679 *249 | 3    A467  8    |
| 8    3    B467 | 5    *467    1   | 9     2   B467  |
| 9   C467   2   | 8    *467    3   |C467   1    5    |
'----------------'------------------'-----------------'

C-marked cells are a RT (Cb67 see each other via b4), so must be A- and B-marked cells.
467b8A \ r47c5 => –467r4c5, then 6.6 skfr

Marek

[denis_berthier]

.
Thanks for your solution. Mine is based on different impossible patterns.

hidden-pairs-in-a-row: r9{n3 n8}{c4 c6} ==> r9c6≠4, r9c4≠7, r9c4≠6
hidden-pairs-in-a-row: r7{n1 n5}{c1 c2} ==> r7c2≠7, r7c2≠6, r7c2≠4, r7c1≠7, r7c1≠6, r7c1≠4

The easy tridagon part:

Code: Select all

Trid-OR3-relation for digits 4, 6 and 7 in blocks:
        b4, with cells (marked #): r4c3, r5c2, r6c1
        b6, with cells (marked #): r4c8, r5c9, r6c7
        b7, with cells (marked #): r7c3, r9c2, r8c1
        b9, with cells (marked #): r7c8, r9c7, r8c9
with 3 guardians (in cells marked @): n8r4c3 n8r6c1 n8r8c1
   +----------------------+----------------------+----------------------+
   ! 1578   2      3      ! 4      59     6      ! 57     78     179    !
   ! 4567   4567   467    ! 1      8      2359   ! 24567  3467   4679   !
   ! 14568  14568  9      ! 23     25     7      ! 2456   3468   146    !
   +----------------------+----------------------+----------------------+
   ! 2      46789  4678#@ ! 6789   45679  4589   ! 1      467#   3      !
   ! 3      467#   5      ! 267    1      24     ! 8      9      467#   !
   ! 4678#@ 46789  1      ! 6789   3      489    ! 467#   5      2      !
   +----------------------+----------------------+----------------------+
   ! 15     15     467#   ! 2679   24679  249    ! 3      467#   8      !
   ! 4678#@ 3      4678   ! 5      467    1      ! 9      2      467#   !
   ! 9      467#   2      ! 38     467    38     ! 467#   1      5      !
   +----------------------+----------------------+----------------------+

Trid-OR3-relation between candidates n8r4c3, n8r6c1 and n8r8c1
+ same valence for candidates n8r8c1 and n8r4c3 via c-chain[2]: n8r8c1,n8r8c3,n8r4c3
==> Trid-OR3-relation can be split into two Trid-OR2-relations with respective lists of guardians:
    n8r4c3 n8r6c1  and n8r6c1 n8r8c1 .

Trid-OR2-whip[1]: OR2{{n8r6c1 n8r4c3 | .}} ==> r6c2≠8
Trid-OR2-whip[1]: OR2{{n8r6c1 n8r4c3 | .}} ==> r4c2≠8
singles ==> r3c2=8, r1c8=8, r7c2=1, r7c1=5, r2c2=5, r4c6=5


Three other impossible patterns, with the numbers of guardians immediately reduced by ORk-splitting rules.
Remember that SudoRules can only identify non-degenerate patterns and therefore has to identify them as soon as possible.

Code: Select all

EL10c28-OR5-relation for digits: 4, 6 and 7
   in cells (marked #): (r7c5 r7c8 r9c2 r9c5 r9c7 r6c2 r5c2 r4c3 r4c5 r4c8)
   with 5 guardians (in cells marked @) : n2r7c5 n9r7c5 n9r6c2 n8r4c3 n9r4c5
   +-------------------------+-------------------------+-------------------------+
   ! 17      2       3       ! 4       59      6       ! 57      8       179     !
   ! 467     5       467     ! 1       8       239     ! 2467    3467    4679    !
   ! 146     8       9       ! 23      25      7       ! 2456    346     146     !
   +-------------------------+-------------------------+-------------------------+
   ! 2       4679    4678#@  ! 6789    4679#@  5       ! 1       467#    3       !
   ! 3       467#    5       ! 267     1       24      ! 8       9       467     !
   ! 4678    4679#@  1       ! 6789    3       489     ! 467     5       2       !
   +-------------------------+-------------------------+-------------------------+
   ! 5       1       467     ! 2679    24679#@ 249     ! 3       467#    8       !
   ! 4678    3       4678    ! 5       467     1       ! 9       2       467     !
   ! 9       467#    2       ! 38      467#    38      ! 467#    1       5       !
   +-------------------------+-------------------------+-------------------------+


EL13c175-OR4-relation for digits: 4, 6 and 7
   in cells (marked #): (r8c5 r8c9 r8c3 r9c5 r9c7 r9c2 r5c9 r5c2 r6c7 r6c1 r4c5 r4c8 r4c2)
   with 4 guardians (in cells marked @) : n8r8c3 n8r6c1 n9r4c5 n9r4c2
   +----------------------+----------------------+----------------------+
   ! 17     2      3      ! 4      59     6      ! 57     8      179    !
   ! 467    5      467    ! 1      8      239    ! 2467   3467   4679   !
   ! 146    8      9      ! 23     25     7      ! 2456   346    146    !
   +----------------------+----------------------+----------------------+
   ! 2      4679#@ 4678   ! 6789   4679#@ 5      ! 1      467#   3      !
   ! 3      467#   5      ! 267    1      24     ! 8      9      467#   !
   ! 4678#@ 4679   1      ! 6789   3      489    ! 467#   5      2      !
   +----------------------+----------------------+----------------------+
   ! 5      1      467    ! 2679   24679  249    ! 3      467    8      !
   ! 4678   3      4678#@ ! 5      467#   1      ! 9      2      467#   !
   ! 9      467#   2      ! 38     467#   38     ! 467#   1      5      !
   +----------------------+----------------------+----------------------+

EL13c175-OR4-relation between candidates n8r8c3, n8r6c1, n9r4c5 and n9r4c2
+ same valence for candidates n8r8c3 and n8r6c1 via c-chain[2]: n8r8c3,n8r8c1,n8r6c1
==> EL13c175-OR4-relation can be split into two EL13c175-OR3-relations with respective lists of guardians:
    n8r6c1 n9r4c5 n9r4c2  and n8r8c3 n9r4c5 n9r4c2 .

EL13c175-OR3-whip[2]: OR3{{n9r4c2 n9r4c5 | n8r6c1}} - b5n8{r6c4 .} ==> r4c4≠9

Code: Select all

EL10c28-OR5-relation between candidates n2r7c5, n9r7c5, n9r6c2, n8r4c3 and n9r4c5
+ same valence for candidates n9r4c5 and n9r6c2 via c-chain[2]: n9r4c5,n9r4c2,n9r6c2
==> EL10c28-OR5-relation can be split into two EL10c28-OR4-relations with respective lists of guardians:
    n2r7c5 n9r7c5 n9r6c2 n8r4c3  and n2r7c5 n9r7c5 n8r4c3 n9r4c5 .


EL14c93-OR4-relation for digits: 4, 6 and 7
   in cells (marked #): (r7c8 r9c5 r9c2 r9c7 r8c5 r8c3 r8c9 r5c2 r5c9 r6c1 r6c2 r6c7 r4c5 r4c8)
   with 4 guardians (in cells marked @) : n8r8c3 n8r6c1 n9r6c2 n9r4c5
   +----------------------+----------------------+----------------------+
   ! 17     2      3      ! 4      59     6      ! 57     8      179    !
   ! 467    5      467    ! 1      8      239    ! 2467   3467   4679   !
   ! 146    8      9      ! 23     25     7      ! 2456   346    146    !
   +----------------------+----------------------+----------------------+
   ! 2      4679   4678   ! 678    4679#@ 5      ! 1      467#   3      !
   ! 3      467#   5      ! 267    1      24     ! 8      9      467#   !
   ! 4678#@ 4679#@ 1      ! 6789   3      489    ! 467#   5      2      !
   +----------------------+----------------------+----------------------+
   ! 5      1      467    ! 2679   24679  249    ! 3      467#   8      !
   ! 4678   3      4678#@ ! 5      467#   1      ! 9      2      467#   !
   ! 9      467#   2      ! 38     467#   38     ! 467#   1      5      !
   +----------------------+----------------------+----------------------+

EL14c93-OR4-relation between candidates n8r8c3, n8r6c1, n9r6c2 and n9r4c5
+ same valence for candidates n8r8c3 and n8r6c1 via c-chain[2]: n8r8c3,n8r8c1,n8r6c1
==> EL14c93-OR4-relation can be split into two EL14c93-OR3-relations with respective lists of guardians:
    n8r6c1 n9r6c2 n9r4c5  and n8r8c3 n9r6c2 n9r4c5 .

EL14c93-OR3-relation between candidates n8r8c3, n9r6c2 and n9r4c5
+ same valence for candidates n9r6c2 and n9r4c5 via c-chain[2]: n9r6c2,n9r4c2,n9r4c5
==> EL14c93-OR3-relation can be split into two EL14c93-OR2-relations with respective lists of guardians:
    n8r8c3 n9r4c5  and n8r8c3 n9r6c2 .

EL14c93-OR3-relation between candidates n8r6c1, n9r6c2 and n9r4c5
+ same valence for candidates n9r6c2 and n9r4c5 via c-chain[2]: n9r6c2,n9r4c2,n9r4c5
==> EL14c93-OR3-relation can be split into two EL14c93-OR2-relations with respective lists of guardians:
    n8r6c1 n9r4c5  and n8r6c1 n9r6c2 .

biv-chain[3]: r1c7{n7 n5} - r1c5{n5 n9} - b3n9{r1c9 r2c9} ==> r2c9≠7


Now come three EL14c93-OR2-whips[4], with the same EL14c93-OR2-relation at different places:

EL14c93-OR2-ctr-whip[4]: c5n7{r9 r4} - c8n7{r4 r2} - c3n7{r2 r8} - OR2{{n9r4c5 n8r8c3 | .}} ==> r7c4≠7
whip[1]: b8n7{r9c5 .} ==> r4c5≠7
EL14c93-OR2-whip[4]: b2n9{r2c6 r1c5} - OR2{{n9r4c5 | n8r6c1}} - c6n8{r6 r9} - c6n3{r9 .} ==> r2c6≠2
hidden-single-in-a-row ==> r2c7=2
EL14c93-OR2-whip[4]: OR2{{n9r4c5 | n8r6c1}} - c6n8{r6 r9} - c6n3{r9 r2} - b2n9{r2c6 .} ==> r7c5≠9

Note that:
- until now, we have used only very short chains and ORk-chains;
- at this point, the puzzle is in W7 and could be solved without using again any impossible pattern;
- FAPP, the puzzle could be considered as solved;
but why stop the fun here?

Code: Select all

Resolution state RS2:
   +----------------+----------------+----------------+
   ! 17   2    3    ! 4    59   6    ! 57   8    179  !
   ! 467  5    467  ! 1    8    39   ! 2    3467 469  !
   ! 146  8    9    ! 23   25   7    ! 456  346  146  !
   +----------------+----------------+----------------+
   ! 2    4679 4678 ! 678  469  5    ! 1    467  3    !
   ! 3    467  5    ! 267  1    24   ! 8    9    467  !
   ! 4678 4679 1    ! 6789 3    489  ! 467  5    2    !
   +----------------+----------------+----------------+
   ! 5    1    467  ! 269  2467 249  ! 3    467  8    !
   ! 4678 3    4678 ! 5    467  1    ! 9    2    467  !
   ! 9    467  2    ! 38   467  38   ! 467  1    5    !
   +----------------+----------------+----------------+
At least one candidate of a previous EL10c28-OR4-relation between candidates n2r7c5 n9r7c5 n9r6c2 n8r4c3 has just been eliminated.
There remains an EL10c28-OR3-relation between candidates: n2r7c5 n9r6c2 n8r4c3

biv-chain[3]: r7n9{c4 c6} - r2c6{n9 n3} - r3c4{n3 n2} ==> r7c4≠2
EL14c93-OR2-whip[4]: OR2{{n8r6c1 | n9r4c5}} - b2n9{r1c5 r2c6} - c6n3{r2 r9} - r9n8{c6 .} ==> r6c4≠8
EL14c93-OR2-whip[5]: OR2{{n8r6c1 | n9r4c5}} - r1c5{n9 n5} - r3n5{c5 c7} - c7n4{r3 r9} - c2n4{r9 .} ==> r6c1≠4
EL14c93-OR2-whip[5]: OR2{{n8r6c1 | n9r4c5}} - r1c5{n9 n5} - r3n5{c5 c7} - c7n6{r3 r9} - c2n6{r9 .} ==> r6c1≠6
t-whip[3]: c3n8{r8 r4} - r6c1{n8 n7} - b1n7{r1c1 .} ==> r8c3≠7
z-chain[4]: r7c4{n6 n9} - r6c4{n9 n7} - r6c1{n7 n8} - b5n8{r6c6 .} ==> r4c4≠6
biv-chain[5]: r4c4{n7 n8} - c6n8{r6 r9} - c6n3{r9 r2} - b2n9{r2c6 r1c5} - r4n9{c5 c2} ==> r4c2≠7
whip[6]: c5n2{r7 r3} - r3c4{n2 n3} - r9c4{n3 n8} - r4c4{n8 n7} - c3n7{r4 r2} - c8n7{r2 .} ==> r7c5≠7
z-chain[5]: r7n7{c8 c3} - r4n7{c3 c4} - r4n8{c4 c3} - r6c1{n8 n7} - r1n7{c1 .} ==> r2c8≠7
whip[1]: r2n7{c3 .} ==> r1c1≠7
naked-single ==> r1c1=1
hidden-single-in-a-block ==> r3c9=1
t-whip[5]: b1n7{r2c1 r2c3} - r7n7{c3 c8} - r4n7{c8 c4} - b5n8{r4c4 r6c6} - r6c1{n8 .} ==> r8c1≠7
biv-chain[6]: c8n7{r7 r4} - r4c4{n7 n8} - c6n8{r6 r9} - c6n3{r9 r2} - r2n9{c6 c9} - r1c9{n9 n7} ==> r8c9≠7
hidden-single-in-a-row ==> r8c5=7
whip[5]: r8n6{c3 c9} - r5n6{c9 c4} - r5n2{c4 c6} - r7n2{c6 c5} - r7n6{c5 .} ==> r9c2≠6
whip[1]: c2n6{r6 .} ==> r4c3≠6
z-chain[3]: b8n4{r7c6 r9c5} - r9c2{n4 n7} - b9n7{r9c7 .} ==> r7c8≠4
EL10c28-OR3-whip[4]: r9c2{n7 n4} - b4n4{r4c2 r4c3} - OR3{{n8r4c3 n9r6c2 | n2r7c5}} - c5n4{r7 .} ==> r6c2≠7
EL14c93-OR2-whip[5]: r8n8{c3 c1} - r6n8{c1 c6} - OR2{{n8r6c1 | n9r6c2}} - r6n4{c2 c7} - b9n4{r9c7 .} ==> r8c3≠4
finned-x-wing-in-rows: n4{r8 r3}{c1 c9} ==> r2c9≠4
biv-chain[3]: r2c9{n6 n9} - r2c6{n9 n3} - b3n3{r2c8 r3c8} ==> r3c8≠6
EL14c93-OR2-ctr-whip[4]: c5n6{r9 r4} - c8n6{r4 r2} - c3n6{r2 r8} - OR2{{n8r8c3 n9r4c5 | .}} ==> r7c4≠6
naked-single ==> r7c4=9
whip[1]: b8n6{r9c5 .} ==> r4c5≠6
naked-pairs-in-a-column: c6{r5 r7}{n2 n4} ==> r6c6≠4
biv-chain[3]: r4c5{n4 n9} - c2n9{r4 r6} - r6n4{c2 c7} ==> r4c8≠4
whip[1]: c8n4{r3 .} ==> r3c7≠4
naked-pairs-in-a-column: c8{r4 r7}{n6 n7} ==> r2c8≠6
finned-swordfish-in-rows: n4{r9 r6 r4}{c5 c7 c2} ==> r5c2≠4
biv-chain[3]: c7n4{r6 r9} - r8c9{n4 n6} - b3n6{r2c9 r3c7} ==> r6c7≠6
biv-chain[3]: r3c1{n4 n6} - c7n6{r3 r9} - b9n4{r9c7 r8c9} ==> r8c1≠4
stte