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[Leren]

Code: Select all

*-----------*
|.8.|...|...|
|3..|82.|.4.|
|..2|.5.|...|
|---+---+---|
|...|.9.|...|
|.6.|3..|.2.|
|..3|..1|5..|
|---+---+---|
|..8|43.|.6.|
|4..|...|..7|
|6..|..9|.8.|
*-----------*
.8.......3..82..4...2.5........9.....6.3...2...3..15....843..6.4.......76....9.8.   

[marek stefanik]

Code: Select all

                                           /5 7?    /19 7?          /159
+------------------------+------------------------+------------------------+
| 1579B   8       145679 | 1679a   1467    3467   | 123679  13579   123569 |
| 3       1579T   15679  | 8       2       67     | 1679    4       1569   |  /7?
| 179B    1479    2      | 1679a   5       3467   | 136789  1379    13689  |
+------------------------+------------------------+------------------------+
| 12578   12457   1457   | 2567    9       245678 | 134678  137     13468  |
| 15789T  6       14579T | 3       478*    4578   | 14789   2       1489   |  /7?
| 2789    2479    3      | 267     4678    1      | 5       79      4689   |
+------------------------+------------------------+------------------------+
| 12579   12579T  8      | 4       3       257*   | 129     6       1259   |
| 4       12359   159B   | 1256b   168     2568   | 1239    1359    7      |
| 6       12357   157B   | 1257b   17c     9      | 1234    8       12345  |
+------------------------+------------------------+------------------------+

Almost Double JE: (1579) r13c1 r5c3 r7c2, (1579) r89c3 r5c1 r2c2
cover houses of the digit 7: r25r7c6 or c67r5c5

Kraken (7b2):
(79–1)r13c4 = r89c4 – (1=7)r9c5 – r5c5 = DJE
r1c5 – r5c5 = DJE
r123c6 – r7c6 = DJE

Double JE has been proven, except for cover-house eliminations on 7 (as there are two possible sets without common eliminations).
stte

Marek

[denis_berthier]

.
SER 10.7
It is neither in T&E(1) nor in gT&E(1) — i.e. it can't be solved by braids or g-braids alone.
It is not in T&E(S2) — i.e. it can't be solved by S2-braids alone.
It it in T&E(B2), the immediate next level after gT&E(1) — i.e. it can be solved by B2-braids (not coded in SudoRules, because of combinatorial explosion and because they would be illegible anyway).

Code: Select all

Resolution state after Singles and whips[1]:
   +----------------------+----------------------+----------------------+
   ! 1579   8      145679 ! 1679   1467   3467   ! 123679 13579  123569 !
   ! 3      1579   15679  ! 8      2      67     ! 1679   4      1569   !
   ! 179    1479   2      ! 1679   5      3467   ! 136789 1379   13689  !
   +----------------------+----------------------+----------------------+
   ! 12578  12457  1457   ! 2567   9      245678 ! 134678 137    13468  !
   ! 15789  6      14579  ! 3      478    4578   ! 14789  2      1489   !
   ! 2789   2479   3      ! 267    4678   1      ! 5      79     4689   !
   +----------------------+----------------------+----------------------+
   ! 12579  12579  8      ! 4      3      257    ! 129    6      1259   !
   ! 4      12359  159    ! 1256   168    2568   ! 1239   1359   7      !
   ! 6      12357  157    ! 1257   17     9      ! 1234   8      12345  !
   +----------------------+----------------------+----------------------+

whip[10]: r9c5{n1 n7} - r9c3{n7 n5} - r9c4{n5 n2} - r7c6{n2 n5} - c4n5{r8 r4} - c2n5{r4 r2} - c9n5{r2 r1} - r1n2{c9 c7} - r8n2{c7 c2} - c2n3{r8 .} ==> r9c2 ≠ 1
g-whip[13]: r9c5{n1 n7} - r9c3{n7 n5} - r9c4{n5 n2} - r7c6{n2 n5} - r5n5{c6 c1} - c2n5{r4 r2} - c9n5{r2 r1} - c9n2{r1 r7} - r7n1{c9 c123} - r8c3{n1 n9} - r7n9{c2 c7} - r5n9{c7 c9} - r2n9{c9 .} ==> r9c9 ≠ 1
g-whip[13]: r9c5{n1 n7} - r9c3{n7 n5} - r9c4{n5 n2} - r7c6{n2 n5} - r5n5{c6 c1} - c2n5{r4 r2} - c9n5{r2 r1} - c9n2{r1 r7} - r7n1{c9 c123} - r8c3{n1 n9} - r7n9{c2 c7} - r5n9{c7 c9} - r2n9{c9 .} ==> r9c7 ≠ 1

The last 2 eliminations can also be obtained by braids:
whip[10]: r9c5{n1 n7} - r9c3{n7 n5} - r9c4{n5 n2} - r7c6{n2 n5} - c4n5{r8 r4} - c2n5{r4 r2} - c9n5{r2 r1} - r1n2{c9 c7} - r8n2{c7 c2} - c2n3{r8 .} ==> r9c2 ≠ 1
braid[13]: r9c5{n1 n7} - r9c3{n7 n5} - r9c4{n5 n2} - r7c6{n2 n5} - r5n5{c6 c1} - c2n5{r4 r2} - c9n5{r2 r1} - r1n2{c9 c7} - r8n2{c7 c2} - r7c7{n1 n9} - b7n9{r7c1 r8c3} - r2n9{c2 c9} - r5n9{c9 .} ==> r9c9 ≠ 1
braid[13]: r9c5{n1 n7} - r9c3{n7 n5} - r9c4{n5 n2} - r7c6{n2 n5} - r5n5{c6 c1} - c2n5{r4 r2} - c9n5{r2 r1} - r1n2{c9 c7} - r8n2{c7 c2} - r7c7{n1 n9} - b7n9{r7c1 r8c3} - r2n9{c2 c9} - r5n9{c9 .} ==> r9c7 ≠ 1


Sudoku Explainer has the same first 3 eliminations as SudoRules (with the 2nd and 3rd exchanged).
Notice that I said "same elimination", not "same rule". The SE view of chains as chains of inferences makes them totally illegible and it makes it impossible to write them in shorter ways.

Dynamic Contradiction Forcing Chains
With this solving technique, we will prove the two following assertions:
If R9C2 contains the value 1, then R1C9 must contain the value 5
If R9C2 contains the value 1, then R1C9 cannot contain the value 5
Because the same assumption yields to contradictory results, we can conclude that the assumption is false, that is, R9C2 cannot contain the value 1.
Each assertion is proved by a different chain of simple rules. The chains can be dynamic, which means that the conclusions of multiple sub-chains must be combined in some cases.
The details of each chain are given below. Use the view selector below the grid to switch between the graphical illustrations of the two different chains.
Chain 1: If R9C2 contains the value 1, then R1C9 cannot contain the value 5 (View 1): (1) If R9C2 contains the value 1, then R9C5 cannot contain the value 1 (the value can occur only once in the row) (2) If R9C5 does not contain the value 1, then R9C5 must contain the value 7 (only remaining possible value in the cell) (3) If R9C5 contains the value 7, then R9C3 cannot contain the value 7 (the value can occur only once in the row) (4) If R9C2 contains the value 1 (initial assumption), then R9C3 cannot contain the value 1 (the value can occur only once in the block) (5) If R9C3 does not contain the value 1 and R9C3 does not contain the value 7 (3), then R9C3 must contain the value 5 (only remaining possible value in the cell) (6) If R9C3 contains the value 5, then R9C4 cannot contain the value 5 (the value can occur only once in the row) (7) If R9C2 contains the value 1 (initial assumption), then R9C4 cannot contain the value 1 (the value can occur only once in the row) (8) If R9C5 contains the value 7 (2), then R9C4 cannot contain the value 7 (the value can occur only once in the block) (9) If R9C4 does not contain the value 7, R9C4 does not contain the value 1 (7) and R9C4 does not contain the value 5 (6), then R9C4 must contain the value 2 (only remaining possible value in the cell) (10) If R9C4 contains the value 2, then R7C6 cannot contain the value 2 (the value can occur only once in the block) (11) If R9C5 contains the value 7 (2), then R7C6 cannot contain the value 7 (the value can occur only once in the block) (12) If R7C6 does not contain the value 7 and R7C6 does not contain the value 2 (10), then R7C6 must contain the value 5 (only remaining possible value in the cell) (13) If R7C6 contains the value 5, then R5C6 cannot contain the value 5 (the value can occur only once in the column) (14) If R9C3 contains the value 5 (5), then R5C3 cannot contain the value 5 (the value can occur only once in the column) (15) If R5C3 does not contain the value 5 and R5C6 does not contain the value 5 (13), then R5C1 must contain the value 5 (only remaining possible position in the row) (16) If R5C1 contains the value 5, then R1C1 cannot contain the value 5 (the value can occur only once in the column) (17) If R9C3 contains the value 5 (5), then R1C3 cannot contain the value 5 (the value can occur only once in the column) (18) If R7C6 contains the value 5 (12), then R8C6 cannot contain the value 5 (the value can occur only once in the block) (19) If R9C2 contains the value 1 (initial assumption), then R9C2 cannot contain the value 3 (the cell can contain only one value) (20) If R9C2 does not contain the value 3, then R8C2 must contain the value 3 (only remaining possible position in the block) (21) If R8C2 contains the value 3, then R8C2 cannot contain the value 5 (the cell can contain only one value) (22) If R9C3 contains the value 5 (5), then R8C3 cannot contain the value 5 (the value can occur only once in the block) (23) If R7C6 contains the value 5 (12), then R8C4 cannot contain the value 5 (the value can occur only once in the block) (24) If R8C4 does not contain the value 5, R8C3 does not contain the value 5 (22), R8C2 does not contain the value 5 (21) and R8C6 does not contain the value 5 (18), then R8C8 must contain the value 5 (only remaining possible position in the row) (25) If R8C8 contains the value 5, then R1C8 cannot contain the value 5 (the value can occur only once in the column) (26) If R1C8 does not contain the value 5, R1C3 does not contain the value 5 (17) and R1C1 does not contain the value 5 (16), then R1C9 must contain the value 5 (only remaining possible position in the row)
Chain 2: If R1C9 must contain the value 5, then R1C9 cannot contain the value 5 (View 2): (1) If R9C2 contains the value 1, then R9C5 cannot contain the value 1 (the value can occur only once in the row) (2) If R9C5 does not contain the value 1, then R9C5 must contain the value 7 (only remaining possible value in the cell) (3) If R9C5 contains the value 7, then R9C3 cannot contain the value 7 (the value can occur only once in the row) (4) If R9C2 contains the value 1 (initial assumption), then R9C3 cannot contain the value 1 (the value can occur only once in the block) (5) If R9C3 does not contain the value 1 and R9C3 does not contain the value 7 (3), then R9C3 must contain the value 5 (only remaining possible value in the cell) (6) If R9C3 contains the value 5, then R9C4 cannot contain the value 5 (the value can occur only once in the row) (7) If R9C2 contains the value 1 (initial assumption), then R9C4 cannot contain the value 1 (the value can occur only once in the row) (8) If R9C5 contains the value 7 (2), then R9C4 cannot contain the value 7 (the value can occur only once in the block) (9) If R9C4 does not contain the value 7, R9C4 does not contain the value 1 (7) and R9C4 does not contain the value 5 (6), then R9C4 must contain the value 2 (only remaining possible value in the cell) (10) If R9C4 contains the value 2, then R8C6 cannot contain the value 2 (the value can occur only once in the block) (11) If R9C2 contains the value 1 (initial assumption), then R9C2 cannot contain the value 3 (the cell can contain only one value) (12) If R9C2 does not contain the value 3, then R8C2 must contain the value 3 (only remaining possible position in the block) (13) If R8C2 contains the value 3, then R8C2 cannot contain the value 2 (the cell can contain only one value) (14) If R9C4 contains the value 2 (9), then R8C4 cannot contain the value 2 (the value can occur only once in the block) (15) If R8C4 does not contain the value 2, R8C2 does not contain the value 2 (13) and R8C6 does not contain the value 2 (10), then R8C7 must contain the value 2 (only remaining possible position in the row) (16) If R8C7 contains the value 2, then R7C9 cannot contain the value 2 (the value can occur only once in the block) (17) If R9C4 contains the value 2 (9), then R9C9 cannot contain the value 2 (the value can occur only once in the row) (18) If R9C9 does not contain the value 2 and R7C9 does not contain the value 2 (16), then R1C9 must contain the value 2 (only remaining possible position in the column) (19) If R1C9 contains the value 2, then R1C9 cannot contain the value 5 (the cell can contain only one value)

Dynamic Contradiction Forcing Chains
With this solving technique, we will prove the two following assertions:
If R9C7 contains the value 1, then R5C9 must contain the value 9
If R9C7 contains the value 1, then R5C9 cannot contain the value 9
Because the same assumption yields to contradictory results, we can conclude that the assumption is false, that is, R9C7 cannot contain the value 1.
Each assertion is proved by a different chain of simple rules. The chains can be dynamic, which means that the conclusions of multiple sub-chains must be combined in some cases.
The details of each chain are given below. Use the view selector below the grid to switch between the graphical illustrations of the two different chains.
Chain 1: If R9C7 contains the value 1, then R5C9 cannot contain the value 9 (View 1): (1) If R9C7 contains the value 1, then R9C7 cannot contain the value 4 (the cell can contain only one value) (2) If R9C7 does not contain the value 4, then R9C9 must contain the value 4 (only remaining possible position in the block) (3) If R9C9 contains the value 4, then R9C9 cannot contain the value 3 (the cell can contain only one value) (4) If R9C7 contains the value 1 (initial assumption), then R9C7 cannot contain the value 3 (the cell can contain only one value) (5) If R9C7 does not contain the value 3 and R9C9 does not contain the value 3 (3), then R9C2 must contain the value 3 (only remaining possible position in the row) (6) If R9C2 contains the value 3, then R9C2 cannot contain the value 2 (the cell can contain only one value) (7) If R9C7 contains the value 1 (initial assumption), then R9C7 cannot contain the value 2 (the cell can contain only one value) (8) If R9C9 contains the value 4 (2), then R9C9 cannot contain the value 2 (the cell can contain only one value) (9) If R9C9 does not contain the value 2, R9C7 does not contain the value 2 (7) and R9C2 does not contain the value 2 (6), then R9C4 must contain the value 2 (only remaining possible position in the row) (10) If R9C4 contains the value 2, then R7C6 cannot contain the value 2 (the value can occur only once in the block) (11) If R9C7 contains the value 1 (initial assumption), then R9C5 cannot contain the value 1 (the value can occur only once in the row) (12) If R9C5 does not contain the value 1, then R9C5 must contain the value 7 (only remaining possible value in the cell) (13) If R9C5 contains the value 7, then R7C6 cannot contain the value 7 (the value can occur only once in the block) (14) If R7C6 does not contain the value 7 and R7C6 does not contain the value 2 (10), then R7C6 must contain the value 5 (only remaining possible value in the cell) (15) If R7C6 contains the value 5, then R5C6 cannot contain the value 5 (the value can occur only once in the column) (16) If R9C5 contains the value 7 (12), then R9C3 cannot contain the value 7 (the value can occur only once in the row) (17) If R9C7 contains the value 1 (initial assumption), then R9C3 cannot contain the value 1 (the value can occur only once in the row) (18) If R9C3 does not contain the value 1 and R9C3 does not contain the value 7 (16), then R9C3 must contain the value 5 (only remaining possible value in the cell) (19) If R9C3 contains the value 5, then R5C3 cannot contain the value 5 (the value can occur only once in the column) (20) If R5C3 does not contain the value 5 and R5C6 does not contain the value 5 (15), then R5C1 must contain the value 5 (only remaining possible position in the row) (21) If R5C1 contains the value 5, then R1C1 cannot contain the value 5 (the value can occur only once in the column) (22) If R9C3 contains the value 5 (18), then R1C3 cannot contain the value 5 (the value can occur only once in the column) (23) If R7C6 contains the value 5 (14), then R8C6 cannot contain the value 5 (the value can occur only once in the block) (24) If R9C3 contains the value 5 (18), then R8C2 cannot contain the value 5 (the value can occur only once in the block) (25) If R9C3 contains the value 5 (18), then R8C3 cannot contain the value 5 (the value can occur only once in the block) (26) If R7C6 contains the value 5 (14), then R8C4 cannot contain the value 5 (the value can occur only once in the block) (27) If R8C4 does not contain the value 5, R8C3 does not contain the value 5 (25), R8C2 does not contain the value 5 (24) and R8C6 does not contain the value 5 (23), then R8C8 must contain the value 5 (only remaining possible position in the row) (28) If R8C8 contains the value 5, then R1C8 cannot contain the value 5 (the value can occur only once in the column) (29) If R1C8 does not contain the value 5, R1C3 does not contain the value 5 (22) and R1C1 does not contain the value 5 (21), then R1C9 must contain the value 5 (only remaining possible position in the row) (30) If R1C9 contains the value 5, then R1C9 cannot contain the value 2 (the cell can contain only one value) (31) If R1C9 does not contain the value 2, then R1C7 must contain the value 2 (only remaining possible position in the block) (32) If R1C7 contains the value 2, then R7C7 cannot contain the value 2 (the value can occur only once in the column) (33) If R9C7 contains the value 1 (initial assumption), then R7C7 cannot contain the value 1 (the value can occur only once in the block) (34) If R7C7 does not contain the value 1 and R7C7 does not contain the value 2 (32), then R7C7 must contain the value 9 (only remaining possible value in the cell) (35) If R7C7 contains the value 9, then R5C7 cannot contain the value 9 (the value can occur only once in the column) (36) If R5C1 contains the value 5 (20), then R5C1 cannot contain the value 9 (the cell can contain only one value) (37) If R8C8 contains the value 5 (27), then R8C8 cannot contain the value 3 (the cell can contain only one value) (38) If R9C9 does not contain the value 3 (3), R9C7 does not contain the value 3 (4) and R8C8 does not contain the value 3, then R8C7 must contain the value 3 (only remaining possible position in the block) (39) If R8C7 contains the value 3, then R8C7 cannot contain the value 2 (the cell can contain only one value) (40) If R9C4 contains the value 2 (9), then R8C4 cannot contain the value 2 (the value can occur only once in the block) (41) If R9C4 contains the value 2 (9), then R8C6 cannot contain the value 2 (the value can occur only once in the block) (42) If R8C6 does not contain the value 2, R8C4 does not contain the value 2 (40) and R8C7 does not contain the value 2 (39), then R8C2 must contain the value 2 (only remaining possible position in the row) (43) If R8C2 contains the value 2, then R8C2 cannot contain the value 9 (the cell can contain only one value) (44) If R8C7 contains the value 3 (38), then R8C7 cannot contain the value 9 (the cell can contain only one value) (45) If R8C8 contains the value 5 (27), then R8C8 cannot contain the value 9 (the cell can contain only one value) (46) If R8C8 does not contain the value 9, R8C7 does not contain the value 9 (44) and R8C2 does not contain the value 9 (43), then R8C3 must contain the value 9 (only remaining possible position in the row) (47) If R8C3 contains the value 9, then R5C3 cannot contain the value 9 (the value can occur only once in the column) (48) If R5C3 does not contain the value 9, R5C1 does not contain the value 9 (36) and R5C7 does not contain the value 9 (35), then R5C9 must contain the value 9 (only remaining possible position in the row)
Chain 2: If R5C9 must contain the value 9, then R5C9 cannot contain the value 9 (View 2): (1) If R9C7 contains the value 1, then R9C7 cannot contain the value 4 (the cell can contain only one value) (2) If R9C7 does not contain the value 4, then R9C9 must contain the value 4 (only remaining possible position in the block) (3) If R9C9 contains the value 4, then R9C9 cannot contain the value 3 (the cell can contain only one value) (4) If R9C7 contains the value 1 (initial assumption), then R9C7 cannot contain the value 3 (the cell can contain only one value) (5) If R9C7 does not contain the value 3 and R9C9 does not contain the value 3 (3), then R9C2 must contain the value 3 (only remaining possible position in the row) (6) If R9C2 contains the value 3, then R9C2 cannot contain the value 2 (the cell can contain only one value) (7) If R9C7 contains the value 1 (initial assumption), then R9C7 cannot contain the value 2 (the cell can contain only one value) (8) If R9C9 contains the value 4 (2), then R9C9 cannot contain the value 2 (the cell can contain only one value) (9) If R9C9 does not contain the value 2, R9C7 does not contain the value 2 (7) and R9C2 does not contain the value 2 (6), then R9C4 must contain the value 2 (only remaining possible position in the row) (10) If R9C4 contains the value 2, then R7C6 cannot contain the value 2 (the value can occur only once in the block) (11) If R9C7 contains the value 1 (initial assumption), then R9C5 cannot contain the value 1 (the value can occur only once in the row) (12) If R9C5 does not contain the value 1, then R9C5 must contain the value 7 (only remaining possible value in the cell) (13) If R9C5 contains the value 7, then R7C6 cannot contain the value 7 (the value can occur only once in the block) (14) If R7C6 does not contain the value 7 and R7C6 does not contain the value 2 (10), then R7C6 must contain the value 5 (only remaining possible value in the cell) (15) If R7C6 contains the value 5, then R5C6 cannot contain the value 5 (the value can occur only once in the column) (16) If R9C5 contains the value 7 (12), then R9C3 cannot contain the value 7 (the value can occur only once in the row) (17) If R9C7 contains the value 1 (initial assumption), then R9C3 cannot contain the value 1 (the value can occur only once in the row) (18) If R9C3 does not contain the value 1 and R9C3 does not contain the value 7 (16), then R9C3 must contain the value 5 (only remaining possible value in the cell) (19) If R9C3 contains the value 5, then R5C3 cannot contain the value 5 (the value can occur only once in the column) (20) If R5C3 does not contain the value 5 and R5C6 does not contain the value 5 (15), then R5C1 must contain the value 5 (only remaining possible position in the row) (21) If R5C1 contains the value 5, then R1C1 cannot contain the value 5 (the value can occur only once in the column) (22) If R9C3 contains the value 5 (18), then R1C3 cannot contain the value 5 (the value can occur only once in the column) (23) If R7C6 contains the value 5 (14), then R8C6 cannot contain the value 5 (the value can occur only once in the block) (24) If R9C3 contains the value 5 (18), then R8C2 cannot contain the value 5 (the value can occur only once in the block) (25) If R9C3 contains the value 5 (18), then R8C3 cannot contain the value 5 (the value can occur only once in the block) (26) If R7C6 contains the value 5 (14), then R8C4 cannot contain the value 5 (the value can occur only once in the block) (27) If R8C4 does not contain the value 5, R8C3 does not contain the value 5 (25), R8C2 does not contain the value 5 (24) and R8C6 does not contain the value 5 (23), then R8C8 must contain the value 5 (only remaining possible position in the row) (28) If R8C8 contains the value 5, then R1C8 cannot contain the value 5 (the value can occur only once in the column) (29) If R1C8 does not contain the value 5, R1C3 does not contain the value 5 (22) and R1C1 does not contain the value 5 (21), then R1C9 must contain the value 5 (only remaining possible position in the row) (30) If R1C9 contains the value 5, then R1C9 cannot contain the value 2 (the cell can contain only one value) (31) If R1C9 does not contain the value 2, then R1C7 must contain the value 2 (only remaining possible position in the block) (32) If R1C7 contains the value 2, then R7C7 cannot contain the value 2 (the value can occur only once in the column) (33) If R9C7 contains the value 1 (initial assumption), then R7C7 cannot contain the value 1 (the value can occur only once in the block) (34) If R7C7 does not contain the value 1 and R7C7 does not contain the value 2 (32), then R7C7 must contain the value 9 (only remaining possible value in the cell) (35) If R7C7 contains the value 9, then R2C7 cannot contain the value 9 (the value can occur only once in the column) (36) If R9C3 contains the value 5 (18), then R9C4 cannot contain the value 5 (the value can occur only once in the row) (37) If R9C4 does not contain the value 5 and R8C4 does not contain the value 5 (26), then R4C4 must contain the value 5 (only remaining possible position in the column) (38) If R4C4 contains the value 5, then R4C2 cannot contain the value 5 (the value can occur only once in the row) (39) If R9C3 contains the value 5 (18), then R7C2 cannot contain the value 5 (the value can occur only once in the block) (40) If R9C2 contains the value 3 (5), then R9C2 cannot contain the value 5 (the cell can contain only one value) (41) If R9C2 does not contain the value 5, R8C2 does not contain the value 5 (24), R7C2 does not contain the value 5 (39) and R4C2 does not contain the value 5 (38), then R2C2 must contain the value 5 (only remaining possible position in the column) (42) If R2C2 contains the value 5, then R2C2 cannot contain the value 9 (the cell can contain only one value) (43) If R8C8 contains the value 5 (27), then R8C8 cannot contain the value 3 (the cell can contain only one value) (44) If R9C9 does not contain the value 3 (3), R9C7 does not contain the value 3 (4) and R8C8 does not contain the value 3, then R8C7 must contain the value 3 (only remaining possible position in the block) (45) If R8C7 contains the value 3, then R8C7 cannot contain the value 2 (the cell can contain only one value) (46) If R9C4 contains the value 2 (9), then R8C4 cannot contain the value 2 (the value can occur only once in the block) (47) If R9C4 contains the value 2 (9), then R8C6 cannot contain the value 2 (the value can occur only once in the block) (48) If R8C6 does not contain the value 2, R8C4 does not contain the value 2 (46) and R8C7 does not contain the value 2 (45), then R8C2 must contain the value 2 (only remaining possible position in the row) (49) If R8C2 contains the value 2, then R8C2 cannot contain the value 9 (the cell can contain only one value) (50) If R8C7 contains the value 3 (44), then R8C7 cannot contain the value 9 (the cell can contain only one value) (51) If R8C8 contains the value 5 (27), then R8C8 cannot contain the value 9 (the cell can contain only one value) (52) If R8C8 does not contain the value 9, R8C7 does not contain the value 9 (50) and R8C2 does not contain the value 9 (49), then R8C3 must contain the value 9 (only remaining possible position in the row) (53) If R8C3 contains the value 9, then R2C3 cannot contain the value 9 (the value can occur only once in the column) (54) If R2C3 does not contain the value 9, R2C2 does not contain the value 9 (42) and R2C7 does not contain the value 9 (35), then R2C9 must contain the value 9 (only remaining possible position in the row) (55) If R2C9 contains the value 9, then R5C9 cannot contain the value 9 (the value can occur only once in the column)

Dynamic Contradiction Forcing Chains
With this solving technique, we will prove the two following assertions:
If R9C9 contains the value 1, then R5C9 must contain the value 9
If R9C9 contains the value 1, then R5C9 cannot contain the value 9
Because the same assumption yields to contradictory results, we can conclude that the assumption is false, that is, R9C9 cannot contain the value 1.
Each assertion is proved by a different chain of simple rules. The chains can be dynamic, which means that the conclusions of multiple sub-chains must be combined in some cases.
The details of each chain are given below. Use the view selector below the grid to switch between the graphical illustrations of the two different chains.
Chain 1: If R9C9 contains the value 1, then R5C9 cannot contain the value 9 (View 1): (1) If R9C9 contains the value 1, then R9C9 cannot contain the value 4 (the cell can contain only one value) (2) If R9C9 does not contain the value 4, then R9C7 must contain the value 4 (only remaining possible position in the block) (3) If R9C7 contains the value 4, then R9C7 cannot contain the value 3 (the cell can contain only one value) (4) If R9C9 contains the value 1 (initial assumption), then R9C9 cannot contain the value 3 (the cell can contain only one value) (5) If R9C9 does not contain the value 3 and R9C7 does not contain the value 3 (3), then R9C2 must contain the value 3 (only remaining possible position in the row) (6) If R9C2 contains the value 3, then R9C2 cannot contain the value 2 (the cell can contain only one value) (7) If R9C7 contains the value 4 (2), then R9C7 cannot contain the value 2 (the cell can contain only one value) (8) If R9C9 contains the value 1 (initial assumption), then R9C9 cannot contain the value 2 (the cell can contain only one value) (9) If R9C9 does not contain the value 2, R9C7 does not contain the value 2 (7) and R9C2 does not contain the value 2 (6), then R9C4 must contain the value 2 (only remaining possible position in the row) (10) If R9C4 contains the value 2, then R7C6 cannot contain the value 2 (the value can occur only once in the block) (11) If R9C9 contains the value 1 (initial assumption), then R9C5 cannot contain the value 1 (the value can occur only once in the row) (12) If R9C5 does not contain the value 1, then R9C5 must contain the value 7 (only remaining possible value in the cell) (13) If R9C5 contains the value 7, then R7C6 cannot contain the value 7 (the value can occur only once in the block) (14) If R7C6 does not contain the value 7 and R7C6 does not contain the value 2 (10), then R7C6 must contain the value 5 (only remaining possible value in the cell) (15) If R7C6 contains the value 5, then R5C6 cannot contain the value 5 (the value can occur only once in the column) (16) If R9C5 contains the value 7 (12), then R9C3 cannot contain the value 7 (the value can occur only once in the row) (17) If R9C9 contains the value 1 (initial assumption), then R9C3 cannot contain the value 1 (the value can occur only once in the row) (18) If R9C3 does not contain the value 1 and R9C3 does not contain the value 7 (16), then R9C3 must contain the value 5 (only remaining possible value in the cell) (19) If R9C3 contains the value 5, then R5C3 cannot contain the value 5 (the value can occur only once in the column) (20) If R5C3 does not contain the value 5 and R5C6 does not contain the value 5 (15), then R5C1 must contain the value 5 (only remaining possible position in the row) (21) If R5C1 contains the value 5, then R1C1 cannot contain the value 5 (the value can occur only once in the column) (22) If R9C3 contains the value 5 (18), then R1C3 cannot contain the value 5 (the value can occur only once in the column) (23) If R7C6 contains the value 5 (14), then R8C6 cannot contain the value 5 (the value can occur only once in the block) (24) If R9C3 contains the value 5 (18), then R8C2 cannot contain the value 5 (the value can occur only once in the block) (25) If R9C3 contains the value 5 (18), then R8C3 cannot contain the value 5 (the value can occur only once in the block) (26) If R7C6 contains the value 5 (14), then R8C4 cannot contain the value 5 (the value can occur only once in the block) (27) If R8C4 does not contain the value 5, R8C3 does not contain the value 5 (25), R8C2 does not contain the value 5 (24) and R8C6 does not contain the value 5 (23), then R8C8 must contain the value 5 (only remaining possible position in the row) (28) If R8C8 contains the value 5, then R1C8 cannot contain the value 5 (the value can occur only once in the column) (29) If R1C8 does not contain the value 5, R1C3 does not contain the value 5 (22) and R1C1 does not contain the value 5 (21), then R1C9 must contain the value 5 (only remaining possible position in the row) (30) If R1C9 contains the value 5, then R1C9 cannot contain the value 2 (the cell can contain only one value) (31) If R1C9 does not contain the value 2, then R1C7 must contain the value 2 (only remaining possible position in the block) (32) If R1C7 contains the value 2, then R7C7 cannot contain the value 2 (the value can occur only once in the column) (33) If R9C9 contains the value 1 (initial assumption), then R7C7 cannot contain the value 1 (the value can occur only once in the block) (34) If R7C7 does not contain the value 1 and R7C7 does not contain the value 2 (32), then R7C7 must contain the value 9 (only remaining possible value in the cell) (35) If R7C7 contains the value 9, then R5C7 cannot contain the value 9 (the value can occur only once in the column) (36) If R5C1 contains the value 5 (20), then R5C1 cannot contain the value 9 (the cell can contain only one value) (37) If R8C8 contains the value 5 (27), then R8C8 cannot contain the value 3 (the cell can contain only one value) (38) If R9C9 does not contain the value 3 (4), R9C7 does not contain the value 3 (3) and R8C8 does not contain the value 3, then R8C7 must contain the value 3 (only remaining possible position in the block) (39) If R8C7 contains the value 3, then R8C7 cannot contain the value 2 (the cell can contain only one value) (40) If R9C4 contains the value 2 (9), then R8C4 cannot contain the value 2 (the value can occur only once in the block) (41) If R9C4 contains the value 2 (9), then R8C6 cannot contain the value 2 (the value can occur only once in the block) (42) If R8C6 does not contain the value 2, R8C4 does not contain the value 2 (40) and R8C7 does not contain the value 2 (39), then R8C2 must contain the value 2 (only remaining possible position in the row) (43) If R8C2 contains the value 2, then R8C2 cannot contain the value 9 (the cell can contain only one value) (44) If R8C7 contains the value 3 (38), then R8C7 cannot contain the value 9 (the cell can contain only one value) (45) If R8C8 contains the value 5 (27), then R8C8 cannot contain the value 9 (the cell can contain only one value) (46) If R8C8 does not contain the value 9, R8C7 does not contain the value 9 (44) and R8C2 does not contain the value 9 (43), then R8C3 must contain the value 9 (only remaining possible position in the row) (47) If R8C3 contains the value 9, then R5C3 cannot contain the value 9 (the value can occur only once in the column) (48) If R5C3 does not contain the value 9, R5C1 does not contain the value 9 (36) and R5C7 does not contain the value 9 (35), then R5C9 must contain the value 9 (only remaining possible position in the row)
Chain 2: If R5C9 must contain the value 9, then R5C9 cannot contain the value 9 (View 2): (1) If R9C9 contains the value 1, then R9C9 cannot contain the value 4 (the cell can contain only one value) (2) If R9C9 does not contain the value 4, then R9C7 must contain the value 4 (only remaining possible position in the block) (3) If R9C7 contains the value 4, then R9C7 cannot contain the value 3 (the cell can contain only one value) (4) If R9C9 contains the value 1 (initial assumption), then R9C9 cannot contain the value 3 (the cell can contain only one value) (5) If R9C9 does not contain the value 3 and R9C7 does not contain the value 3 (3), then R9C2 must contain the value 3 (only remaining possible position in the row) (6) If R9C2 contains the value 3, then R9C2 cannot contain the value 2 (the cell can contain only one value) (7) If R9C7 contains the value 4 (2), then R9C7 cannot contain the value 2 (the cell can contain only one value) (8) If R9C9 contains the value 1 (initial assumption), then R9C9 cannot contain the value 2 (the cell can contain only one value) (9) If R9C9 does not contain the value 2, R9C7 does not contain the value 2 (7) and R9C2 does not contain the value 2 (6), then R9C4 must contain the value 2 (only remaining possible position in the row) (10) If R9C4 contains the value 2, then R7C6 cannot contain the value 2 (the value can occur only once in the block) (11) If R9C9 contains the value 1 (initial assumption), then R9C5 cannot contain the value 1 (the value can occur only once in the row) (12) If R9C5 does not contain the value 1, then R9C5 must contain the value 7 (only remaining possible value in the cell) (13) If R9C5 contains the value 7, then R7C6 cannot contain the value 7 (the value can occur only once in the block) (14) If R7C6 does not contain the value 7 and R7C6 does not contain the value 2 (10), then R7C6 must contain the value 5 (only remaining possible value in the cell) (15) If R7C6 contains the value 5, then R5C6 cannot contain the value 5 (the value can occur only once in the column) (16) If R9C5 contains the value 7 (12), then R9C3 cannot contain the value 7 (the value can occur only once in the row) (17) If R9C9 contains the value 1 (initial assumption), then R9C3 cannot contain the value 1 (the value can occur only once in the row) (18) If R9C3 does not contain the value 1 and R9C3 does not contain the value 7 (16), then R9C3 must contain the value 5 (only remaining possible value in the cell) (19) If R9C3 contains the value 5, then R5C3 cannot contain the value 5 (the value can occur only once in the column) (20) If R5C3 does not contain the value 5 and R5C6 does not contain the value 5 (15), then R5C1 must contain the value 5 (only remaining possible position in the row) (21) If R5C1 contains the value 5, then R1C1 cannot contain the value 5 (the value can occur only once in the column) (22) If R9C3 contains the value 5 (18), then R1C3 cannot contain the value 5 (the value can occur only once in the column) (23) If R7C6 contains the value 5 (14), then R8C6 cannot contain the value 5 (the value can occur only once in the block) (24) If R9C3 contains the value 5 (18), then R8C2 cannot contain the value 5 (the value can occur only once in the block) (25) If R9C3 contains the value 5 (18), then R8C3 cannot contain the value 5 (the value can occur only once in the block) (26) If R7C6 contains the value 5 (14), then R8C4 cannot contain the value 5 (the value can occur only once in the block) (27) If R8C4 does not contain the value 5, R8C3 does not contain the value 5 (25), R8C2 does not contain the value 5 (24) and R8C6 does not contain the value 5 (23), then R8C8 must contain the value 5 (only remaining possible position in the row) (28) If R8C8 contains the value 5, then R1C8 cannot contain the value 5 (the value can occur only once in the column) (29) If R1C8 does not contain the value 5, R1C3 does not contain the value 5 (22) and R1C1 does not contain the value 5 (21), then R1C9 must contain the value 5 (only remaining possible position in the row) (30) If R1C9 contains the value 5, then R1C9 cannot contain the value 2 (the cell can contain only one value) (31) If R1C9 does not contain the value 2, then R1C7 must contain the value 2 (only remaining possible position in the block) (32) If R1C7 contains the value 2, then R7C7 cannot contain the value 2 (the value can occur only once in the column) (33) If R9C9 contains the value 1 (initial assumption), then R7C7 cannot contain the value 1 (the value can occur only once in the block) (34) If R7C7 does not contain the value 1 and R7C7 does not contain the value 2 (32), then R7C7 must contain the value 9 (only remaining possible value in the cell) (35) If R7C7 contains the value 9, then R2C7 cannot contain the value 9 (the value can occur only once in the column) (36) If R9C3 contains the value 5 (18), then R9C4 cannot contain the value 5 (the value can occur only once in the row) (37) If R9C4 does not contain the value 5 and R8C4 does not contain the value 5 (26), then R4C4 must contain the value 5 (only remaining possible position in the column) (38) If R4C4 contains the value 5, then R4C2 cannot contain the value 5 (the value can occur only once in the row) (39) If R9C3 contains the value 5 (18), then R7C2 cannot contain the value 5 (the value can occur only once in the block) (40) If R9C2 contains the value 3 (5), then R9C2 cannot contain the value 5 (the cell can contain only one value) (41) If R9C2 does not contain the value 5, R8C2 does not contain the value 5 (24), R7C2 does not contain the value 5 (39) and R4C2 does not contain the value 5 (38), then R2C2 must contain the value 5 (only remaining possible position in the column) (42) If R2C2 contains the value 5, then R2C2 cannot contain the value 9 (the cell can contain only one value) (43) If R8C8 contains the value 5 (27), then R8C8 cannot contain the value 3 (the cell can contain only one value) (44) If R9C9 does not contain the value 3 (4), R9C7 does not contain the value 3 (3) and R8C8 does not contain the value 3, then R8C7 must contain the value 3 (only remaining possible position in the block) (45) If R8C7 contains the value 3, then R8C7 cannot contain the value 2 (the cell can contain only one value) (46) If R9C4 contains the value 2 (9), then R8C4 cannot contain the value 2 (the value can occur only once in the block) (47) If R9C4 contains the value 2 (9), then R8C6 cannot contain the value 2 (the value can occur only once in the block) (48) If R8C6 does not contain the value 2, R8C4 does not contain the value 2 (46) and R8C7 does not contain the value 2 (45), then R8C2 must contain the value 2 (only remaining possible position in the row) (49) If R8C2 contains the value 2, then R8C2 cannot contain the value 9 (the cell can contain only one value) (50) If R8C7 contains the value 3 (44), then R8C7 cannot contain the value 9 (the cell can contain only one value) (51) If R8C8 contains the value 5 (27), then R8C8 cannot contain the value 9 (the cell can contain only one value) (52) If R8C8 does not contain the value 9, R8C7 does not contain the value 9 (50) and R8C2 does not contain the value 9 (49), then R8C3 must contain the value 9 (only remaining possible position in the row) (53) If R8C3 contains the value 9, then R2C3 cannot contain the value 9 (the value can occur only once in the column) (54) If R2C3 does not contain the value 9, R2C2 does not contain the value 9 (42) and R2C7 does not contain the value 9 (35), then R2C9 must contain the value 9 (only remaining possible position in the row) (55) If R2C9 contains the value 9, then R5C9 cannot contain the value 9 (the value can occur only once in the column)

And it continues with still more complex nets.


I'm curious to know the "mystery" of this puzzle.

[Leren]

Code: Select all

*------------------------------------------------------------------*
|b1579    8       46-1579 | 1679 1467 3467   | 123679 13579 123569 |
| 3      T1579    6-1579  | 8    2    67     | 1679   4     1569   | < 7
|b179     1479    2       | 1679 5    3467   | 136789 1379  13689  |
|-------------------------+------------------+---------------------|
| 12578   247-15  1457    | 2567 9    245678 | 134678 137   13468  |
|T15789   6      t14579   | 3    478  4578   | 14789  2     1489   | < 7
| 2789    247-9   3       | 267  4678 1      | 5      79    4689   |
|-------------------------+------------------+---------------------|
| 2-1579 t12579   8       | 4    3    257    | 129    6     1259   | < 7
| 4       12359  B159     | 1256 168  2568   | 1239   1359  7      |
| 6       12357  B157     | 1257 17   9      | 1234   8     12345  |
*------------------------------------------------------------------*
                                      5        19           159
Almost Double JExocet (1589) r13c1 r5c3 r7c2, r89c3, r2c2 r5c1, Rogue Digit 7; 15 Eliminations : - 1579 r12c3, r7c1, - 15 r4c2, - 9 r6c2; btte

Hi Denis,

This is an old topic that dates from 2013, which is about the last time I looked at it, but it's still in my solver, and I have some collections of puzzles with this move.

The argument says that if you have all the conditions for a double J Exocet, except for 1 digit that does not satisfy an S cell Truth pattern (called the Rogue digit) then you can still eliminate all Base digits in the Base band from cells that see all 4 Base cells, and you can eliminate all non-Rogue digits from all cells that see all 4 Target cells. The good digits are 159 and the Rogue digit is 7 here.

I've long since forgotten the details of why this works, but you might like to read a post by me here and the surrounding posts if you want further information. David P Bird may have something about it in his JExocet Compendium.

Leren

[eleven]

I can't see, why the 7 can't be in both base pairs. Can you explain, please ?

[Leren]

Hi eleven.

The answer is, that this has to checked for. Reading my solver's code from 8 years ago, in the general case, I do it by "Scenario Testing" ie I put the rogue digit in both base cell pairs and check for a contradiction. If this sounds like T&E I'd have to agree with you. The J Exocet compendium says something similar. For this puzzle it would say that proving r46c2 <> 7 would be "an elimination goal".

Leren

[denis_berthier]

this has to checked for. Reading my solver's code from 8 years ago, in the general case, I do it by "Scenario Testing" ie I put the rogue digit in both base cell pairs and check for a contradiction. If this sounds like T&E I'd have to agree with you. The J Exocet compendium says something similar. For this puzzle it would say that proving r46c2 <> 7 would be "an elimination goal".
Leren

David P. Bird introduced the J-Exocet as a precisely defined pattern in order to palliate the vague, T&E-ish notion of an exocet.
It seems to me that re-introducing this T&E-ish flavour in association with the name J-Exocet is a step backwards.


BTW, for people who have Sukaku Explainer running, how does r46c2≠7 change the rating?

[Leren]

Hi denis,

That was the main reason I didn't bother to post a solution when there were no responses. There may be some puzzles where the required proof isn't too hard, and I'm sure that's what David had in mind, whereas my "Scenario testing" approach was just a cheap way of coding the move that would work in a large number of cases. Even ordinary J Exocets suffer from this problem to some extent. The line covering method is great, because you know there is an underlying theorem that proves the required digit property. There are also some easy in band chains that work if the line cover is not there. Failing that some people say that it is OK if the chains wander all over the board but only use the one digit - mmmaybe. Next you get into complicated multi digit chains .... how far do you go before a line of reasonableness is crossed ? In my solver I have an option called "Allow full digit Expansions" to prove an Exocet, which I normally have turned off, for obvious reasons.

Hey, why not turn on the full digit expansion option and see what turns up - well, here goes :

Code: Select all

*----------------------------------------------------------------*
|b1579   8       145679 | 1679 1467 3467   | 123679 13579 123569 |
| 3      1579    15679  | 8    2    67     | 1679   4     1569   |
|b179    1479    2      | 1679 5    3467   | 136789 1379  13689  |
|-----------------------+------------------+---------------------|
| 12578  12457  m157-4  | 2567 9    245678 | 134678 137   13468  |
| 15789  6      t1579-4 | 3    478  4578   | 14789  2     1489   |
| 2789   2479    3      | 267  4678 1      | 5      79    4689   |
|-----------------------+------------------+---------------------|
| 12579 t157-29  8      | 4    3    257    | 129    6     1259   |
| 4     m1359 2  159    | 1256 168  2568   | 1239   1359  7      |
| 6     m1357-2  157    | 1257 17   9      | 1234   8     12345  |
*----------------------------------------------------------------*

Exocet : r1c1 r3c1 r5c3 r7c2 1579. Eliminate non base digits from Target cells => - 4 r5c3, - 2 r7c2. Mirror node exclusions - 4 r4c3, - 9 r7c2, - 2 r89c2; btte

How did I establish the required J Exocet property for digit 7 ? I used a full digit expansion (complicated chain) and, for all I know, I might have even fully solved the puzzle. Brilliant

Leren

[denis_berthier]

Hi denis,
That was the main reason I didn't bother to post a solution when there were no responses. There may be some puzzles where the required proof isn't too hard, and I'm sure that's what David had in mind, whereas my "Scenario testing" approach was just a cheap way of coding the move that would work in a large number of cases. Even ordinary J Exocets suffer from this problem to some extent. The line covering method is great, because you know there is an underlying theorem that proves the required digit property. There are also some easy in band chains that work if the line cover is not there. Failing that some people say that it is OK if the chains wander all over the board but only use the one digit - mmmaybe. Next you get into complicated multi digit chains .... how far do you go before a line of reasonableness is crossed ? In my solver I have an option called "Allow full digit Expansions" to prove an Exocet, which I normally have turned off, for obvious reasons.
Leren

Hi Leren,
In my approach and in its SudoRules implementation, the question doesn't arise in the same way: if a rule can be defined in purely logic terms then it can be implemented as a CLIPS Rule; otherwise, it cannot. Nothing can be hidden in T&E-ish code, as is the case in Exocets. And not much can be written in set-cover form, because of obvious combinatorial explosion.

Adding "wandering" chains is not impossible; but they come with a cost: their length must be added to the Excocet own length (which depends on its own size but is already quite long); the result is a longer pattern with lower priority. As pure J-Excocets are already a very rare pattern(*), those with extra "wandering" parts will be still rarer (if not granted higher priority than they deserve). This is without considering the cost of coding them, with a null RoI.

In SudoRules, I have only the pure J-Exocets.


(*)I've run a test for all the exotic patterns, the generic ones (oddagons) and the Sudoku-specific ones (sk-loop, J-Exocets) all together. The result is, none of the W ratings in the 21375 puzzles in the cbg-000 collection is changed by adding these rules; i.e. none of these rules has any significant effect on solving puzzles non specifically designed to be have these patterns.

[Leren]

HI denis,

Just cross-posted with you and enabled the full digit expansion option. J Exocet solving in no-holds barred mode.

Leren

[eleven]

Ok, i thought, i would not have understood a hidden property.
But what is painfully missing in the move is a proof, that 7 can't be in both base pairs.
Something like this:

Code: Select all

+-------------------------+-------------------------+-------------------------+
| 1579    8       145679  | 1679   b1467  b 3467    | 123679  13579   123569  |
| 3      #1579   #15679   | 8       2     b*67      |#1679    4       1569    |
| 179     1479    2       | 1679    5     b 3467    | 136789  1379    13689   |
+-------------------------+-------------------------+-------------------------+
| 12578   12457   1457    | 2567    9       245678  | 134678  137     13468   |
|*15789   6      #14579   | 3      *478    *4578    |#14789   2       1489    |
| 2789    2479    3       | 267     4678    1       | 5       79      4689    |
+-------------------------+-------------------------+-------------------------+
|*12579  #12579   8       | 4       3     c#257     | 129     6       1259    |
| 4       12359   159     | 1256    168     2568    | 1239    1359    7       |
| 6       12357   157     | 1257   a17      9       | 1234    8       12345   |
+-------------------------+-------------------------+-------------------------+

Finned swordfish 7r257c237 (-7r46c2), fins r2c6, r5c156, r7c1
7r25c6 - r7c6 = r7c12 (=> -7r9c3)
7r57c1 (=> -7r13c1)
7r5c5 - (7=1)r9c5 - (1=3467)b2 - r7c6 = r7c12 (=> -7r9c3)

In all cases 7 can't be in both base pairs => the exocet can be used.

Now this is a very complex move with the exocet, but still a very short solution for an extreme puzzle.

[Leren]

Do I see a puzzle series here? I've got a batch of about 2,000 of these ADE puzzles. I could give you the ADE move but that would not be the puzzle.

The puzzle would be to prove that the Rogue digit(s) can't appear in both sets of base cells. Worth a thought.

Leren