Rating ranges - Githubissues

newman-iknewit commented 4 years ago

More a curiosity than an issue. The following puzzle:

........1.....2.3...4.5.6....57......1...8...7..59.4....96..5...3........4.....82

Has a difficulty rating of 11.0 (Dynamic Contradiction Forcing Chains (+ Forcing Chains)), which is above the range listed in the wiki for Dynamic (+Forcing). Just wondering if the ranges on the wiki need updating. (I'm using version 1.15.10, but get the same result rating with Explainer 1.2.1 mode.)

SudokuMonster commented 4 years ago

Thanks @newman-iknewit , As you mentioned, it is an issue with documentation rather than a bug. I'll update the difficulty ratings

Thanks again

newman-iknewit commented 4 years ago

Cool, thanks! I just found another one, in the other direction:

........1.....2.3...4.5.6.....1...7...6..3.2..8..4.9...48.......657.....9...6.5..

10.7/10.7/10.0 - Dynamic Region Forcing Chains (+ Multiple Forcing Chains)

SudokuMonster commented 4 years ago

Thanks,

I will leave this issue open for a while in case there are more of these around. The quoted ranges have been based on one user's collated observations. In theory we may have lower or higher ratings! I'll update when examples extend these ranges.

I posted about this as well on the New Sudoku Players forum: http://forum.enjoysudoku.com/s-e-records-in-the-patterns-game-t6695-15.html#p292807

newman-iknewit commented 4 years ago

Here's a couple more:

........1.....2.....1345.....36....7.2.....8.9.....5....6.7...3.8.....2.5.....9.. 9.7/9.7/9.3 - Dynamic Contradiction Forcing Chains

........1.......23..1..245...3..5..2.4..6....7..8.......5..9.1..8..7....6..4..... 11.5/11.5/11.2 - Dynamic Contradiction Forcing Chains (+ Multiple Forcing Chains) minlex form of 1.......7.2.4...6...3...5...4.2.9.......46.9....5.......7...1...8.9...2.5.......3;11.50;11.50;11.20;col;H6;133;21;

I'll post them in the forum too, but it may be a bit before they show up (still a new user). [edit]Never mind, I guess I don't need approval any more![/edit]

newman-iknewit commented 4 years ago

Found a better +MFC: ..............1..2.13.4..5...6..3....751...3.8...5.....47....1..5.7..9..6..4..5.. ED=11.6/1.2/1.2 (DCFC+MFC)

newman-iknewit commented 4 years ago

After stepping through it, this may be just due to the batch solving. Applying everything 11.4 and below still got to a solution. (I was using new ratings/order, vs. 1.2.1 for the original rating, so maybe that made a difference.)

SudokuMonster commented 4 years ago

That explains it!!!

newman-iknewit commented 4 years ago

I found another one, and stepping through with 1.2.1 mode shows that it does need the 11.6.

........1..2..3....4.15..6......17...5.63..8.9.........3......84.5.8.6..6..4...3. ED=11.6/1.2/1.2 (DCFC+MFC)

Here's the candidate grid after eliminating candidates from the lower rated 11.X +MFCs.

+----------------------+----------------------+----------------------+ | 35 6789 3679 | 2789 24679 246789 | 234589 2479 1 | | 158 16789 2 | 789 469 3 | 4589 4579 4579 | | 378 4 3789 | 1 5 2789 | 2389 6 2379 | +----------------------+----------------------+----------------------+ | 238 268 3468 | 2589 249 1 | 7 2459 3456 | | 127 5 147 | 6 3 2479 | 1249 8 249 | | 9 12678 134678 | 2578 247 24578 | 12345 1245 3456 | +----------------------+----------------------+----------------------+ | 127 3 179 | 2579 12679 56 | 2459 124579 8 | | 4 1279 5 | 3 8 279 | 6 1279 279 | | 6 12789 1789 | 4 1279 2579 | 1259 3 257 | +----------------------+----------------------+----------------------+

I'd be curious whether there are any lower rated +DFCs that solve this.

newman-iknewit commented 4 years ago

I haven't stepped through this one yet, but:

........1..2..3..4.5..6.27.....8.52.5..9.6...8.......6..76..4...6.....1.2...7.8.. ED=11.7/1.2/1.2 (DCFC+MFC)

(Like the 11.6s, skfr_ER for this is 11.2.)

newman-iknewit commented 4 years ago

Yep, it gets to the 11.7 using 1.2.1 mode. Here's the pencilmark grid:

+----------------------+----------------------+----------------------+ | 34679 3489 34689 | 24578 2459 245789 | 369 358 1 | | 1679 1789 2 | 1578 159 3 | 69 589 4 | | 1349 5 13489 | 148 6 1489 | 2 7 389 | +----------------------+----------------------+----------------------+ | 134679 13479 13469 | 1347 8 147 | 5 2 379 | | 5 12347 134 | 9 1234 6 | 137 348 378 | | 8 2379 1349 | 123457 12345 12457 | 1379 349 6 | +----------------------+----------------------+----------------------+ | 139 138 7 | 6 1359 1258 | 4 39 25 | | 349 6 34589 | 23458 2349 2489 | 379 1 23579 | | 2 1349 13459 | 1345 7 1459 | 8 6 359 | +----------------------+----------------------+----------------------+

Is there any way to check for the existence of +DFC chains from this grid short of rebuilding SE?

newman-iknewit commented 4 years ago

........1.....2.3...4.5.6....756.....6..4.8..9....3.....87..5..1......4.2.......9 ED=11.1/11.1/2.6 (DCFC+FC)

SudokuMonster commented 4 years ago

Yep, it gets to the 11.7 using 1.2.1 mode. Here's the pencilmark grid:

+----------------------+----------------------+----------------------+ | 34679 3489 34689 | 24578 2459 245789 | 369 358 1 | | 1679 1789 2 | 1578 159 3 | 69 589 4 | | 1349 5 13489 | 148 6 1489 | 2 7 389 | +----------------------+----------------------+----------------------+ | 134679 13479 13469 | 1347 8 147 | 5 2 379 | | 5 12347 134 | 9 1234 6 | 137 348 378 | | 8 2379 1349 | 123457 12345 12457 | 1379 349 6 | +----------------------+----------------------+----------------------+ | 139 138 7 | 6 1359 1258 | 4 39 25 | | 349 6 34589 | 23458 2349 2489 | 379 1 23579 | | 2 1349 13459 | 1345 7 1459 | 8 6 359 | +----------------------+----------------------+----------------------+

Is there any way to check for the existence of +DFC chains from this grid short of rebuilding SE?

You should be able to paste the PM grid which should continue exactly from that point (unless I misunderstood your question)

newman-iknewit commented 4 years ago

When I "Get all hints" at this point, it only returns +MFC chains. My question is whether there is a way to force it to also look for +DFC chains - obviously it will do so if there are no +MFC chains available, but in this case there's the 11.7 so that's all that shows up. (Having a separate checkbox for +FC, +MFC, and +DFC would work; currently, there is Dynamic Forcing Chains, Dynamic Forcing Chains (+), and Nested Forcing Chains - I assume the latter covers all of D*FC+FC/+MFC/+DFC?)

Just curious if there is a +DFC with a lower rating in this case.

SudokuMonster commented 4 years ago

Did you try "nexr step"?

icecream17 commented 3 years ago

There's a sukaku grid at http://forum.enjoysudoku.com/pencilmark-only-sudoku-t4929.html with only candidates:

+--------------------------------------------------------------------------+
| 1237    1256    36789  | 25789   4789    4569   | 1679    23457   1348   |
| 2456    13489   3459   | 1356    3679    1247   | 2478    5789    12368  |
| 2358    24789   1267   | 2489    123457  13468  | 34569   4568    1279   |
|------------------------+------------------------+------------------------|
| 1569    2578    3457   | 145678  3459    2356   | 234689  1236    146789 |
| 3679    12348   2489   | 1236    56789   1569   | 2357    1489    4579   |
| 14689   1567    234678 | 1237    12489   34578  | 1235    5789    6789   |
|------------------------+------------------------+------------------------|
| 24789   3678    1469   | 1346    1246    1357   | 5689    1256789 2345   |
| 1238    4569    125678 | 3579    2347    2689   | 1479    1348    2356   |
| 3457    3789    1256   | 23489   14568   256789 | 1578    23569   1457   |
+--------------------------------------------------------------------------+

It ends up with a 27 view contradiction forcing chain

the big step

``` Dynamic Contradiction Forcing Chains With this solving technique, we will prove the two following assertions: If H9 contains the value 5, then G2 must contain the value 4 If H9 contains the value 5, then G2 cannot contain the value 4 Because the same assumption yields to contradictory results, we can conclude that the assumption is false, that is, H9 cannot contain the value 5. 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 H9 contains the value 5, then G2 cannot contain the value 4 (View 1): (1) If H9 contains the value 5, then C9 cannot contain the value 5 (the value can occur only once in the row) (2) If H9 contains the value 5 (initial assumption), then I9 cannot contain the value 5 (the value can occur only once in the block) (3) If H9 contains the value 5 (initial assumption), then I7 cannot contain the value 5 (the value can occur only once in the block) (4) If H9 contains the value 5 (initial assumption), then I8 cannot contain the value 5 (the value can occur only once in the block) (5) If I8 does not contain the value 5, I7 does not contain the value 5 (3) and I9 does not contain the value 5 (2), then I5 must contain the value 5 (only remaining possible position in the column) (6) If I5 contains the value 5, then G6 cannot contain the value 5 (the value can occur only once in the block) (7) If H9 contains the value 5 (initial assumption), then G7 cannot contain the value 5 (the value can occur only once in the block) (8) If H9 contains the value 5 (initial assumption), then H7 cannot contain the value 5 (the value can occur only once in the block) (9) If H7 does not contain the value 5, G7 does not contain the value 5 (7) and I7 does not contain the value 5 (3), then F7 must contain the value 5 (only remaining possible position in the row) (10) If F7 contains the value 5, then F6 cannot contain the value 5 (the value can occur only once in the column) (11) If H9 contains the value 5 (initial assumption), then H6 cannot contain the value 5 (the value can occur only once in the column) (12) If H6 does not contain the value 5, F6 does not contain the value 5 (10) and G6 does not contain the value 5 (6), then B6 must contain the value 5 (only remaining possible position in the row) (13) If B6 contains the value 5, then A4 cannot contain the value 5 (the value can occur only once in the block) (14) If H9 contains the value 5 (initial assumption), then H3 cannot contain the value 5 (the value can occur only once in the column) (15) If H9 contains the value 5 (initial assumption), then H1 cannot contain the value 5 (the value can occur only once in the column) (16) If H9 contains the value 5 (initial assumption), then H2 cannot contain the value 5 (the value can occur only once in the column) (17) If H2 does not contain the value 5, H1 does not contain the value 5 (15) and H3 does not contain the value 5 (14), then G3 must contain the value 5 (only remaining possible position in the block) (18) If G3 contains the value 5, then A3 cannot contain the value 5 (the value can occur only once in the row) (19) If H9 contains the value 5 (initial assumption), then A9 cannot contain the value 5 (the value can occur only once in the row) (20) If A9 does not contain the value 5, A3 does not contain the value 5 (18) and A4 does not contain the value 5 (13), then A2 must contain the value 5 (only remaining possible position in the column) (21) If A2 contains the value 5, then A2 cannot contain the value 6 (the cell can contain only one value) (22) If B6 contains the value 5 (12), then B8 cannot contain the value 5 (the value can occur only once in the column) (23) If C9 does not contain the value 5 (1), A9 does not contain the value 5 (19) and B8 does not contain the value 5, then C8 must contain the value 5 (only remaining possible position in the block) (24) If C8 contains the value 5, then C8 cannot contain the value 1 (the cell can contain only one value) (25) If H9 contains the value 5 (initial assumption), then H9 cannot contain the value 2 (the cell can contain only one value) (26) If B6 contains the value 5 (12), then B6 cannot contain the value 1 (the cell can contain only one value) (27) If F7 contains the value 5 (9), then F7 cannot contain the value 1 (the cell can contain only one value) (28) If A2 contains the value 5 (20), then A2 cannot contain the value 2 (the cell can contain only one value) (29) If A2 does not contain the value 2, F7 does not contain the value 1 (27), B6 does not contain the value 1 (26), H9 does not contain the value 2 (25), C8 does not contain the value 1 (24), A2 does not contain the value 6 (21) and C9 does not contain the value 5 (1), then H7 cannot contain the value 1 (Contradiction Forcing Chain: H7.1 on ==> C9.2 both on & off) (30) If C8 contains the value 5 (23), then C8 cannot contain the value 8 (the cell can contain only one value) (31) If G3 contains the value 5 (17), then G3 cannot contain the value 4 (the cell can contain only one value) (32) If G3 contains the value 5 (17), then G3 cannot contain the value 6 (the cell can contain only one value) (33) If G3 contains the value 5 (17), then G3 cannot contain the value 3 (the cell can contain only one value) (34) If C8 contains the value 5 (23), then C8 cannot contain the value 2 (the cell can contain only one value) (35) If H9 does not contain the value 2 (25), C8 does not contain the value 2, B6 does not contain the value 1 (26), F7 does not contain the value 1 (27), G3 does not contain the value 3 (33), G3 does not contain the value 6 (32), A2 does not contain the value 2 (28), G3 does not contain the value 4 (31), C8 does not contain the value 1 (24) and C8 does not contain the value 8 (30), then G9 cannot contain the value 1 (Contradiction Forcing Chain: G9.1 on ==> F8.8 both on & off) (36) If A2 does not contain the value 6 (21), then C6 cannot contain the value 6 (Claiming: Cells A4,A5,A6: 6 in column and block) (37) If C8 contains the value 5 (23), then C8 cannot contain the value 6 (the cell can contain only one value) (38) If F7 contains the value 5 (9), then F4 cannot contain the value 5 (the value can occur only once in the column) (39) If H9 contains the value 5 (initial assumption), then H9 cannot contain the value 3 (the cell can contain only one value) (40) If C8 contains the value 5 (23), then C8 cannot contain the value 7 (the cell can contain only one value) (41) If B6 contains the value 5 (12), then B1 cannot contain the value 5 (the value can occur only once in the column) (42) If H9 contains the value 5 (initial assumption), then G9 cannot contain the value 5 (the value can occur only once in the block) (43) If I5 contains the value 5 (5), then I5 cannot contain the value 7 (the cell can contain only one value) (44) If I5 does not contain the value 7, B6 does not contain the value 1 (26), F7 does not contain the value 1 (27), H9 does not contain the value 2 (25), A2 does not contain the value 2 (28), G9 does not contain the value 5 (42), C8 does not contain the value 8 (30), C8 does not contain the value 2 (34), C8 does not contain the value 1 (24), B1 does not contain the value 5 (41), C8 does not contain the value 7 (40), H9 does not contain the value 3 (39), F4 does not contain the value 5 (38), C8 does not contain the value 6 (37), C6 does not contain the value 6 (36) and G3 does not contain the value 6 (32), then I9 cannot contain the value 1 (Contradiction Forcing Chain: I9.1 on ==> D5.6 both on & off) (45) If I9 does not contain the value 1, G9 does not contain the value 1 (35) and H7 does not contain the value 1 (29), then A8 cannot contain the value 1 (Pointing: Cells G8,H8: 1 in block and row) (46) If F7 contains the value 5 (9), then F1 cannot contain the value 5 (the value can occur only once in the column) (47) If H1 does not contain the value 5 (15), F1 does not contain the value 5 and B1 does not contain the value 5 (41), then D1 must contain the value 5 (only remaining possible position in the row) (48) If D1 contains the value 5, then D1 cannot contain the value 7 (the cell can contain only one value) (49) If G3 does not contain the value 3 (33), then H4 cannot contain the value 3 (Claiming: Cells G4,G5,G6: 3 in column and block) (50) If D1 contains the value 5 (47), then D1 cannot contain the value 8 (the cell can contain only one value) (51) If G9 does not contain the value 5 (42), A3 does not contain the value 5 (18), C8 does not contain the value 8 (30), D1 does not contain the value 8, B8 does not contain the value 5 (22), H9 does not contain the value 3 (39), H4 does not contain the value 3 (49), I8 does not contain the value 5 (4), D1 does not contain the value 7 (48), C9 does not contain the value 5 (1) and H9 does not contain the value 2 (25), then A8 cannot contain the value 3 (Contradiction Forcing Chain: A8.3 on ==> E6.2 both on & off) (52) If H9 contains the value 5 (initial assumption), then H9 cannot contain the value 6 (the cell can contain only one value) (53) If D1 contains the value 5 (47), then D1 cannot contain the value 2 (the cell can contain only one value) (54) If F7 contains the value 5 (9), then F7 cannot contain the value 7 (the cell can contain only one value) (55) If F7 does not contain the value 7, I5 does not contain the value 7 (43), A9 does not contain the value 5 (19), G9 does not contain the value 5 (42), D1 does not contain the value 2 (53), A3 does not contain the value 5 (18), I9 does not contain the value 5 (2), H9 does not contain the value 3 (39), H4 does not contain the value 3 (49), C8 does not contain the value 2 (34), I7 does not contain the value 5 (3), C8 does not contain the value 1 (24) and H9 does not contain the value 6 (52), then A8 cannot contain the value 8 (Contradiction Forcing Chain: A8.8 on ==> E7.4 both on & off) (56) If A8 does not contain the value 8, A8 does not contain the value 3 (51) and A8 does not contain the value 1 (45), then A8 must contain the value 2 (only remaining possible value in the cell) (57) If A8 contains the value 2, then A3 cannot contain the value 2 (the value can occur only once in the column) (58) If A8 contains the value 2 (56), then A1 cannot contain the value 2 (the value can occur only once in the column) (59) If G3 does not contain the value 4 (31), D1 does not contain the value 2 (53), H9 does not contain the value 3 (39), H4 does not contain the value 3 (49) and H9 does not contain the value 2 (25), then B3 cannot contain the value 2 (Contradiction Forcing Chain: B3.2 on ==> E8.4 both on & off) (60) If D1 contains the value 5 (47), then D1 cannot contain the value 9 (the cell can contain only one value) (61) If F7 contains the value 5 (9), then F7 cannot contain the value 3 (the cell can contain only one value) (62) If F7 does not contain the value 3, H9 does not contain the value 3 (39), H4 does not contain the value 3 (49), C8 does not contain the value 1 (24), D1 does not contain the value 2 (53), C8 does not contain the value 7 (40), H9 does not contain the value 2 (25), D1 does not contain the value 9 (60) and D1 does not contain the value 8 (50), then C3 cannot contain the value 2 (Contradiction Forcing Chain: C3.2 on ==> D4.8 both on & off) (63) If C3 does not contain the value 2, B3 does not contain the value 2 (59), A2 does not contain the value 2 (28), A1 does not contain the value 2 (58) and A3 does not contain the value 2 (57), then B1 must contain the value 2 (only remaining possible position in the block) (64) If B1 contains the value 2, then B1 cannot contain the value 6 (the cell can contain only one value) (65) If B6 contains the value 5 (12), then B6 cannot contain the value 6 (the cell can contain only one value) (66) If F7 does not contain the value 1 (27), B8 does not contain the value 5 (22), C6 does not contain the value 6 (36), B6 does not contain the value 6, H9 does not contain the value 6 (52) and H9 does not contain the value 3 (39), then B7 cannot contain the value 6 (Contradiction Forcing Chain: B7.6 on ==> G8.4 both on & off) (67) If B7 does not contain the value 6, B6 does not contain the value 6 (65) and B1 does not contain the value 6 (64), then B8 must contain the value 6 (only remaining possible position in the column) (68) If B8 contains the value 6, then C7 cannot contain the value 6 (the value can occur only once in the block) (69) If A8 contains the value 2 (56), then C9 cannot contain the value 2 (the value can occur only once in the block) (70) If F7 does not contain the value 3 (61), B6 does not contain the value 6 (65), H9 does not contain the value 3 (39), H4 does not contain the value 3 (49), D1 does not contain the value 2 (53), A2 does not contain the value 2 (28), B8 does not contain the value 5 (22), D1 does not contain the value 9 (60) and D1 does not contain the value 8 (50), then C9 cannot contain the value 6 (Contradiction Forcing Chain: C9.6 on ==> D4.8 both on & off) (71) If C9 does not contain the value 6, C9 does not contain the value 5 (1) and C9 does not contain the value 2 (69), then C9 must contain the value 1 (only remaining possible value in the cell) (72) If C9 contains the value 1, then C7 cannot contain the value 1 (the value can occur only once in the block) (73) If A2 contains the value 5 (20), then A2 cannot contain the value 4 (the cell can contain only one value) (74) If G3 contains the value 5 (17), then E3 cannot contain the value 5 (the value can occur only once in the row) (75) If I5 contains the value 5 (5), then E5 cannot contain the value 5 (the value can occur only once in the row) (76) If H9 contains the value 5 (initial assumption), then E9 cannot contain the value 5 (the value can occur only once in the row) (77) If E9 does not contain the value 5, E5 does not contain the value 5 (75) and E3 does not contain the value 5 (74), then E4 must contain the value 5 (only remaining possible position in the column) (78) If E4 contains the value 5, then E4 cannot contain the value 4 (the cell can contain only one value) (79) If I5 contains the value 5 (5), then I5 cannot contain the value 4 (the cell can contain only one value) (80) If F7 contains the value 5 (9), then F5 cannot contain the value 5 (the value can occur only once in the column) (81) If F1 does not contain the value 5 (46), F5 does not contain the value 5, H9 does not contain the value 3 (39), H4 does not contain the value 3 (49), I5 does not contain the value 4 (79), G3 does not contain the value 4 (31), C6 does not contain the value 6 (36), B6 does not contain the value 6 (65), E4 does not contain the value 4 (78), A2 does not contain the value 4 (73) and B6 does not contain the value 1 (26), then C7 cannot contain the value 4 (Contradiction Forcing Chain: C7.4 on ==> B5.1 both on & off) (82) If C7 does not contain the value 4, C7 does not contain the value 1 (72) and C7 does not contain the value 6 (68), then C7 must contain the value 9 (only remaining possible value in the cell) (83) If C7 contains the value 9, then H7 cannot contain the value 9 (the value can occur only once in the row) (84) If C7 contains the value 9 (82), then G7 cannot contain the value 9 (the value can occur only once in the row) (85) If H9 contains the value 5 (initial assumption), then H9 cannot contain the value 9 (the cell can contain only one value) (86) If H9 does not contain the value 9, G7 does not contain the value 9 (84) and H7 does not contain the value 9 (83), then G8 must contain the value 9 (only remaining possible position in the block) (87) If G8 contains the value 9, then G8 cannot contain the value 4 (the cell can contain only one value) (88) If G8 contains the value 9 (86), then G8 cannot contain the value 1 (the cell can contain only one value) (89) If I9 does not contain the value 1 (44), G9 does not contain the value 1 (35), H7 does not contain the value 1 (29) and G8 does not contain the value 1, then H8 must contain the value 1 (only remaining possible position in the block) (90) If H8 contains the value 1, then H8 cannot contain the value 4 (the cell can contain only one value) (91) If H8 does not contain the value 4 and G8 does not contain the value 4 (87), then I1 cannot contain the value 4 (Pointing: Cells I7,I9: 4 in block and column) (92) If G3 does not contain the value 4 (31) and A2 does not contain the value 4 (73), then B8 cannot contain the value 4 (Region Forcing Chains: 4 in column ==> B8.4 off) (93) If G8 does not contain the value 4 (87), B8 does not contain the value 4 and H8 does not contain the value 4 (90), then E8 must contain the value 4 (only remaining possible position in the row) (94) If E8 contains the value 4, then E1 cannot contain the value 4 (the value can occur only once in the column) (95) If B8 contains the value 6 (67), then I8 cannot contain the value 6 (the value can occur only once in the row) (96) If A8 contains the value 2 (56), then I8 cannot contain the value 2 (the value can occur only once in the row) (97) If I8 does not contain the value 5 (4), I8 does not contain the value 2 and I8 does not contain the value 6 (95), then I8 must contain the value 3 (only remaining possible value in the cell) (98) If I8 contains the value 3, then H8 cannot contain the value 3 (the value can occur only once in the block) (99) If H9 does not contain the value 3 (39), H4 does not contain the value 3 (49) and H8 does not contain the value 3, then H1 must contain the value 3 (only remaining possible position in the column) (100) If H1 contains the value 3, then H1 cannot contain the value 4 (the cell can contain only one value) (101) If H1 does not contain the value 4, E1 does not contain the value 4 (94) and I1 does not contain the value 4 (91), then F1 must contain the value 4 (only remaining possible position in the row) (102) If F1 contains the value 4, then D3 cannot contain the value 4 (the value can occur only once in the block) (103) If E8 contains the value 4 (93), then D7 cannot contain the value 4 (the value can occur only once in the block) (104) If E8 contains the value 4 (93), then D9 cannot contain the value 4 (the value can occur only once in the block) (105) If D9 does not contain the value 4, D7 does not contain the value 4 (103) and D3 does not contain the value 4 (102), then D4 must contain the value 4 (only remaining possible position in the column) (106) If D4 contains the value 4, then G4 cannot contain the value 4 (the value can occur only once in the row) (107) If G8 does not contain the value 4 (87), G3 does not contain the value 4 (31) and G4 does not contain the value 4, then G2 must contain the value 4 (only remaining possible position in the column) Chain 2: If G2 must contain the value 4, then G2 cannot contain the value 4 (View 2): (1) If H9 contains the value 5, then H3 cannot contain the value 5 (the value can occur only once in the column) (2) If H9 contains the value 5 (initial assumption), then H1 cannot contain the value 5 (the value can occur only once in the column) (3) If H9 contains the value 5 (initial assumption), then H2 cannot contain the value 5 (the value can occur only once in the column) (4) If H2 does not contain the value 5, H1 does not contain the value 5 (2) and H3 does not contain the value 5 (1), then G3 must contain the value 5 (only remaining possible position in the block) (5) If G3 contains the value 5, then G3 cannot contain the value 4 (the cell can contain only one value) (6) If H9 contains the value 5 (initial assumption), then I9 cannot contain the value 5 (the value can occur only once in the block) (7) If H9 contains the value 5 (initial assumption), then I7 cannot contain the value 5 (the value can occur only once in the block) (8) If H9 contains the value 5 (initial assumption), then I8 cannot contain the value 5 (the value can occur only once in the block) (9) If I8 does not contain the value 5, I7 does not contain the value 5 (7) and I9 does not contain the value 5 (6), then I5 must contain the value 5 (only remaining possible position in the column) (10) If I5 contains the value 5, then G6 cannot contain the value 5 (the value can occur only once in the block) (11) If H9 contains the value 5 (initial assumption), then G7 cannot contain the value 5 (the value can occur only once in the block) (12) If H9 contains the value 5 (initial assumption), then H7 cannot contain the value 5 (the value can occur only once in the block) (13) If H7 does not contain the value 5, G7 does not contain the value 5 (11) and I7 does not contain the value 5 (7), then F7 must contain the value 5 (only remaining possible position in the row) (14) If F7 contains the value 5, then F6 cannot contain the value 5 (the value can occur only once in the column) (15) If H9 contains the value 5 (initial assumption), then H6 cannot contain the value 5 (the value can occur only once in the column) (16) If H6 does not contain the value 5, F6 does not contain the value 5 (14) and G6 does not contain the value 5 (10), then B6 must contain the value 5 (only remaining possible position in the row) (17) If B6 contains the value 5, then A4 cannot contain the value 5 (the value can occur only once in the block) (18) If G3 contains the value 5 (4), then A3 cannot contain the value 5 (the value can occur only once in the row) (19) If H9 contains the value 5 (initial assumption), then A9 cannot contain the value 5 (the value can occur only once in the row) (20) If A9 does not contain the value 5, A3 does not contain the value 5 (18) and A4 does not contain the value 5 (17), then A2 must contain the value 5 (only remaining possible position in the column) (21) If A2 contains the value 5, then A2 cannot contain the value 2 (the cell can contain only one value) (22) If H9 contains the value 5 (initial assumption), then H9 cannot contain the value 9 (the cell can contain only one value) (23) If B6 contains the value 5 (16), then B1 cannot contain the value 5 (the value can occur only once in the column) (24) If F7 contains the value 5 (13), then F1 cannot contain the value 5 (the value can occur only once in the column) (25) If H1 does not contain the value 5 (2), F1 does not contain the value 5 and B1 does not contain the value 5 (23), then D1 must contain the value 5 (only remaining possible position in the row) (26) If D1 contains the value 5, then D1 cannot contain the value 9 (the cell can contain only one value) (27) If B6 contains the value 5 (16), then B6 cannot contain the value 6 (the cell can contain only one value) (28) If A2 contains the value 5 (20), then A2 cannot contain the value 6 (the cell can contain only one value) (29) If A2 does not contain the value 6, then C6 cannot contain the value 6 (Claiming: Cells A4,A5,A6: 6 in column and block) (30) If C6 does not contain the value 6, B6 does not contain the value 6 (27), D1 does not contain the value 9 (26), H9 does not contain the value 9 (22), A2 does not contain the value 2 (21) and G3 does not contain the value 4 (5), then I2 cannot contain the value 6 (Contradiction Forcing Chain: I2.6 on ==> H4.6 both on & off) (31) If H9 contains the value 5 (initial assumption), then C9 cannot contain the value 5 (the value can occur only once in the row) (32) If B6 contains the value 5 (16), then B8 cannot contain the value 5 (the value can occur only once in the column) (33) If C9 does not contain the value 5 (31), A9 does not contain the value 5 (19) and B8 does not contain the value 5, then C8 must contain the value 5 (only remaining possible position in the block) (34) If C8 contains the value 5, then C8 cannot contain the value 1 (the cell can contain only one value) (35) If H9 contains the value 5 (initial assumption), then H9 cannot contain the value 2 (the cell can contain only one value) (36) If B6 contains the value 5 (16), then B6 cannot contain the value 1 (the cell can contain only one value) (37) If F7 contains the value 5 (13), then F7 cannot contain the value 1 (the cell can contain only one value) (38) If A2 does not contain the value 2 (21), F7 does not contain the value 1, B6 does not contain the value 1 (36), H9 does not contain the value 2 (35), C8 does not contain the value 1 (34), A2 does not contain the value 6 (28) and C9 does not contain the value 5 (31), then H7 cannot contain the value 1 (Contradiction Forcing Chain: H7.1 on ==> C9.2 both on & off) (39) If C8 contains the value 5 (33), then C8 cannot contain the value 8 (the cell can contain only one value) (40) If G3 contains the value 5 (4), then G3 cannot contain the value 6 (the cell can contain only one value) (41) If G3 contains the value 5 (4), then G3 cannot contain the value 3 (the cell can contain only one value) (42) If C8 contains the value 5 (33), then C8 cannot contain the value 2 (the cell can contain only one value) (43) If H9 does not contain the value 2 (35), C8 does not contain the value 2, B6 does not contain the value 1 (36), F7 does not contain the value 1 (37), G3 does not contain the value 3 (41), G3 does not contain the value 6 (40), A2 does not contain the value 2 (21), G3 does not contain the value 4 (5), C8 does not contain the value 1 (34) and C8 does not contain the value 8 (39), then G9 cannot contain the value 1 (Contradiction Forcing Chain: G9.1 on ==> F8.8 both on & off) (44) If C8 contains the value 5 (33), then C8 cannot contain the value 6 (the cell can contain only one value) (45) If F7 contains the value 5 (13), then F4 cannot contain the value 5 (the value can occur only once in the column) (46) If H9 contains the value 5 (initial assumption), then H9 cannot contain the value 3 (the cell can contain only one value) (47) If C8 contains the value 5 (33), then C8 cannot contain the value 7 (the cell can contain only one value) (48) If H9 contains the value 5 (initial assumption), then G9 cannot contain the value 5 (the value can occur only once in the block) (49) If I5 contains the value 5 (9), then I5 cannot contain the value 7 (the cell can contain only one value) (50) If I5 does not contain the value 7, B6 does not contain the value 1 (36), F7 does not contain the value 1 (37), H9 does not contain the value 2 (35), A2 does not contain the value 2 (21), G9 does not contain the value 5 (48), C8 does not contain the value 8 (39), C8 does not contain the value 2 (42), C8 does not contain the value 1 (34), B1 does not contain the value 5 (23), C8 does not contain the value 7 (47), H9 does not contain the value 3 (46), F4 does not contain the value 5 (45), C8 does not contain the value 6 (44), C6 does not contain the value 6 (29) and G3 does not contain the value 6 (40), then I9 cannot contain the value 1 (Contradiction Forcing Chain: I9.1 on ==> D5.6 both on & off) (51) If I9 does not contain the value 1, G9 does not contain the value 1 (43) and H7 does not contain the value 1 (38), then A8 cannot contain the value 1 (Pointing: Cells G8,H8: 1 in block and row) (52) If D1 contains the value 5 (25), then D1 cannot contain the value 7 (the cell can contain only one value) (53) If G3 does not contain the value 3 (41), then H4 cannot contain the value 3 (Claiming: Cells G4,G5,G6: 3 in column and block) (54) If D1 contains the value 5 (25), then D1 cannot contain the value 8 (the cell can contain only one value) (55) If G9 does not contain the value 5 (48), A3 does not contain the value 5 (18), C8 does not contain the value 8 (39), D1 does not contain the value 8, B8 does not contain the value 5 (32), H9 does not contain the value 3 (46), H4 does not contain the value 3 (53), I8 does not contain the value 5 (8), D1 does not contain the value 7 (52), C9 does not contain the value 5 (31) and H9 does not contain the value 2 (35), then A8 cannot contain the value 3 (Contradiction Forcing Chain: A8.3 on ==> E6.2 both on & off) (56) If H9 contains the value 5 (initial assumption), then H9 cannot contain the value 6 (the cell can contain only one value) (57) If D1 contains the value 5 (25), then D1 cannot contain the value 2 (the cell can contain only one value) (58) If F7 contains the value 5 (13), then F7 cannot contain the value 7 (the cell can contain only one value) (59) If F7 does not contain the value 7, I5 does not contain the value 7 (49), A9 does not contain the value 5 (19), G9 does not contain the value 5 (48), D1 does not contain the value 2 (57), A3 does not contain the value 5 (18), I9 does not contain the value 5 (6), H9 does not contain the value 3 (46), H4 does not contain the value 3 (53), C8 does not contain the value 2 (42), I7 does not contain the value 5 (7), C8 does not contain the value 1 (34) and H9 does not contain the value 6 (56), then A8 cannot contain the value 8 (Contradiction Forcing Chain: A8.8 on ==> E7.4 both on & off) (60) If A8 does not contain the value 8, A8 does not contain the value 3 (55) and A8 does not contain the value 1 (51), then A8 must contain the value 2 (only remaining possible value in the cell) (61) If A8 contains the value 2, then A3 cannot contain the value 2 (the value can occur only once in the column) (62) If A8 contains the value 2 (60), then A1 cannot contain the value 2 (the value can occur only once in the column) (63) If G3 does not contain the value 4 (5), D1 does not contain the value 2 (57), H9 does not contain the value 3 (46), H4 does not contain the value 3 (53) and H9 does not contain the value 2 (35), then B3 cannot contain the value 2 (Contradiction Forcing Chain: B3.2 on ==> E8.4 both on & off) (64) If F7 contains the value 5 (13), then F7 cannot contain the value 3 (the cell can contain only one value) (65) If F7 does not contain the value 3, H9 does not contain the value 3 (46), H4 does not contain the value 3 (53), C8 does not contain the value 1 (34), D1 does not contain the value 2 (57), C8 does not contain the value 7 (47), H9 does not contain the value 2 (35), D1 does not contain the value 9 (26) and D1 does not contain the value 8 (54), then C3 cannot contain the value 2 (Contradiction Forcing Chain: C3.2 on ==> D4.8 both on & off) (66) If C3 does not contain the value 2, B3 does not contain the value 2 (63), A2 does not contain the value 2 (21), A1 does not contain the value 2 (62) and A3 does not contain the value 2 (61), then B1 must contain the value 2 (only remaining possible position in the block) (67) If B1 contains the value 2, then B1 cannot contain the value 6 (the cell can contain only one value) (68) If F7 does not contain the value 1 (37), B8 does not contain the value 5 (32), C6 does not contain the value 6 (29), B6 does not contain the value 6 (27), H9 does not contain the value 6 (56) and H9 does not contain the value 3 (46), then B7 cannot contain the value 6 (Contradiction Forcing Chain: B7.6 on ==> G8.4 both on & off) (69) If B7 does not contain the value 6, B6 does not contain the value 6 (27) and B1 does not contain the value 6 (67), then B8 must contain the value 6 (only remaining possible position in the column) (70) If B8 contains the value 6, then I8 cannot contain the value 6 (the value can occur only once in the row) (71) If I8 does not contain the value 6 and I2 does not contain the value 6 (30), then H4 cannot contain the value 6 (Claiming: Cells I4,I6: 6 in column and block) (72) If B8 contains the value 6 (69), then C7 cannot contain the value 6 (the value can occur only once in the block) (73) If A8 contains the value 2 (60), then C9 cannot contain the value 2 (the value can occur only once in the block) (74) If F7 does not contain the value 3 (64), B6 does not contain the value 6 (27), H9 does not contain the value 3 (46), H4 does not contain the value 3 (53), D1 does not contain the value 2 (57), A2 does not contain the value 2 (21), B8 does not contain the value 5 (32), D1 does not contain the value 9 (26) and D1 does not contain the value 8 (54), then C9 cannot contain the value 6 (Contradiction Forcing Chain: C9.6 on ==> D4.8 both on & off) (75) If C9 does not contain the value 6, C9 does not contain the value 5 (31) and C9 does not contain the value 2 (73), then C9 must contain the value 1 (only remaining possible value in the cell) (76) If C9 contains the value 1, then C7 cannot contain the value 1 (the value can occur only once in the block) (77) If A2 contains the value 5 (20), then A2 cannot contain the value 4 (the cell can contain only one value) (78) If G3 contains the value 5 (4), then E3 cannot contain the value 5 (the value can occur only once in the row) (79) If I5 contains the value 5 (9), then E5 cannot contain the value 5 (the value can occur only once in the row) (80) If H9 contains the value 5 (initial assumption), then E9 cannot contain the value 5 (the value can occur only once in the row) (81) If E9 does not contain the value 5, E5 does not contain the value 5 (79) and E3 does not contain the value 5 (78), then E4 must contain the value 5 (only remaining possible position in the column) (82) If E4 contains the value 5, then E4 cannot contain the value 4 (the cell can contain only one value) (83) If I5 contains the value 5 (9), then I5 cannot contain the value 4 (the cell can contain only one value) (84) If F7 contains the value 5 (13), then F5 cannot contain the value 5 (the value can occur only once in the column) (85) If F1 does not contain the value 5 (24), F5 does not contain the value 5, H9 does not contain the value 3 (46), H4 does not contain the value 3 (53), I5 does not contain the value 4 (83), G3 does not contain the value 4 (5), C6 does not contain the value 6 (29), B6 does not contain the value 6 (27), E4 does not contain the value 4 (82), A2 does not contain the value 4 (77) and B6 does not contain the value 1 (36), then C7 cannot contain the value 4 (Contradiction Forcing Chain: C7.4 on ==> B5.1 both on & off) (86) If C7 does not contain the value 4, C7 does not contain the value 1 (76) and C7 does not contain the value 6 (72), then C7 must contain the value 9 (only remaining possible value in the cell) (87) If C7 contains the value 9, then H7 cannot contain the value 9 (the value can occur only once in the row) (88) If C7 contains the value 9 (86), then G7 cannot contain the value 9 (the value can occur only once in the row) (89) If H9 does not contain the value 9 (22), G7 does not contain the value 9 and H7 does not contain the value 9 (87), then G8 must contain the value 9 (only remaining possible position in the block) (90) If G8 contains the value 9, then G8 cannot contain the value 1 (the cell can contain only one value) (91) If I9 does not contain the value 1 (50), G9 does not contain the value 1 (43), H7 does not contain the value 1 (38) and G8 does not contain the value 1, then H8 must contain the value 1 (only remaining possible position in the block) (92) If H8 contains the value 1, then H4 cannot contain the value 1 (the value can occur only once in the column) (93) If H4 does not contain the value 3 (53), H4 does not contain the value 1 and H4 does not contain the value 6 (71), then H4 must contain the value 2 (only remaining possible value in the cell) (94) If H4 contains the value 2, then G6 cannot contain the value 2 (the value can occur only once in the block) (95) If H4 contains the value 2 (93), then G4 cannot contain the value 2 (the value can occur only once in the block) (96) If H4 contains the value 2 (93), then G5 cannot contain the value 2 (the value can occur only once in the block) (97) If G5 does not contain the value 2, G4 does not contain the value 2 (95) and G6 does not contain the value 2 (94), then G2 must contain the value 2 (only remaining possible position in the column) (98) If G2 contains the value 2, then G2 cannot contain the value 4 (the cell can contain only one value) Nested Forcing Chains details (Note that each Nested Forcing Chain relies on the fact that some candidates have been excluded by the main Forcing Chain): Nested Region Forcing Chains: 4 in column ==> B8.4 off Chain 3: If G2 contains the value 4, then B8 cannot contain the value 4 (View 3): (1) If G2 contains the value 4, then C2 cannot contain the value 4 (the value can occur only once in the row) (2) If G2 contains the value 4 (initial assumption), then B2 cannot contain the value 4 (the value can occur only once in the row) (3) If B2 does not contain the value 4 and C2 does not contain the value 4 (1), then B3 must contain the value 4 (only remaining possible position in the block) (4) If B3 contains the value 4, then B8 cannot contain the value 4 (the value can occur only once in the column) Chain 4: If G4 contains the value 4, then B8 cannot contain the value 4 (View 4): (1) If G4 contains the value 4, then D4 cannot contain the value 4 (the value can occur only once in the row) (2) If D4 does not contain the value 4, then A6 cannot contain the value 4 (Pointing: Cells E6,F6: 4 in block and row) (3) If A6 does not contain the value 4, then B8 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) Chain 5: If G8 contains the value 4, then B8 cannot contain the value 4 (View 5): (1) If G8 contains the value 4, then B8 cannot contain the value 4 (the value can occur only once in the row) Nested Contradiction Forcing Chain: C7.4 on ==> B5.1 both on & off Chain 6: If C7 contains the value 4, then B5 must contain the value 1 (View 6): (1) If C7 contains the value 4, then A9 cannot contain the value 4 (the value can occur only once in the block) (2) If C7 contains the value 4 (initial assumption), then A7 cannot contain the value 4 (the value can occur only once in the block) (3) If A7 does not contain the value 4 and A9 does not contain the value 4 (1), then A6 must contain the value 4 (only remaining possible position in the column) (4) If A6 contains the value 4, then A6 cannot contain the value 6 (the cell can contain only one value) (5) If A6 does not contain the value 6, then I6 must contain the value 6 (only remaining possible position in the row) (6) If I6 contains the value 6, then I8 cannot contain the value 6 (the value can occur only once in the column) (7) If I8 does not contain the value 6, then D7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (8) If I8 does not contain the value 6 (6), then E7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (9) If C7 contains the value 4 (initial assumption), then I7 cannot contain the value 4 (the value can occur only once in the row) (10) If I8 does not contain the value 6 (6) and I7 does not contain the value 4, then H8 cannot contain the value 3 (Naked Pair: Cells I7,I8: 2,3 in block) (11) If H8 does not contain the value 3, then H1 must contain the value 3 (only remaining possible position in the column) (12) If H1 contains the value 3, then H1 cannot contain the value 2 (the cell can contain only one value) (13) If I8 does not contain the value 6 (6) and I7 does not contain the value 4 (9), then I2 cannot contain the value 2 (Naked Pair: Cells I7,I8: 2,3 in column) (14) If I8 does not contain the value 6 (6) and I7 does not contain the value 4 (9), then I3 cannot contain the value 2 (Naked Pair: Cells I7,I8: 2,3 in column) (15) If I3 does not contain the value 2, I2 does not contain the value 2 (13) and H1 does not contain the value 2 (12), then G2 must contain the value 2 (only remaining possible position in the block) (16) If G2 contains the value 2, then G2 cannot contain the value 4 (the cell can contain only one value) (17) If A6 contains the value 4 (3), then B5 cannot contain the value 4 (the value can occur only once in the block) (18) If C7 contains the value 4 (initial assumption), then C5 cannot contain the value 4 (the value can occur only once in the column) (19) If C5 does not contain the value 4 and B5 does not contain the value 4 (17), then H5 must contain the value 4 (only remaining possible position in the row) (20) If H5 contains the value 4, then H1 cannot contain the value 4 (the value can occur only once in the column) (21) If H5 contains the value 4 (19), then H3 cannot contain the value 4 (the value can occur only once in the column) (22) If H3 does not contain the value 4, H1 does not contain the value 4 (20) and G2 does not contain the value 4 (16), then I1 must contain the value 4 (only remaining possible position in the block) (23) If I1 contains the value 4, then I9 cannot contain the value 4 (the value can occur only once in the column) (24) If A6 contains the value 4 (3), then F6 cannot contain the value 4 (the value can occur only once in the row) (25) If A6 contains the value 4 (3), then E6 cannot contain the value 4 (the value can occur only once in the row) (26) If E6 does not contain the value 4 and F6 does not contain the value 4 (24), then D4 must contain the value 4 (only remaining possible position in the block) (27) If D4 contains the value 4, then D9 cannot contain the value 4 (the value can occur only once in the column) (28) If D9 does not contain the value 4, A9 does not contain the value 4 (1) and I9 does not contain the value 4 (23), then E9 must contain the value 4 (only remaining possible position in the row) (29) If E9 contains the value 4, then E9 cannot contain the value 6 (the cell can contain only one value) (30) If E9 does not contain the value 6, E7 does not contain the value 6 (8) and D7 does not contain the value 6 (7), then F4 cannot contain the value 6 (Pointing: Cells F8,F9: 6 in block and column) (31) If D4 contains the value 4 (26), then D4 cannot contain the value 6 (the cell can contain only one value) (32) If I6 contains the value 6 (5), then G4 cannot contain the value 6 (the value can occur only once in the block) (33) If I6 contains the value 6 (5), then H4 cannot contain the value 6 (the value can occur only once in the block) (34) If I6 contains the value 6 (5), then I4 cannot contain the value 6 (the value can occur only once in the block) (35) If I4 does not contain the value 6, H4 does not contain the value 6 (33), G4 does not contain the value 6 (32), D4 does not contain the value 6 (31) and F4 does not contain the value 6 (30), then A4 must contain the value 6 (only remaining possible position in the row) (36) If A4 contains the value 6, then A4 cannot contain the value 1 (the cell can contain only one value) (37) If A6 contains the value 4 (3), then A6 cannot contain the value 1 (the cell can contain only one value) (38) If A6 does not contain the value 1 and A4 does not contain the value 1 (36), then B5 must contain the value 1 (only remaining possible position in the block) Chain 7: If C7 contains the value 4, then B5 cannot contain the value 1 (View 7): (1) If C7 contains the value 4, then A9 cannot contain the value 4 (the value can occur only once in the block) (2) If C7 contains the value 4 (initial assumption), then A7 cannot contain the value 4 (the value can occur only once in the block) (3) If A7 does not contain the value 4 and A9 does not contain the value 4 (1), then A6 must contain the value 4 (only remaining possible position in the column) (4) If A6 contains the value 4, then A6 cannot contain the value 6 (the cell can contain only one value) (5) If A6 does not contain the value 6, then I6 must contain the value 6 (only remaining possible position in the row) (6) If I6 contains the value 6, then I8 cannot contain the value 6 (the value can occur only once in the column) (7) If I8 does not contain the value 6, then D7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (8) If I8 does not contain the value 6 (6), then E7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (9) If C7 contains the value 4 (initial assumption), then I7 cannot contain the value 4 (the value can occur only once in the row) (10) If I8 does not contain the value 6 (6) and I7 does not contain the value 4, then H8 cannot contain the value 3 (Naked Pair: Cells I7,I8: 2,3 in block) (11) If H8 does not contain the value 3, then H1 must contain the value 3 (only remaining possible position in the column) (12) If H1 contains the value 3, then H1 cannot contain the value 2 (the cell can contain only one value) (13) If I8 does not contain the value 6 (6) and I7 does not contain the value 4 (9), then I2 cannot contain the value 2 (Naked Pair: Cells I7,I8: 2,3 in column) (14) If I8 does not contain the value 6 (6) and I7 does not contain the value 4 (9), then I3 cannot contain the value 2 (Naked Pair: Cells I7,I8: 2,3 in column) (15) If I3 does not contain the value 2, I2 does not contain the value 2 (13) and H1 does not contain the value 2 (12), then G2 must contain the value 2 (only remaining possible position in the block) (16) If G2 contains the value 2, then G2 cannot contain the value 4 (the cell can contain only one value) (17) If A6 contains the value 4 (3), then B5 cannot contain the value 4 (the value can occur only once in the block) (18) If C7 contains the value 4 (initial assumption), then C5 cannot contain the value 4 (the value can occur only once in the column) (19) If C5 does not contain the value 4 and B5 does not contain the value 4 (17), then H5 must contain the value 4 (only remaining possible position in the row) (20) If H5 contains the value 4, then H1 cannot contain the value 4 (the value can occur only once in the column) (21) If H5 contains the value 4 (19), then H3 cannot contain the value 4 (the value can occur only once in the column) (22) If H3 does not contain the value 4, H1 does not contain the value 4 (20) and G2 does not contain the value 4 (16), then I1 must contain the value 4 (only remaining possible position in the block) (23) If I1 contains the value 4, then I9 cannot contain the value 4 (the value can occur only once in the column) (24) If A6 contains the value 4 (3), then F6 cannot contain the value 4 (the value can occur only once in the row) (25) If A6 contains the value 4 (3), then E6 cannot contain the value 4 (the value can occur only once in the row) (26) If E6 does not contain the value 4 and F6 does not contain the value 4 (24), then D4 must contain the value 4 (only remaining possible position in the block) (27) If D4 contains the value 4, then D9 cannot contain the value 4 (the value can occur only once in the column) (28) If D9 does not contain the value 4, A9 does not contain the value 4 (1) and I9 does not contain the value 4 (23), then E9 must contain the value 4 (only remaining possible position in the row) (29) If E9 contains the value 4, then E9 cannot contain the value 6 (the cell can contain only one value) (30) If E9 does not contain the value 6, E7 does not contain the value 6 (8) and D7 does not contain the value 6 (7), then F5 cannot contain the value 6 (Pointing: Cells F8,F9: 6 in block and column) (31) If I1 contains the value 4 (22), then F1 cannot contain the value 4 (the value can occur only once in the row) (32) If I6 contains the value 6 (5), then I2 cannot contain the value 6 (the value can occur only once in the column) (33) If I2 does not contain the value 6, then F1 cannot contain the value 6 (Claiming: Cells D2,E2: 6 in row and block) (34) If F1 does not contain the value 6 and F1 does not contain the value 4 (31), then F1 must contain the value 9 (only remaining possible value in the cell) (35) If F1 contains the value 9, then F5 cannot contain the value 9 (the value can occur only once in the column) (36) If F5 does not contain the value 9 and F5 does not contain the value 6 (30), then F5 must contain the value 1 (only remaining possible value in the cell) (37) If F5 contains the value 1, then B5 cannot contain the value 1 (the value can occur only once in the row) Nested Contradiction Forcing Chain: H7.1 on ==> C9.2 both on & off Chain 8: If H7 contains the value 1, then C9 must contain the value 2 (View 8): (1) If H7 contains the value 1, then H8 cannot contain the value 1 (the value can occur only once in the block) (2) If H7 contains the value 1 (initial assumption), then G8 cannot contain the value 1 (the value can occur only once in the block) (3) If G8 does not contain the value 1 and H8 does not contain the value 1 (1), then A8 must contain the value 1 (only remaining possible position in the row) (4) If A8 contains the value 1, then A6 cannot contain the value 1 (the value can occur only once in the column) (5) If A8 contains the value 1 (3), then A4 cannot contain the value 1 (the value can occur only once in the column) (6) If A4 does not contain the value 1 and A6 does not contain the value 1 (4), then B5 must contain the value 1 (only remaining possible position in the block) (7) If B5 contains the value 1, then F5 cannot contain the value 1 (the value can occur only once in the row) (8) If A8 contains the value 1 (3), then C9 cannot contain the value 1 (the value can occur only once in the block) (9) If H7 contains the value 1 (initial assumption), then C7 cannot contain the value 1 (the value can occur only once in the row) (10) If C7 does not contain the value 1 and C9 does not contain the value 1 (8), then C3 must contain the value 1 (only remaining possible position in the column) (11) If C3 contains the value 1, then F3 cannot contain the value 1 (the value can occur only once in the row) (12) If F3 does not contain the value 1 and F5 does not contain the value 1 (7), then F2 must contain the value 1 (only remaining possible position in the column) (13) If F2 contains the value 1, then F2 cannot contain the value 2 (the cell can contain only one value) (14) If F2 does not contain the value 2, then H1 cannot contain the value 2 (Claiming: Cells G2,I2: 2 in row and block) (15) If H7 contains the value 1 (initial assumption), then H7 cannot contain the value 2 (the cell can contain only one value) (16) If H7 does not contain the value 2 and H1 does not contain the value 2 (14), then H4 must contain the value 2 (only remaining possible position in the column) (17) If H4 contains the value 2, then B4 cannot contain the value 2 (the value can occur only once in the row) (18) If F2 does not contain the value 2 (13), then B3 cannot contain the value 2 (Pointing: Cells D3,E3: 2 in block and row) (19) If B5 contains the value 1 (6), then B5 cannot contain the value 2 (the cell can contain only one value) (20) If B5 does not contain the value 2, B3 does not contain the value 2 (18) and B4 does not contain the value 2 (17), then B1 must contain the value 2 (only remaining possible position in the column) (21) If B1 contains the value 2, then B1 cannot contain the value 6 (the cell can contain only one value) (22) If C3 contains the value 1 (10), then C3 cannot contain the value 6 (the cell can contain only one value) (23) If C3 does not contain the value 6 and B1 does not contain the value 6 (21), then C1 must contain the value 6 (only remaining possible position in the block) (24) If C1 contains the value 6, then C9 cannot contain the value 6 (the value can occur only once in the column) (25) If C9 does not contain the value 1 (8) and C9 does not contain the value 6, then C9 must contain the value 2 (only remaining possible value in the cell) Chain 9: If H7 contains the value 1, then C9 cannot contain the value 2 (View 9): (1) If H7 contains the value 1, then H8 cannot contain the value 1 (the value can occur only once in the block) (2) If H7 contains the value 1 (initial assumption), then G8 cannot contain the value 1 (the value can occur only once in the block) (3) If G8 does not contain the value 1 and H8 does not contain the value 1 (1), then A8 must contain the value 1 (only remaining possible position in the row) (4) If A8 contains the value 1, then A6 cannot contain the value 1 (the value can occur only once in the column) (5) If A8 contains the value 1 (3), then A4 cannot contain the value 1 (the value can occur only once in the column) (6) If A4 does not contain the value 1 and A6 does not contain the value 1 (4), then B5 must contain the value 1 (only remaining possible position in the block) (7) If B5 contains the value 1, then F5 cannot contain the value 1 (the value can occur only once in the row) (8) If A8 contains the value 1 (3), then C9 cannot contain the value 1 (the value can occur only once in the block) (9) If H7 contains the value 1 (initial assumption), then C7 cannot contain the value 1 (the value can occur only once in the row) (10) If C7 does not contain the value 1 and C9 does not contain the value 1 (8), then C3 must contain the value 1 (only remaining possible position in the column) (11) If C3 contains the value 1, then F3 cannot contain the value 1 (the value can occur only once in the row) (12) If F3 does not contain the value 1 and F5 does not contain the value 1 (7), then F2 must contain the value 1 (only remaining possible position in the column) (13) If F2 contains the value 1, then F2 cannot contain the value 2 (the cell can contain only one value) (14) If F2 does not contain the value 2, then H1 cannot contain the value 2 (Claiming: Cells G2,I2: 2 in row and block) (15) If H7 contains the value 1 (initial assumption), then H7 cannot contain the value 2 (the cell can contain only one value) (16) If H7 does not contain the value 2 and H1 does not contain the value 2 (14), then H4 must contain the value 2 (only remaining possible position in the column) (17) If H4 contains the value 2, then B4 cannot contain the value 2 (the value can occur only once in the row) (18) If F2 does not contain the value 2 (13), then B3 cannot contain the value 2 (Pointing: Cells D3,E3: 2 in block and row) (19) If B5 contains the value 1 (6), then B5 cannot contain the value 2 (the cell can contain only one value) (20) If B5 does not contain the value 2, B3 does not contain the value 2 (18) and B4 does not contain the value 2 (17), then B1 must contain the value 2 (only remaining possible position in the column) (21) If B1 contains the value 2, then A1 cannot contain the value 2 (the value can occur only once in the block) (22) If F2 does not contain the value 2 (13), then A3 cannot contain the value 2 (Pointing: Cells D3,E3: 2 in block and row) (23) If A8 contains the value 1 (3), then A8 cannot contain the value 2 (the cell can contain only one value) (24) If A8 does not contain the value 2, A3 does not contain the value 2 (22) and A1 does not contain the value 2 (21), then A7 must contain the value 2 (only remaining possible position in the column) (25) If A7 contains the value 2, then C9 cannot contain the value 2 (the value can occur only once in the block) Nested Contradiction Forcing Chain: G9.1 on ==> F8.8 both on & off Chain 10: If G9 contains the value 1, then F8 must contain the value 8 (View 10): (1) If G9 contains the value 1, then H8 cannot contain the value 1 (the value can occur only once in the block) (2) If G9 contains the value 1 (initial assumption), then G8 cannot contain the value 1 (the value can occur only once in the block) (3) If G8 does not contain the value 1 and H8 does not contain the value 1 (1), then A8 must contain the value 1 (only remaining possible position in the row) (4) If A8 contains the value 1, then A6 cannot contain the value 1 (the value can occur only once in the column) (5) If A8 contains the value 1 (3), then A4 cannot contain the value 1 (the value can occur only once in the column) (6) If A4 does not contain the value 1 and A6 does not contain the value 1 (4), then B5 must contain the value 1 (only remaining possible position in the block) (7) If B5 contains the value 1, then H5 cannot contain the value 1 (the value can occur only once in the row) (8) If G9 contains the value 1 (initial assumption), then H7 cannot contain the value 1 (the value can occur only once in the block) (9) If H8 does not contain the value 1 (1), H7 does not contain the value 1 and H5 does not contain the value 1 (7), then H4 must contain the value 1 (only remaining possible position in the column) (10) If H4 contains the value 1, then H4 cannot contain the value 2 (the cell can contain only one value) (11) If H4 does not contain the value 2, then G2 cannot contain the value 2 (Pointing: Cells G4,G5,G6: 2 in block and column) (12) If B5 contains the value 1 (6), then F5 cannot contain the value 1 (the value can occur only once in the row) (13) If A8 contains the value 1 (3), then C7 cannot contain the value 1 (the value can occur only once in the block) (14) If G9 contains the value 1 (initial assumption), then C9 cannot contain the value 1 (the value can occur only once in the row) (15) If C9 does not contain the value 1 and C7 does not contain the value 1 (13), then C3 must contain the value 1 (only remaining possible position in the column) (16) If C3 contains the value 1, then F3 cannot contain the value 1 (the value can occur only once in the row) (17) If F3 does not contain the value 1 and F5 does not contain the value 1 (12), then F2 must contain the value 1 (only remaining possible position in the column) (18) If F2 contains the value 1, then F2 cannot contain the value 2 (the cell can contain only one value) (19) If F2 does not contain the value 2 and G2 does not contain the value 2 (11), then I2 must contain the value 2 (only remaining possible position in the row) (20) If I2 contains the value 2, then I2 cannot contain the value 6 (the cell can contain only one value) (21) If C3 contains the value 1 (15), then C3 cannot contain the value 6 (the cell can contain only one value) (22) If C3 does not contain the value 6, then G1 cannot contain the value 6 (Pointing: Cells B1,C1: 6 in block and row) (23) If G1 does not contain the value 6 and I2 does not contain the value 6 (20), then H3 must contain the value 6 (only remaining possible position in the block) (24) If H3 contains the value 6, then H3 cannot contain the value 4 (the cell can contain only one value) (25) If I2 contains the value 2 (19), then I2 cannot contain the value 3 (the cell can contain only one value) (26) If C3 contains the value 1 (15), then B1 cannot contain the value 1 (the value can occur only once in the block) (27) If A8 contains the value 1 (3), then A1 cannot contain the value 1 (the value can occur only once in the column) (28) If G9 contains the value 1 (initial assumption), then G1 cannot contain the value 1 (the value can occur only once in the column) (29) If G1 does not contain the value 1, A1 does not contain the value 1 (27) and B1 does not contain the value 1 (26), then I1 must contain the value 1 (only remaining possible position in the row) (30) If I1 contains the value 1, then I1 cannot contain the value 3 (the cell can contain only one value) (31) If I1 does not contain the value 3 and I2 does not contain the value 3 (25), then H1 must contain the value 3 (only remaining possible position in the block) (32) If H1 contains the value 3, then H1 cannot contain the value 4 (the cell can contain only one value) (33) If I1 contains the value 1 (29), then I1 cannot contain the value 4 (the cell can contain only one value) (34) If I1 does not contain the value 4, H1 does not contain the value 4 (32) and H3 does not contain the value 4 (24), then G2 must contain the value 4 (only remaining possible position in the block) (35) If G2 contains the value 4, then G2 cannot contain the value 8 (the cell can contain only one value) (36) If I1 contains the value 1 (29), then I1 cannot contain the value 8 (the cell can contain only one value) (37) If I2 contains the value 2 (19), then I2 cannot contain the value 8 (the cell can contain only one value) (38) If H3 contains the value 6 (23), then H3 cannot contain the value 8 (the cell can contain only one value) (39) If H3 does not contain the value 8, I2 does not contain the value 8 (37), I1 does not contain the value 8 (36) and G2 does not contain the value 8 (35), then H2 must contain the value 8 (only remaining possible position in the block) (40) If H2 contains the value 8, then H8 cannot contain the value 8 (the value can occur only once in the column) (41) If A8 contains the value 1 (3), then A8 cannot contain the value 8 (the cell can contain only one value) (42) If A8 does not contain the value 8 and H8 does not contain the value 8 (40), then F8 must contain the value 8 (only remaining possible position in the row) Chain 11: If G9 contains the value 1, then F8 cannot contain the value 8 (View 11): (1) If G9 contains the value 1, then H8 cannot contain the value 1 (the value can occur only once in the block) (2) If G9 contains the value 1 (initial assumption), then G8 cannot contain the value 1 (the value can occur only once in the block) (3) If G8 does not contain the value 1 and H8 does not contain the value 1 (1), then A8 must contain the value 1 (only remaining possible position in the row) (4) If A8 contains the value 1, then A6 cannot contain the value 1 (the value can occur only once in the column) (5) If A8 contains the value 1 (3), then A4 cannot contain the value 1 (the value can occur only once in the column) (6) If A4 does not contain the value 1 and A6 does not contain the value 1 (4), then B5 must contain the value 1 (only remaining possible position in the block) (7) If B5 contains the value 1, then F5 cannot contain the value 1 (the value can occur only once in the row) (8) If A8 contains the value 1 (3), then C7 cannot contain the value 1 (the value can occur only once in the block) (9) If G9 contains the value 1 (initial assumption), then C9 cannot contain the value 1 (the value can occur only once in the row) (10) If C9 does not contain the value 1 and C7 does not contain the value 1 (8), then C3 must contain the value 1 (only remaining possible position in the column) (11) If C3 contains the value 1, then F3 cannot contain the value 1 (the value can occur only once in the row) (12) If F3 does not contain the value 1 and F5 does not contain the value 1 (7), then F2 must contain the value 1 (only remaining possible position in the column) (13) If F2 contains the value 1, then F2 cannot contain the value 2 (the cell can contain only one value) (14) If F2 does not contain the value 2, then H1 cannot contain the value 2 (Claiming: Cells G2,I2: 2 in row and block) (15) If B5 contains the value 1 (6), then H5 cannot contain the value 1 (the value can occur only once in the row) (16) If G9 contains the value 1 (initial assumption), then H7 cannot contain the value 1 (the value can occur only once in the block) (17) If H8 does not contain the value 1 (1), H7 does not contain the value 1 and H5 does not contain the value 1 (15), then H4 must contain the value 1 (only remaining possible position in the column) (18) If H4 contains the value 1, then H4 cannot contain the value 2 (the cell can contain only one value) (19) If H4 does not contain the value 2 and H1 does not contain the value 2 (14), then H7 must contain the value 2 (only remaining possible position in the column) (20) If H7 contains the value 2, then A7 cannot contain the value 2 (the value can occur only once in the row) (21) If A8 contains the value 1 (3), then A8 cannot contain the value 2 (the cell can contain only one value) (22) If A8 does not contain the value 2 and A7 does not contain the value 2 (20), then C9 must contain the value 2 (only remaining possible position in the block) (23) If C9 contains the value 2, then C6 cannot contain the value 2 (the value can occur only once in the column) (24) If B5 contains the value 1 (6), then B5 cannot contain the value 2 (the cell can contain only one value) (25) If C9 contains the value 2 (22), then C5 cannot contain the value 2 (the value can occur only once in the column) (26) If C5 does not contain the value 2, B5 does not contain the value 2 (24) and C6 does not contain the value 2 (23), then B4 must contain the value 2 (only remaining possible position in the block) (27) If B4 contains the value 2, then F4 cannot contain the value 2 (the value can occur only once in the row) (28) If C9 contains the value 2 (22), then F9 cannot contain the value 2 (the value can occur only once in the row) (29) If F9 does not contain the value 2, F2 does not contain the value 2 (13) and F4 does not contain the value 2 (27), then F8 must contain the value 2 (only remaining possible position in the column) (30) If F8 contains the value 2, then F8 cannot contain the value 8 (the cell can contain only one value) Nested Contradiction Forcing Chain: I9.1 on ==> D5.6 both on & off Chain 12: If I9 contains the value 1, then D5 must contain the value 6 (View 12): (1) If I9 contains the value 1, then H8 cannot contain the value 1 (the value can occur only once in the block) (2) If I9 contains the value 1 (initial assumption), then G8 cannot contain the value 1 (the value can occur only once in the block) (3) If G8 does not contain the value 1 and H8 does not contain the value 1 (1), then A8 must contain the value 1 (only remaining possible position in the row) (4) If A8 contains the value 1, then A6 cannot contain the value 1 (the value can occur only once in the column) (5) If A8 contains the value 1 (3), then A4 cannot contain the value 1 (the value can occur only once in the column) (6) If A4 does not contain the value 1 and A6 does not contain the value 1 (4), then B5 must contain the value 1 (only remaining possible position in the block) (7) If B5 contains the value 1, then F5 cannot contain the value 1 (the value can occur only once in the row) (8) If A8 contains the value 1 (3), then C7 cannot contain the value 1 (the value can occur only once in the block) (9) If I9 contains the value 1 (initial assumption), then C9 cannot contain the value 1 (the value can occur only once in the row) (10) If C9 does not contain the value 1 and C7 does not contain the value 1 (8), then C3 must contain the value 1 (only remaining possible position in the column) (11) If C3 contains the value 1, then F3 cannot contain the value 1 (the value can occur only once in the row) (12) If F3 does not contain the value 1 and F5 does not contain the value 1 (7), then F2 must contain the value 1 (only remaining possible position in the column) (13) If F2 contains the value 1, then F2 cannot contain the value 2 (the cell can contain only one value) (14) If F2 does not contain the value 2, then H1 cannot contain the value 2 (Claiming: Cells G2,I2: 2 in row and block) (15) If B5 contains the value 1 (6), then H5 cannot contain the value 1 (the value can occur only once in the row) (16) If I9 contains the value 1 (initial assumption), then I4 cannot contain the value 1 (the value can occur only once in the column) (17) If I9 contains the value 1 (initial assumption), then I3 cannot contain the value 1 (the value can occur only once in the column) (18) If I9 contains the value 1 (initial assumption), then I1 cannot contain the value 1 (the value can occur only once in the column) (19) If I9 contains the value 1 (initial assumption), then I2 cannot contain the value 1 (the value can occur only once in the column) (20) If I2 does not contain the value 1, I1 does not contain the value 1 (18) and I3 does not contain the value 1 (17), then G1 must contain the value 1 (only remaining possible position in the block) (21) If G1 contains the value 1, then G6 cannot contain the value 1 (the value can occur only once in the column) (22) If G6 does not contain the value 1, I4 does not contain the value 1 (16) and H5 does not contain the value 1 (15), then H4 must contain the value 1 (only remaining possible position in the block) (23) If H4 contains the value 1, then H4 cannot contain the value 2 (the cell can contain only one value) (24) If H4 does not contain the value 2 and H1 does not contain the value 2 (14), then H7 must contain the value 2 (only remaining possible position in the column) (25) If H7 contains the value 2, then A7 cannot contain the value 2 (the value can occur only once in the row) (26) If A8 contains the value 1 (3), then A8 cannot contain the value 2 (the cell can contain only one value) (27) If A8 does not contain the value 2 and A7 does not contain the value 2 (25), then C9 must contain the value 2 (only remaining possible position in the block) (28) If C9 contains the value 2, then C6 cannot contain the value 2 (the value can occur only once in the column) (29) If B5 contains the value 1 (6), then B5 cannot contain the value 2 (the cell can contain only one value) (30) If C9 contains the value 2 (27), then C5 cannot contain the value 2 (the value can occur only once in the column) (31) If C5 does not contain the value 2, B5 does not contain the value 2 (29) and C6 does not contain the value 2 (28), then B4 must contain the value 2 (only remaining possible position in the block) (32) If B4 contains the value 2, then F4 cannot contain the value 2 (the value can occur only once in the row) (33) If C9 contains the value 2 (27), then F9 cannot contain the value 2 (the value can occur only once in the row) (34) If F9 does not contain the value 2, F2 does not contain the value 2 (13) and F4 does not contain the value 2 (32), then F8 must contain the value 2 (only remaining possible position in the column) (35) If F8 contains the value 2, then F8 cannot contain the value 8 (the cell can contain only one value) (36) If A8 contains the value 1 (3), then A8 cannot contain the value 8 (the cell can contain only one value) (37) If A8 does not contain the value 8 and F8 does not contain the value 8 (35), then H8 must contain the value 8 (only remaining possible position in the row) (38) If H8 contains the value 8, then H8 cannot contain the value 3 (the cell can contain only one value) (39) If F8 contains the value 2 (34), then F8 cannot contain the value 6 (the cell can contain only one value) (40) If F2 does not contain the value 2 (13), then A3 cannot contain the value 2 (Pointing: Cells D3,E3: 2 in block and row) (41) If A8 does not contain the value 2 (26), A3 does not contain the value 2 and A7 does not contain the value 2 (25), then A1 must contain the value 2 (only remaining possible position in the column) (42) If A1 contains the value 2, then B1 cannot contain the value 2 (the value can occur only once in the block) (43) If G1 contains the value 1 (20), then B1 cannot contain the value 1 (the value can occur only once in the row) (44) If B1 does not contain the value 1 and B1 does not contain the value 2 (42), then B1 must contain the value 6 (only remaining possible value in the cell) (45) If B1 contains the value 6, then B8 cannot contain the value 6 (the value can occur only once in the column) (46) If B8 does not contain the value 6 and F8 does not contain the value 6 (39), then I8 must contain the value 6 (only remaining possible position in the row) (47) If I8 contains the value 6, then I8 cannot contain the value 3 (the cell can contain only one value) (48) If I8 does not contain the value 3 and H8 does not contain the value 3 (38), then I7 must contain the value 3 (only remaining possible position in the block) (49) If I7 contains the value 3, then I7 cannot contain the value 4 (the cell can contain only one value) (50) If H8 contains the value 8 (37), then H8 cannot contain the value 4 (the cell can contain only one value) (51) If I9 contains the value 1 (initial assumption), then I9 cannot contain the value 4 (the cell can contain only one value) (52) If I9 does not contain the value 4, H8 does not contain the value 4 (50) and I7 does not contain the value 4 (49), then G8 must contain the value 4 (only remaining possible position in the block) (53) If G8 contains the value 4, then E8 cannot contain the value 4 (the value can occur only once in the row) (54) If F8 contains the value 2 (34), then E8 cannot contain the value 2 (the value can occur only once in the block) (55) If F2 contains the value 1 (12), then F2 cannot contain the value 7 (the cell can contain only one value) (56) If F2 does not contain the value 7, then E8 cannot contain the value 7 (Pointing: Cells E1,E2,E3: 7 in block and column) (57) If E8 does not contain the value 7, E8 does not contain the value 2 (54) and E8 does not contain the value 4 (53), then E8 must contain the value 3 (only remaining possible value in the cell) (58) If E8 contains the value 3, then D9 cannot contain the value 3 (the value can occur only once in the block) (59) If B1 contains the value 6 (44), then C1 cannot contain the value 6 (the value can occur only once in the block) (60) If C3 contains the value 1 (10), then C3 cannot contain the value 6 (the cell can contain only one value) (61) If C9 contains the value 2 (27), then C9 cannot contain the value 6 (the cell can contain only one value) (62) If C9 does not contain the value 6, C3 does not contain the value 6 (60) and C1 does not contain the value 6 (59), then C7 must contain the value 6 (only remaining possible position in the column) (63) If C7 contains the value 6, then G7 cannot contain the value 6 (the value can occur only once in the row) (64) If G1 contains the value 1 (20), then G1 cannot contain the value 6 (the cell can contain only one value) (65) If G1 does not contain the value 6 and G7 does not contain the value 6 (63), then G4 must contain the value 6 (only remaining possible position in the column) (66) If G4 contains the value 6, then F4 cannot contain the value 6 (the value can occur only once in the row) (67) If F4 does not contain the value 2 (32) and F4 does not contain the value 6, then F4 must contain the value 3 (only remaining possible value in the cell) (68) If F4 contains the value 3, then D5 cannot contain the value 3 (the value can occur only once in the block) (69) If F4 contains the value 3 (67), then D6 cannot contain the value 3 (the value can occur only once in the block) (70) If I7 contains the value 3 (48), then D7 cannot contain the value 3 (the value can occur only once in the row) (71) If H8 contains the value 8 (37), then G9 cannot contain the value 8 (the value can occur only once in the block) (72) If I9 contains the value 1 (initial assumption), then G9 cannot contain the value 1 (the value can occur only once in the block) (73) If G9 does not contain the value 1 and G9 does not contain the value 8 (71), then G9 must contain the value 7 (only remaining possible value in the cell) (74) If G9 contains the value 7, then G8 cannot contain the value 7 (the value can occur only once in the block) (75) If E8 does not contain the value 7 (56) and G8 does not contain the value 7, then D8 must contain the value 7 (only remaining possible position in the row) (76) If D8 contains the value 7, then D8 cannot contain the value 3 (the cell can contain only one value) (77) If D8 does not contain the value 3, D7 does not contain the value 3 (70), D6 does not contain the value 3 (69), D5 does not contain the value 3 (68) and D9 does not contain the value 3 (58), then D2 must contain the value 3 (only remaining possible position in the column) (78) If D2 contains the value 3, then D2 cannot contain the value 6 (the cell can contain only one value) (79) If G4 contains the value 6 (65), then D4 cannot contain the value 6 (the value can occur only once in the row) (80) If C7 contains the value 6 (62), then D7 cannot contain the value 6 (the value can occur only once in the row) (81) If D7 does not contain the value 6, D4 does not contain the value 6 (79) and D2 does not contain the value 6 (78), then D5 must contain the value 6 (only remaining possible position in the column) Chain 13: If I9 contains the value 1, then D5 cannot contain the value 6 (View 13): (1) If I9 contains the value 1, then H8 cannot contain the value 1 (the value can occur only once in the block) (2) If I9 contains the value 1 (initial assumption), then G8 cannot contain the value 1 (the value can occur only once in the block) (3) If G8 does not contain the value 1 and H8 does not contain the value 1 (1), then A8 must contain the value 1 (only remaining possible position in the row) (4) If A8 contains the value 1, then A6 cannot contain the value 1 (the value can occur only once in the column) (5) If A8 contains the value 1 (3), then A4 cannot contain the value 1 (the value can occur only once in the column) (6) If A4 does not contain the value 1 and A6 does not contain the value 1 (4), then B5 must contain the value 1 (only remaining possible position in the block) (7) If B5 contains the value 1, then F5 cannot contain the value 1 (the value can occur only once in the row) (8) If A8 contains the value 1 (3), then C7 cannot contain the value 1 (the value can occur only once in the block) (9) If I9 contains the value 1 (initial assumption), then C9 cannot contain the value 1 (the value can occur only once in the row) (10) If C9 does not contain the value 1 and C7 does not contain the value 1 (8), then C3 must contain the value 1 (only remaining possible position in the column) (11) If C3 contains the value 1, then F3 cannot contain the value 1 (the value can occur only once in the row) (12) If F3 does not contain the value 1 and F5 does not contain the value 1 (7), then F2 must contain the value 1 (only remaining possible position in the column) (13) If F2 contains the value 1, then F2 cannot contain the value 2 (the cell can contain only one value) (14) If F2 does not contain the value 2, then H1 cannot contain the value 2 (Claiming: Cells G2,I2: 2 in row and block) (15) If B5 contains the value 1 (6), then H5 cannot contain the value 1 (the value can occur only once in the row) (16) If I9 contains the value 1 (initial assumption), then I4 cannot contain the value 1 (the value can occur only once in the column) (17) If I9 contains the value 1 (initial assumption), then I3 cannot contain the value 1 (the value can occur only once in the column) (18) If I9 contains the value 1 (initial assumption), then I1 cannot contain the value 1 (the value can occur only once in the column) (19) If I9 contains the value 1 (initial assumption), then I2 cannot contain the value 1 (the value can occur only once in the column) (20) If I2 does not contain the value 1, I1 does not contain the value 1 (18) and I3 does not contain the value 1 (17), then G1 must contain the value 1 (only remaining possible position in the block) (21) If G1 contains the value 1, then G6 cannot contain the value 1 (the value can occur only once in the column) (22) If G6 does not contain the value 1, I4 does not contain the value 1 (16) and H5 does not contain the value 1 (15), then H4 must contain the value 1 (only remaining possible position in the block) (23) If H4 contains the value 1, then H4 cannot contain the value 2 (the cell can contain only one value) (24) If H4 does not contain the value 2 and H1 does not contain the value 2 (14), then H7 must contain the value 2 (only remaining possible position in the column) (25) If H7 contains the value 2, then A7 cannot contain the value 2 (the value can occur only once in the row) (26) If A8 contains the value 1 (3), then A8 cannot contain the value 2 (the cell can contain only one value) (27) If A8 does not contain the value 2 and A7 does not contain the value 2 (25), then C9 must contain the value 2 (only remaining possible position in the block) (28) If C9 contains the value 2, then C6 cannot contain the value 2 (the value can occur only once in the column) (29) If B5 contains the value 1 (6), then B5 cannot contain the value 2 (the cell can contain only one value) (30) If C9 contains the value 2 (27), then C5 cannot contain the value 2 (the value can occur only once in the column) (31) If C5 does not contain the value 2, B5 does not contain the value 2 (29) and C6 does not contain the value 2 (28), then B4 must contain the value 2 (only remaining possible position in the block) (32) If B4 contains the value 2, then F4 cannot contain the value 2 (the value can occur only once in the row) (33) If C9 contains the value 2 (27), then F9 cannot contain the value 2 (the value can occur only once in the row) (34) If F9 does not contain the value 2, F2 does not contain the value 2 (13) and F4 does not contain the value 2 (32), then F8 must contain the value 2 (only remaining possible position in the column) (35) If F8 contains the value 2, then F8 cannot contain the value 8 (the cell can contain only one value) (36) If A8 contains the value 1 (3), then A8 cannot contain the value 8 (the cell can contain only one value) (37) If A8 does not contain the value 8 and F8 does not contain the value 8 (35), then H8 must contain the value 8 (only remaining possible position in the row) (38) If H8 contains the value 8, then G9 cannot contain the value 8 (the value can occur only once in the block) (39) If I9 contains the value 1 (initial assumption), then G9 cannot contain the value 1 (the value can occur only once in the block) (40) If G9 does not contain the value 1 and G9 does not contain the value 8 (38), then G9 must contain the value 7 (only remaining possible value in the cell) (41) If G9 contains the value 7, then G5 cannot contain the value 7 (the value can occur only once in the column) (42) If F2 contains the value 1 (12), then F2 cannot contain the value 7 (the cell can contain only one value) (43) If F2 does not contain the value 7, then E5 cannot contain the value 7 (Pointing: Cells E1,E2,E3: 7 in block and column) (44) If E5 does not contain the value 7 and G5 does not contain the value 7 (41), then A5 must contain the value 7 (only remaining possible position in the row) (45) If A5 contains the value 7, then A5 cannot contain the value 3 (the cell can contain only one value) (46) If B5 contains the value 1 (6), then B5 cannot contain the value 3 (the cell can contain only one value) (47) If F2 does not contain the value 2 (13), then A3 cannot contain the value 2 (Pointing: Cells D3,E3: 2 in block and row) (48) If A8 does not contain the value 2 (26), A3 does not contain the value 2 and A7 does not contain the value 2 (25), then A1 must contain the value 2 (only remaining possible position in the column) (49) If A1 contains the value 2, then B1 cannot contain the value 2 (the value can occur only once in the block) (50) If G1 contains the value 1 (20), then B1 cannot contain the value 1 (the value can occur only once in the row) (51) If B1 does not contain the value 1 and B1 does not contain the value 2 (49), then B1 must contain the value 6 (only remaining possible value in the cell) (52) If B1 contains the value 6, then C1 cannot contain the value 6 (the value can occur only once in the block) (53) If C3 contains the value 1 (10), then C3 cannot contain the value 6 (the cell can contain only one value) (54) If C9 contains the value 2 (27), then C9 cannot contain the value 6 (the cell can contain only one value) (55) If C9 does not contain the value 6, C3 does not contain the value 6 (53) and C1 does not contain the value 6 (52), then C7 must contain the value 6 (only remaining possible position in the column) (56) If C7 contains the value 6, then G7 cannot contain the value 6 (the value can occur only once in the row) (57) If G1 contains the value 1 (20), then G1 cannot contain the value 6 (the cell can contain only one value) (58) If G1 does not contain the value 6 and G7 does not contain the value 6 (56), then G4 must contain the value 6 (only remaining possible position in the column) (59) If G4 contains the value 6, then F4 cannot contain the value 6 (the value can occur only once in the row) (60) If F4 does not contain the value 2 (32) and F4 does not contain the value 6, then F4 must contain the value 3 (only remaining possible value in the cell) (61) If F4 contains the value 3, then D5 cannot contain the value 3 (the value can occur only once in the block) (62) If D5 does not contain the value 3, B5 does not contain the value 3 (46) and A5 does not contain the value 3 (45), then G5 must contain the value 3 (only remaining possible position in the row) (63) If G5 contains the value 3, then G5 cannot contain the value 2 (the cell can contain only one value) (64) If C5 does not contain the value 2 (30), B5 does not contain the value 2 (29) and G5 does not contain the value 2, then D5 must contain the value 2 (only remaining possible position in the row) (65) If D5 contains the value 2, then D5 cannot contain the value 6 (the cell can contain only one value) Nested Contradiction Forcing Chain: B7.6 on ==> G8.4 both on & off Chain 14: If B7 contains the value 6, then G8 must contain the value 4 (View 14): (1) If B7 contains the value 6, then H7 cannot contain the value 6 (the value can occur only once in the row) (2) If B7 contains the value 6 (initial assumption), then G7 cannot contain the value 6 (the value can occur only once in the row) (3) If G7 does not contain the value 6 and H7 does not contain the value 6 (1), then I8 must contain the value 6 (only remaining possible position in the block) (4) If I8 contains the value 6, then I6 cannot contain the value 6 (the value can occur only once in the column) (5) If I6 does not contain the value 6, then A6 must contain the value 6 (only remaining possible position in the row) (6) If A6 contains the value 6, then A6 cannot contain the value 4 (the cell can contain only one value) (7) If A6 does not contain the value 4, then B8 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (8) If B7 contains the value 6 (initial assumption), then B8 cannot contain the value 6 (the value can occur only once in the block) (9) If B8 does not contain the value 6 and B8 does not contain the value 4 (7), then B8 must contain the value 9 (only remaining possible value in the cell) (10) If B8 contains the value 9, then C7 cannot contain the value 9 (the value can occur only once in the block) (11) If A6 does not contain the value 4 (6), then C7 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (12) If B7 contains the value 6 (initial assumption), then C7 cannot contain the value 6 (the value can occur only once in the block) (13) If C7 does not contain the value 6, C7 does not contain the value 4 (11) and C7 does not contain the value 9 (10), then C7 must contain the value 1 (only remaining possible value in the cell) (14) If C7 contains the value 1, then E7 cannot contain the value 1 (the value can occur only once in the row) (15) If I8 contains the value 6 (3), then I8 cannot contain the value 2 (the cell can contain only one value) (16) If I8 does not contain the value 2, then E7 cannot contain the value 2 (Pointing: Cells H7,I7: 2 in block and row) (17) If B7 contains the value 6 (initial assumption), then E7 cannot contain the value 6 (the value can occur only once in the row) (18) If E7 does not contain the value 6, E7 does not contain the value 2 (16) and E7 does not contain the value 1 (14), then E7 must contain the value 4 (only remaining possible value in the cell) (19) If E7 contains the value 4, then D7 cannot contain the value 4 (the value can occur only once in the block) (20) If C7 contains the value 1 (13), then D7 cannot contain the value 1 (the value can occur only once in the row) (21) If B7 contains the value 6 (initial assumption), then D7 cannot contain the value 6 (the value can occur only once in the row) (22) If D7 does not contain the value 6, D7 does not contain the value 1 (20) and D7 does not contain the value 4 (19), then D7 must contain the value 3 (only remaining possible value in the cell) (23) If D7 contains the value 3, then I7 cannot contain the value 3 (the value can occur only once in the row) (24) If I8 contains the value 6 (3), then I8 cannot contain the value 3 (the cell can contain only one value) (25) If I8 does not contain the value 3 and I7 does not contain the value 3 (23), then H8 must contain the value 3 (only remaining possible position in the block) (26) If H8 contains the value 3, then H8 cannot contain the value 4 (the cell can contain only one value) (27) If E7 contains the value 4 (18), then I7 cannot contain the value 4 (the value can occur only once in the row) (28) If E7 contains the value 4 (18), then A7 cannot contain the value 4 (the value can occur only once in the row) (29) If B8 does not contain the value 4 (7), C7 does not contain the value 4 (11) and A7 does not contain the value 4, then A9 must contain the value 4 (only remaining possible position in the block) (30) If A9 contains the value 4, then I9 cannot contain the value 4 (the value can occur only once in the row) (31) If I9 does not contain the value 4, I7 does not contain the value 4 (27) and H8 does not contain the value 4 (26), then G8 must contain the value 4 (only remaining possible position in the block) Chain 15: If B7 contains the value 6, then G8 cannot contain the value 4 (View 15): (1) If B7 contains the value 6, then H7 cannot contain the value 6 (the value can occur only once in the row) (2) If B7 contains the value 6 (initial assumption), then G7 cannot contain the value 6 (the value can occur only once in the row) (3) If G7 does not contain the value 6 and H7 does not contain the value 6 (1), then I8 must contain the value 6 (only remaining possible position in the block) (4) If I8 contains the value 6, then I6 cannot contain the value 6 (the value can occur only once in the column) (5) If I6 does not contain the value 6, then A6 must contain the value 6 (only remaining possible position in the row) (6) If A6 contains the value 6, then A6 cannot contain the value 4 (the cell can contain only one value) (7) If A6 does not contain the value 4, then B8 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (8) If B7 contains the value 6 (initial assumption), then B8 cannot contain the value 6 (the value can occur only once in the block) (9) If B8 does not contain the value 6 and B8 does not contain the value 4 (7), then B8 must contain the value 9 (only remaining possible value in the cell) (10) If B8 contains the value 9, then C7 cannot contain the value 9 (the value can occur only once in the block) (11) If A6 does not contain the value 4 (6), then C7 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (12) If B7 contains the value 6 (initial assumption), then C7 cannot contain the value 6 (the value can occur only once in the block) (13) If C7 does not contain the value 6, C7 does not contain the value 4 (11) and C7 does not contain the value 9 (10), then C7 must contain the value 1 (only remaining possible value in the cell) (14) If C7 contains the value 1, then E7 cannot contain the value 1 (the value can occur only once in the row) (15) If I8 contains the value 6 (3), then I8 cannot contain the value 2 (the cell can contain only one value) (16) If I8 does not contain the value 2, then E7 cannot contain the value 2 (Pointing: Cells H7,I7: 2 in block and row) (17) If B7 contains the value 6 (initial assumption), then E7 cannot contain the value 6 (the value can occur only once in the row) (18) If E7 does not contain the value 6, E7 does not contain the value 2 (16) and E7 does not contain the value 1 (14), then E7 must contain the value 4 (only remaining possible value in the cell) (19) If E7 contains the value 4, then D7 cannot contain the value 4 (the value can occur only once in the block) (20) If C7 contains the value 1 (13), then D7 cannot contain the value 1 (the value can occur only once in the row) (21) If B7 contains the value 6 (initial assumption), then D7 cannot contain the value 6 (the value can occur only once in the row) (22) If D7 does not contain the value 6, D7 does not contain the value 1 (20) and D7 does not contain the value 4 (19), then D7 must contain the value 3 (only remaining possible value in the cell) (23) If D7 contains the value 3, then I7 cannot contain the value 3 (the value can occur only once in the row) (24) If I8 contains the value 6 (3), then I8 cannot contain the value 3 (the cell can contain only one value) (25) If I8 does not contain the value 3 and I7 does not contain the value 3 (23), then H8 must contain the value 3 (only remaining possible position in the block) (26) If H8 contains the value 3, then H8 cannot contain the value 1 (the cell can contain only one value) (27) If C7 contains the value 1 (13), then H7 cannot contain the value 1 (the value can occur only once in the row) (28) If D7 does not contain the value 1 (20) and E7 does not contain the value 1 (14), then E9 must contain the value 1 (only remaining possible position in the block) (29) If E9 contains the value 1, then G9 cannot contain the value 1 (the value can occur only once in the row) (30) If E9 contains the value 1 (28), then I9 cannot contain the value 1 (the value can occur only once in the row) (31) If I9 does not contain the value 1, G9 does not contain the value 1 (29), H7 does not contain the value 1 (27) and H8 does not contain the value 1 (26), then G8 must contain the value 1 (only remaining possible position in the block) (32) If G8 contains the value 1, then G8 cannot contain the value 4 (the cell can contain only one value) Nested Contradiction Forcing Chain: C9.6 on ==> D4.8 both on & off Chain 16: If C9 contains the value 6, then D4 must contain the value 8 (View 16): (1) If C9 contains the value 6, then C9 cannot contain the value 2 (the cell can contain only one value) (2) If C9 does not contain the value 2, then A1 cannot contain the value 2 (Pointing: Cells A7,A8: 2 in block and column) (3) If C9 contains the value 6 (initial assumption), then B8 cannot contain the value 6 (the value can occur only once in the block) (4) If C9 contains the value 6 (initial assumption), then B7 cannot contain the value 6 (the value can occur only once in the block) (5) If B7 does not contain the value 6 and B8 does not contain the value 6 (3), then B1 must contain the value 6 (only remaining possible position in the column) (6) If B1 contains the value 6, then B1 cannot contain the value 2 (the cell can contain only one value) (7) If B1 does not contain the value 2 and A1 does not contain the value 2 (2), then H1 must contain the value 2 (only remaining possible position in the row) (8) If H1 contains the value 2, then I2 cannot contain the value 2 (the value can occur only once in the block) (9) If H1 contains the value 2 (7), then G2 cannot contain the value 2 (the value can occur only once in the block) (10) If G2 does not contain the value 2 and I2 does not contain the value 2 (8), then F2 must contain the value 2 (only remaining possible position in the row) (11) If F2 contains the value 2, then F2 cannot contain the value 7 (the cell can contain only one value) (12) If F2 does not contain the value 7, then E8 cannot contain the value 7 (Pointing: Cells E1,E2,E3: 7 in block and column) (13) If C9 does not contain the value 2 (1), then E8 cannot contain the value 2 (Claiming: Cells D9,F9: 2 in row and block) (14) If H1 contains the value 2 (7), then H1 cannot contain the value 3 (the cell can contain only one value) (15) If H1 does not contain the value 3, then H8 must contain the value 3 (only remaining possible position in the column) (16) If H8 contains the value 3, then E8 cannot contain the value 3 (the value can occur only once in the row) (17) If E8 does not contain the value 3, E8 does not contain the value 2 (13) and E8 does not contain the value 7 (12), then E8 must contain the value 4 (only remaining possible value in the cell) (18) If E8 contains the value 4, then B8 cannot contain the value 4 (the value can occur only once in the row) (19) If B8 does not contain the value 6 (3) and B8 does not contain the value 4, then B8 must contain the value 9 (only remaining possible value in the cell) (20) If B8 contains the value 9, then D8 cannot contain the value 9 (the value can occur only once in the row) (21) If F2 contains the value 2 (10), then F9 cannot contain the value 2 (the value can occur only once in the column) (22) If C9 does not contain the value 2 (1), then E7 cannot contain the value 2 (Claiming: Cells D9,F9: 2 in row and block) (23) If C9 does not contain the value 2 (1), then F8 cannot contain the value 2 (Claiming: Cells D9,F9: 2 in row and block) (24) If F8 does not contain the value 2, E8 does not contain the value 2 (13), E7 does not contain the value 2 (22) and F9 does not contain the value 2 (21), then D9 must contain the value 2 (only remaining possible position in the block) (25) If D9 contains the value 2, then D9 cannot contain the value 9 (the cell can contain only one value) (26) If D9 does not contain the value 9 and D8 does not contain the value 9 (20), then D3 must contain the value 9 (only remaining possible position in the column) (27) If D3 contains the value 9, then D3 cannot contain the value 8 (the cell can contain only one value) (28) If D9 contains the value 2 (24), then D9 cannot contain the value 8 (the cell can contain only one value) (29) If D9 does not contain the value 8 and D3 does not contain the value 8 (27), then D4 must contain the value 8 (only remaining possible position in the column) Chain 17: If C9 contains the value 6, then D4 cannot contain the value 8 (View 17): (1) If C9 contains the value 6, then C9 cannot contain the value 2 (the cell can contain only one value) (2) If C9 does not contain the value 2, then A1 cannot contain the value 2 (Pointing: Cells A7,A8: 2 in block and column) (3) If C9 contains the value 6 (initial assumption), then B8 cannot contain the value 6 (the value can occur only once in the block) (4) If C9 contains the value 6 (initial assumption), then B7 cannot contain the value 6 (the value can occur only once in the block) (5) If B7 does not contain the value 6 and B8 does not contain the value 6 (3), then B1 must contain the value 6 (only remaining possible position in the column) (6) If B1 contains the value 6, then B1 cannot contain the value 2 (the cell can contain only one value) (7) If B1 does not contain the value 2 and A1 does not contain the value 2 (2), then H1 must contain the value 2 (only remaining possible position in the row) (8) If H1 contains the value 2, then I2 cannot contain the value 2 (the value can occur only once in the block) (9) If H1 contains the value 2 (7), then G2 cannot contain the value 2 (the value can occur only once in the block) (10) If G2 does not contain the value 2 and I2 does not contain the value 2 (8), then F2 must contain the value 2 (only remaining possible position in the row) (11) If F2 contains the value 2, then F2 cannot contain the value 7 (the cell can contain only one value) (12) If F2 does not contain the value 7, then E8 cannot contain the value 7 (Pointing: Cells E1,E2,E3: 7 in block and column) (13) If C9 does not contain the value 2 (1), then E8 cannot contain the value 2 (Claiming: Cells D9,F9: 2 in row and block) (14) If H1 contains the value 2 (7), then H1 cannot contain the value 3 (the cell can contain only one value) (15) If H1 does not contain the value 3, then H8 must contain the value 3 (only remaining possible position in the column) (16) If H8 contains the value 3, then E8 cannot contain the value 3 (the value can occur only once in the row) (17) If E8 does not contain the value 3, E8 does not contain the value 2 (13) and E8 does not contain the value 7 (12), then E8 must contain the value 4 (only remaining possible value in the cell) (18) If E8 contains the value 4, then B8 cannot contain the value 4 (the value can occur only once in the row) (19) If B8 does not contain the value 6 (3) and B8 does not contain the value 4, then B8 must contain the value 9 (only remaining possible value in the cell) (20) If B8 contains the value 9, then D8 cannot contain the value 9 (the value can occur only once in the row) (21) If F2 contains the value 2 (10), then F9 cannot contain the value 2 (the value can occur only once in the column) (22) If C9 does not contain the value 2 (1), then E7 cannot contain the value 2 (Claiming: Cells D9,F9: 2 in row and block) (23) If C9 does not contain the value 2 (1), then F8 cannot contain the value 2 (Claiming: Cells D9,F9: 2 in row and block) (24) If F8 does not contain the value 2, E8 does not contain the value 2 (13), E7 does not contain the value 2 (22) and F9 does not contain the value 2 (21), then D9 must contain the value 2 (only remaining possible position in the block) (25) If D9 contains the value 2, then D9 cannot contain the value 9 (the cell can contain only one value) (26) If D9 does not contain the value 9 and D8 does not contain the value 9 (20), then D3 must contain the value 9 (only remaining possible position in the column) (27) If D3 contains the value 9, then D3 cannot contain the value 4 (the cell can contain only one value) (28) If D9 contains the value 2 (24), then D9 cannot contain the value 3 (the cell can contain only one value) (29) If H8 contains the value 3 (15), then D8 cannot contain the value 3 (the value can occur only once in the row) (30) If E8 does not contain the value 3 (16), D8 does not contain the value 3 and D9 does not contain the value 3 (28), then D7 must contain the value 3 (only remaining possible position in the block) (31) If D7 contains the value 3, then D7 cannot contain the value 4 (the cell can contain only one value) (32) If D9 contains the value 2 (24), then D9 cannot contain the value 4 (the cell can contain only one value) (33) If D9 does not contain the value 4, D7 does not contain the value 4 (31) and D3 does not contain the value 4 (27), then D4 must contain the value 4 (only remaining possible position in the column) (34) If D4 contains the value 4, then D4 cannot contain the value 8 (the cell can contain only one value) Nested Contradiction Forcing Chain: B3.2 on ==> E8.4 both on & off Chain 18: If B3 contains the value 2, then E8 must contain the value 4 (View 18): (1) If B3 contains the value 2, then B4 cannot contain the value 2 (the value can occur only once in the column) (2) If B3 contains the value 2 (initial assumption), then B5 cannot contain the value 2 (the value can occur only once in the column) (3) If B5 does not contain the value 2 and B4 does not contain the value 2 (1), then C9 cannot contain the value 2 (Pointing: Cells C5,C6: 2 in block and column) (4) If B3 contains the value 2 (initial assumption), then E3 cannot contain the value 2 (the value can occur only once in the row) (5) If B3 contains the value 2 (initial assumption), then D3 cannot contain the value 2 (the value can occur only once in the row) (6) If D3 does not contain the value 2 and E3 does not contain the value 2 (4), then F2 must contain the value 2 (only remaining possible position in the block) (7) If F2 contains the value 2, then F9 cannot contain the value 2 (the value can occur only once in the column) (8) If F9 does not contain the value 2 and C9 does not contain the value 2 (3), then D9 must contain the value 2 (only remaining possible position in the row) (9) If D9 contains the value 2, then E8 cannot contain the value 2 (the value can occur only once in the block) (10) If B3 contains the value 2 (initial assumption), then B1 cannot contain the value 2 (the value can occur only once in the block) (11) If B3 contains the value 2 (initial assumption), then A1 cannot contain the value 2 (the value can occur only once in the block) (12) If A1 does not contain the value 2 and B1 does not contain the value 2 (10), then H1 must contain the value 2 (only remaining possible position in the row) (13) If H1 contains the value 2, then H1 cannot contain the value 3 (the cell can contain only one value) (14) If H1 does not contain the value 3, then H8 must contain the value 3 (only remaining possible position in the column) (15) If H8 contains the value 3, then E8 cannot contain the value 3 (the value can occur only once in the row) (16) If F2 contains the value 2 (6), then F2 cannot contain the value 7 (the cell can contain only one value) (17) If F2 does not contain the value 7, then E8 cannot contain the value 7 (Pointing: Cells E1,E2,E3: 7 in block and column) (18) If E8 does not contain the value 7, E8 does not contain the value 3 (15) and E8 does not contain the value 2 (9), then E8 must contain the value 4 (only remaining possible value in the cell) Chain 19: If B3 contains the value 2, then E8 cannot contain the value 4 (View 19): (1) If B3 contains the value 2, then B3 cannot contain the value 4 (the cell can contain only one value) (2) If B3 does not contain the value 4, then G2 cannot contain the value 4 (Pointing: Cells B2,C2: 4 in block and row) (3) If B3 contains the value 2 (initial assumption), then E3 cannot contain the value 2 (the value can occur only once in the row) (4) If B3 contains the value 2 (initial assumption), then D3 cannot contain the value 2 (the value can occur only once in the row) (5) If D3 does not contain the value 2 and E3 does not contain the value 2 (3), then F2 must contain the value 2 (only remaining possible position in the block) (6) If F2 contains the value 2, then F4 cannot contain the value 2 (the value can occur only once in the column) (7) If B3 contains the value 2 (initial assumption), then B4 cannot contain the value 2 (the value can occur only once in the column) (8) If B3 contains the value 2 (initial assumption), then B1 cannot contain the value 2 (the value can occur only once in the block) (9) If B3 contains the value 2 (initial assumption), then A1 cannot contain the value 2 (the value can occur only once in the block) (10) If A1 does not contain the value 2 and B1 does not contain the value 2 (8), then H1 must contain the value 2 (only remaining possible position in the row) (11) If H1 contains the value 2, then H4 cannot contain the value 2 (the value can occur only once in the column) (12) If H4 does not contain the value 2, B4 does not contain the value 2 (7) and F4 does not contain the value 2 (6), then G4 must contain the value 2 (only remaining possible position in the row) (13) If G4 contains the value 2, then G4 cannot contain the value 4 (the cell can contain only one value) (14) If G4 does not contain the value 4 and G2 does not contain the value 4 (2), then G8 must contain the value 4 (only remaining possible position in the column) (15) If G8 contains the value 4, then E8 cannot contain the value 4 (the value can occur only once in the row) Nested Contradiction Forcing Chain: C3.2 on ==> D4.8 both on & off Chain 20: If C3 contains the value 2, then D4 must contain the value 8 (View 20): (1) If C3 contains the value 2, then C3 cannot contain the value 1 (the cell can contain only one value) (2) If C3 does not contain the value 1, then A8 cannot contain the value 1 (Claiming: Cells C7,C9: 1 in column and block) (3) If C3 contains the value 2 (initial assumption), then B1 cannot contain the value 2 (the value can occur only once in the block) (4) If C3 contains the value 2 (initial assumption), then A1 cannot contain the value 2 (the value can occur only once in the block) (5) If A1 does not contain the value 2 and B1 does not contain the value 2 (3), then H1 must contain the value 2 (only remaining possible position in the row) (6) If H1 contains the value 2, then H1 cannot contain the value 3 (the cell can contain only one value) (7) If H1 does not contain the value 3, then H8 must contain the value 3 (only remaining possible position in the column) (8) If H8 contains the value 3, then H8 cannot contain the value 1 (the cell can contain only one value) (9) If H8 does not contain the value 1 and A8 does not contain the value 1 (2), then G8 must contain the value 1 (only remaining possible position in the row) (10) If G8 contains the value 1, then G8 cannot contain the value 7 (the cell can contain only one value) (11) If C3 contains the value 2 (initial assumption), then E3 cannot contain the value 2 (the value can occur only once in the row) (12) If C3 contains the value 2 (initial assumption), then D3 cannot contain the value 2 (the value can occur only once in the row) (13) If D3 does not contain the value 2 and E3 does not contain the value 2 (11), then F2 must contain the value 2 (only remaining possible position in the block) (14) If F2 contains the value 2, then F2 cannot contain the value 7 (the cell can contain only one value) (15) If F2 does not contain the value 7, then E8 cannot contain the value 7 (Pointing: Cells E1,E2,E3: 7 in block and column) (16) If E8 does not contain the value 7 and G8 does not contain the value 7 (10), then D8 must contain the value 7 (only remaining possible position in the row) (17) If D8 contains the value 7, then D8 cannot contain the value 9 (the cell can contain only one value) (18) If F2 contains the value 2 (13), then F9 cannot contain the value 2 (the value can occur only once in the column) (19) If C3 contains the value 2 (initial assumption), then C9 cannot contain the value 2 (the value can occur only once in the column) (20) If C9 does not contain the value 2 and F9 does not contain the value 2 (18), then D9 must contain the value 2 (only remaining possible position in the row) (21) If D9 contains the value 2, then D9 cannot contain the value 9 (the cell can contain only one value) (22) If D9 does not contain the value 9 and D8 does not contain the value 9 (17), then D3 must contain the value 9 (only remaining possible position in the column) (23) If D3 contains the value 9, then D3 cannot contain the value 8 (the cell can contain only one value) (24) If D9 contains the value 2 (20), then D9 cannot contain the value 8 (the cell can contain only one value) (25) If D9 does not contain the value 8 and D3 does not contain the value 8 (23), then D4 must contain the value 8 (only remaining possible position in the column) Chain 21: If C3 contains the value 2, then D4 cannot contain the value 8 (View 21): (1) If C3 contains the value 2, then C3 cannot contain the value 1 (the cell can contain only one value) (2) If C3 does not contain the value 1, then A8 cannot contain the value 1 (Claiming: Cells C7,C9: 1 in column and block) (3) If C3 contains the value 2 (initial assumption), then B1 cannot contain the value 2 (the value can occur only once in the block) (4) If C3 contains the value 2 (initial assumption), then A1 cannot contain the value 2 (the value can occur only once in the block) (5) If A1 does not contain the value 2 and B1 does not contain the value 2 (3), then H1 must contain the value 2 (only remaining possible position in the row) (6) If H1 contains the value 2, then H1 cannot contain the value 3 (the cell can contain only one value) (7) If H1 does not contain the value 3, then H8 must contain the value 3 (only remaining possible position in the column) (8) If H8 contains the value 3, then H8 cannot contain the value 1 (the cell can contain only one value) (9) If H8 does not contain the value 1 and A8 does not contain the value 1 (2), then G8 must contain the value 1 (only remaining possible position in the row) (10) If G8 contains the value 1, then G8 cannot contain the value 7 (the cell can contain only one value) (11) If C3 contains the value 2 (initial assumption), then E3 cannot contain the value 2 (the value can occur only once in the row) (12) If C3 contains the value 2 (initial assumption), then D3 cannot contain the value 2 (the value can occur only once in the row) (13) If D3 does not contain the value 2 and E3 does not contain the value 2 (11), then F2 must contain the value 2 (only remaining possible position in the block) (14) If F2 contains the value 2, then F2 cannot contain the value 7 (the cell can contain only one value) (15) If F2 does not contain the value 7, then E8 cannot contain the value 7 (Pointing: Cells E1,E2,E3: 7 in block and column) (16) If E8 does not contain the value 7 and G8 does not contain the value 7 (10), then D8 must contain the value 7 (only remaining possible position in the row) (17) If D8 contains the value 7, then D8 cannot contain the value 9 (the cell can contain only one value) (18) If F2 contains the value 2 (13), then F9 cannot contain the value 2 (the value can occur only once in the column) (19) If C3 contains the value 2 (initial assumption), then C9 cannot contain the value 2 (the value can occur only once in the column) (20) If C9 does not contain the value 2 and F9 does not contain the value 2 (18), then D9 must contain the value 2 (only remaining possible position in the row) (21) If D9 contains the value 2, then D9 cannot contain the value 9 (the cell can contain only one value) (22) If D9 does not contain the value 9 and D8 does not contain the value 9 (17), then D3 must contain the value 9 (only remaining possible position in the column) (23) If D3 contains the value 9, then D3 cannot contain the value 4 (the cell can contain only one value) (24) If D9 contains the value 2 (20), then D9 cannot contain the value 3 (the cell can contain only one value) (25) If H8 contains the value 3 (7), then D8 cannot contain the value 3 (the value can occur only once in the row) (26) If H8 contains the value 3 (7), then E8 cannot contain the value 3 (the value can occur only once in the row) (27) If E8 does not contain the value 3, D8 does not contain the value 3 (25) and D9 does not contain the value 3 (24), then D7 must contain the value 3 (only remaining possible position in the block) (28) If D7 contains the value 3, then D7 cannot contain the value 4 (the cell can contain only one value) (29) If D9 contains the value 2 (20), then D9 cannot contain the value 4 (the cell can contain only one value) (30) If D9 does not contain the value 4, D7 does not contain the value 4 (28) and D3 does not contain the value 4 (23), then D4 must contain the value 4 (only remaining possible position in the column) (31) If D4 contains the value 4, then D4 cannot contain the value 8 (the cell can contain only one value) Nested Contradiction Forcing Chain: A8.3 on ==> E6.2 both on & off Chain 22: If A8 contains the value 3, then E6 must contain the value 2 (View 22): (1) If A8 contains the value 3, then A9 cannot contain the value 3 (the value can occur only once in the block) (2) If A8 contains the value 3 (initial assumption), then A8 cannot contain the value 1 (the cell can contain only one value) (3) If A8 does not contain the value 1, then I9 cannot contain the value 1 (Claiming: Cells G8,H8: 1 in row and block) (4) If I9 does not contain the value 1 and A9 does not contain the value 3 (1), then G9 cannot contain the value 7 (Naked Pair: Cells A9,I9: 4,7 in row) (5) If A8 does not contain the value 1 (2), then G9 cannot contain the value 1 (Claiming: Cells G8,H8: 1 in row and block) (6) If G9 does not contain the value 1 and G9 does not contain the value 7 (4), then G9 must contain the value 8 (only remaining possible value in the cell) (7) If G9 contains the value 8, then B9 cannot contain the value 8 (the value can occur only once in the row) (8) If A8 contains the value 3 (initial assumption), then B9 cannot contain the value 3 (the value can occur only once in the block) (9) If I9 does not contain the value 1 (3) and A9 does not contain the value 3 (1), then B9 cannot contain the value 7 (Naked Pair: Cells A9,I9: 4,7 in row) (10) If B9 does not contain the value 7, B9 does not contain the value 3 (8) and B9 does not contain the value 8 (7), then B9 must contain the value 9 (only remaining possible value in the cell) (11) If B9 contains the value 9, then B8 cannot contain the value 9 (the value can occur only once in the block) (12) If A8 contains the value 3 (initial assumption), then H8 cannot contain the value 3 (the value can occur only once in the row) (13) If H8 does not contain the value 3, then H1 must contain the value 3 (only remaining possible position in the column) (14) If H1 contains the value 3, then H1 cannot contain the value 2 (the cell can contain only one value) (15) If H1 does not contain the value 2, then A3 cannot contain the value 2 (Claiming: Cells A1,B1: 2 in row and block) (16) If A8 contains the value 3 (initial assumption), then A3 cannot contain the value 3 (the value can occur only once in the column) (17) If A3 does not contain the value 3 and A3 does not contain the value 2 (15), then A3 must contain the value 8 (only remaining possible value in the cell) (18) If A3 contains the value 8, then D3 cannot contain the value 8 (the value can occur only once in the row) (19) If A9 does not contain the value 3 (1) and B9 does not contain the value 3 (8), then D9 must contain the value 3 (only remaining possible position in the row) (20) If D9 contains the value 3, then D9 cannot contain the value 8 (the cell can contain only one value) (21) If D9 does not contain the value 8 and D3 does not contain the value 8 (18), then D4 must contain the value 8 (only remaining possible position in the column) (22) If D4 contains the value 8, then D4 cannot contain the value 4 (the cell can contain only one value) (23) If D4 does not contain the value 4, then A6 cannot contain the value 4 (Pointing: Cells E6,F6: 4 in block and row) (24) If A6 does not contain the value 4, then B8 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (25) If B8 does not contain the value 4 and B8 does not contain the value 9 (11), then B8 must contain the value 6 (only remaining possible value in the cell) (26) If B8 contains the value 6, then I8 cannot contain the value 6 (the value can occur only once in the row) (27) If A8 contains the value 3 (initial assumption), then I8 cannot contain the value 3 (the value can occur only once in the row) (28) If I8 does not contain the value 3 and I8 does not contain the value 6 (26), then I8 must contain the value 2 (only remaining possible value in the cell) (29) If I8 contains the value 2, then H7 cannot contain the value 2 (the value can occur only once in the block) (30) If H1 does not contain the value 2 (14) and H7 does not contain the value 2, then H4 must contain the value 2 (only remaining possible position in the column) (31) If H4 contains the value 2, then G6 cannot contain the value 2 (the value can occur only once in the block) (32) If B8 contains the value 6 (25), then C7 cannot contain the value 6 (the value can occur only once in the block) (33) If A6 does not contain the value 4 (23), then C7 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (34) If B9 contains the value 9 (10), then C7 cannot contain the value 9 (the value can occur only once in the block) (35) If C7 does not contain the value 9, C7 does not contain the value 4 (33) and C7 does not contain the value 6 (32), then C7 must contain the value 1 (only remaining possible value in the cell) (36) If C7 contains the value 1, then C9 cannot contain the value 1 (the value can occur only once in the block) (37) If B8 contains the value 6 (25), then C9 cannot contain the value 6 (the value can occur only once in the block) (38) If C9 does not contain the value 6 and C9 does not contain the value 1 (36), then C9 must contain the value 2 (only remaining possible value in the cell) (39) If C9 contains the value 2, then C6 cannot contain the value 2 (the value can occur only once in the column) (40) If B9 contains the value 9 (10), then F9 cannot contain the value 9 (the value can occur only once in the row) (41) If G9 contains the value 8 (6), then H8 cannot contain the value 8 (the value can occur only once in the block) (42) If A8 contains the value 3 (initial assumption), then A8 cannot contain the value 8 (the cell can contain only one value) (43) If A8 does not contain the value 8 and H8 does not contain the value 8 (41), then F8 must contain the value 8 (only remaining possible position in the row) (44) If F8 contains the value 8, then F8 cannot contain the value 9 (the cell can contain only one value) (45) If D9 contains the value 3 (19), then D9 cannot contain the value 9 (the cell can contain only one value) (46) If D9 does not contain the value 9, F8 does not contain the value 9 (44) and F9 does not contain the value 9 (40), then D8 must contain the value 9 (only remaining possible position in the block) (47) If D8 contains the value 9, then D8 cannot contain the value 7 (the cell can contain only one value) (48) If D4 contains the value 8 (21), then D4 cannot contain the value 7 (the cell can contain only one value) (49) If D4 does not contain the value 7 and D8 does not contain the value 7 (47), then D6 must contain the value 7 (only remaining possible position in the column) (50) If D6 contains the value 7, then D6 cannot contain the value 2 (the cell can contain only one value) (51) If D6 does not contain the value 2, C6 does not contain the value 2 (39) and G6 does not contain the value 2 (31), then E6 must contain the value 2 (only remaining possible position in the row) Chain 23: If A8 contains the value 3, then E6 cannot contain the value 2 (View 23): (1) If A8 contains the value 3, then A9 cannot contain the value 3 (the value can occur only once in the block) (2) If A8 contains the value 3 (initial assumption), then A8 cannot contain the value 1 (the cell can contain only one value) (3) If A8 does not contain the value 1, then I9 cannot contain the value 1 (Claiming: Cells G8,H8: 1 in row and block) (4) If I9 does not contain the value 1 and A9 does not contain the value 3 (1), then G9 cannot contain the value 7 (Naked Pair: Cells A9,I9: 4,7 in row) (5) If A8 does not contain the value 1 (2), then G9 cannot contain the value 1 (Claiming: Cells G8,H8: 1 in row and block) (6) If G9 does not contain the value 1 and G9 does not contain the value 7 (4), then G9 must contain the value 8 (only remaining possible value in the cell) (7) If G9 contains the value 8, then B9 cannot contain the value 8 (the value can occur only once in the row) (8) If A8 contains the value 3 (initial assumption), then B9 cannot contain the value 3 (the value can occur only once in the block) (9) If I9 does not contain the value 1 (3) and A9 does not contain the value 3 (1), then B9 cannot contain the value 7 (Naked Pair: Cells A9,I9: 4,7 in row) (10) If B9 does not contain the value 7, B9 does not contain the value 3 (8) and B9 does not contain the value 8 (7), then B9 must contain the value 9 (only remaining possible value in the cell) (11) If B9 contains the value 9, then B8 cannot contain the value 9 (the value can occur only once in the block) (12) If A8 contains the value 3 (initial assumption), then H8 cannot contain the value 3 (the value can occur only once in the row) (13) If H8 does not contain the value 3, then H1 must contain the value 3 (only remaining possible position in the column) (14) If H1 contains the value 3, then H1 cannot contain the value 2 (the cell can contain only one value) (15) If H1 does not contain the value 2, then A3 cannot contain the value 2 (Claiming: Cells A1,B1: 2 in row and block) (16) If A8 contains the value 3 (initial assumption), then A3 cannot contain the value 3 (the value can occur only once in the column) (17) If A3 does not contain the value 3 and A3 does not contain the value 2 (15), then A3 must contain the value 8 (only remaining possible value in the cell) (18) If A3 contains the value 8, then D3 cannot contain the value 8 (the value can occur only once in the row) (19) If A9 does not contain the value 3 (1) and B9 does not contain the value 3 (8), then D9 must contain the value 3 (only remaining possible position in the row) (20) If D9 contains the value 3, then D9 cannot contain the value 8 (the cell can contain only one value) (21) If D9 does not contain the value 8 and D3 does not contain the value 8 (18), then D4 must contain the value 8 (only remaining possible position in the column) (22) If D4 contains the value 8, then D4 cannot contain the value 4 (the cell can contain only one value) (23) If D4 does not contain the value 4, then A6 cannot contain the value 4 (Pointing: Cells E6,F6: 4 in block and row) (24) If A6 does not contain the value 4, then B8 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (25) If B8 does not contain the value 4 and B8 does not contain the value 9 (11), then B8 must contain the value 6 (only remaining possible value in the cell) (26) If B8 contains the value 6, then C7 cannot contain the value 6 (the value can occur only once in the block) (27) If A6 does not contain the value 4 (23), then C7 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (28) If B9 contains the value 9 (10), then C7 cannot contain the value 9 (the value can occur only once in the block) (29) If C7 does not contain the value 9, C7 does not contain the value 4 (27) and C7 does not contain the value 6 (26), then C7 must contain the value 1 (only remaining possible value in the cell) (30) If C7 contains the value 1, then C9 cannot contain the value 1 (the value can occur only once in the block) (31) If B8 contains the value 6 (25), then C9 cannot contain the value 6 (the value can occur only once in the block) (32) If C9 does not contain the value 6 and C9 does not contain the value 1 (30), then C9 must contain the value 2 (only remaining possible value in the cell) (33) If C9 contains the value 2, then A7 cannot contain the value 2 (the value can occur only once in the block) (34) If B8 contains the value 6 (25), then I8 cannot contain the value 6 (the value can occur only once in the row) (35) If A8 contains the value 3 (initial assumption), then I8 cannot contain the value 3 (the value can occur only once in the row) (36) If I8 does not contain the value 3 and I8 does not contain the value 6 (34), then I8 must contain the value 2 (only remaining possible value in the cell) (37) If I8 contains the value 2, then H7 cannot contain the value 2 (the value can occur only once in the block) (38) If H8 does not contain the value 3 (12) and I8 does not contain the value 3 (35), then I7 must contain the value 3 (only remaining possible position in the block) (39) If I7 contains the value 3, then I7 cannot contain the value 2 (the cell can contain only one value) (40) If I7 does not contain the value 2, H7 does not contain the value 2 (37) and A7 does not contain the value 2 (33), then E7 must contain the value 2 (only remaining possible position in the row) (41) If E7 contains the value 2, then E6 cannot contain the value 2 (the value can occur only once in the column) Nested Contradiction Forcing Chain: A8.8 on ==> E7.4 both on & off Chain 24: If A8 contains the value 8, then E7 must contain the value 4 (View 24): (1) If A8 contains the value 8, then A8 cannot contain the value 1 (the cell can contain only one value) (2) If A8 does not contain the value 1, then G9 cannot contain the value 1 (Claiming: Cells G8,H8: 1 in row and block) (3) If A8 contains the value 8 (initial assumption), then F8 cannot contain the value 8 (the value can occur only once in the row) (4) If F8 does not contain the value 8, then G9 cannot contain the value 8 (Pointing: Cells D9,E9,F9: 8 in block and row) (5) If G9 does not contain the value 8 and G9 does not contain the value 1 (2), then G9 must contain the value 7 (only remaining possible value in the cell) (6) If G9 contains the value 7, then I9 cannot contain the value 7 (the value can occur only once in the block) (7) If A8 does not contain the value 1 (1), then I9 cannot contain the value 1 (Claiming: Cells G8,H8: 1 in row and block) (8) If I9 does not contain the value 1 and I9 does not contain the value 7 (6), then I9 must contain the value 4 (only remaining possible value in the cell) (9) If I9 contains the value 4, then A9 cannot contain the value 4 (the value can occur only once in the row) (10) If G9 contains the value 7 (5), then A9 cannot contain the value 7 (the value can occur only once in the row) (11) If A9 does not contain the value 7 and A9 does not contain the value 4 (9), then A9 must contain the value 3 (only remaining possible value in the cell) (12) If A9 contains the value 3, then A3 cannot contain the value 3 (the value can occur only once in the column) (13) If A8 contains the value 8 (initial assumption), then A3 cannot contain the value 8 (the value can occur only once in the column) (14) If A3 does not contain the value 8 and A3 does not contain the value 3 (12), then A3 must contain the value 2 (only remaining possible value in the cell) (15) If A3 contains the value 2, then A7 cannot contain the value 2 (the value can occur only once in the column) (16) If A8 contains the value 8 (initial assumption), then A8 cannot contain the value 2 (the cell can contain only one value) (17) If A8 does not contain the value 2 and A7 does not contain the value 2 (15), then C9 must contain the value 2 (only remaining possible position in the block) (18) If C9 contains the value 2, then C9 cannot contain the value 1 (the cell can contain only one value) (19) If I9 does not contain the value 1 (7), G9 does not contain the value 1 (2) and C9 does not contain the value 1, then E9 must contain the value 1 (only remaining possible position in the row) (20) If E9 contains the value 1, then E9 cannot contain the value 6 (the cell can contain only one value) (21) If C9 contains the value 2 (17), then C9 cannot contain the value 6 (the cell can contain only one value) (22) If C9 does not contain the value 6 and E9 does not contain the value 6 (20), then F9 must contain the value 6 (only remaining possible position in the row) (23) If F9 contains the value 6, then E7 cannot contain the value 6 (the value can occur only once in the block) (24) If A8 does not contain the value 1 (1) and C9 does not contain the value 1 (18), then C7 must contain the value 1 (only remaining possible position in the block) (25) If C7 contains the value 1, then E7 cannot contain the value 1 (the value can occur only once in the row) (26) If A3 contains the value 2 (14), then B1 cannot contain the value 2 (the value can occur only once in the block) (27) If A3 contains the value 2 (14), then A1 cannot contain the value 2 (the value can occur only once in the block) (28) If A1 does not contain the value 2 and B1 does not contain the value 2 (26), then H1 must contain the value 2 (only remaining possible position in the row) (29) If H1 contains the value 2, then H1 cannot contain the value 3 (the cell can contain only one value) (30) If H1 does not contain the value 3, then H8 must contain the value 3 (only remaining possible position in the column) (31) If H8 contains the value 3, then I7 cannot contain the value 3 (the value can occur only once in the block) (32) If I9 contains the value 4 (8), then I7 cannot contain the value 4 (the value can occur only once in the block) (33) If I7 does not contain the value 4 and I7 does not contain the value 3 (31), then I7 must contain the value 2 (only remaining possible value in the cell) (34) If I7 contains the value 2, then E7 cannot contain the value 2 (the value can occur only once in the row) (35) If E7 does not contain the value 2, E7 does not contain the value 1 (25) and E7 does not contain the value 6 (23), then E7 must contain the value 4 (only remaining possible value in the cell) Chain 25: If A8 contains the value 8, then E7 cannot contain the value 4 (View 25): (1) If A8 contains the value 8, then A8 cannot contain the value 1 (the cell can contain only one value) (2) If A8 does not contain the value 1, then G9 cannot contain the value 1 (Claiming: Cells G8,H8: 1 in row and block) (3) If A8 contains the value 8 (initial assumption), then F8 cannot contain the value 8 (the value can occur only once in the row) (4) If F8 does not contain the value 8, then G9 cannot contain the value 8 (Pointing: Cells D9,E9,F9: 8 in block and row) (5) If G9 does not contain the value 8 and G9 does not contain the value 1 (2), then G9 must contain the value 7 (only remaining possible value in the cell) (6) If G9 contains the value 7, then I9 cannot contain the value 7 (the value can occur only once in the block) (7) If A8 does not contain the value 1 (1), then I9 cannot contain the value 1 (Claiming: Cells G8,H8: 1 in row and block) (8) If I9 does not contain the value 1 and I9 does not contain the value 7 (6), then I9 must contain the value 4 (only remaining possible value in the cell) (9) If I9 contains the value 4, then A9 cannot contain the value 4 (the value can occur only once in the row) (10) If G9 contains the value 7 (5), then A9 cannot contain the value 7 (the value can occur only once in the row) (11) If A9 does not contain the value 7 and A9 does not contain the value 4 (9), then A9 must contain the value 3 (only remaining possible value in the cell) (12) If A9 contains the value 3, then A3 cannot contain the value 3 (the value can occur only once in the column) (13) If A8 contains the value 8 (initial assumption), then A3 cannot contain the value 8 (the value can occur only once in the column) (14) If A3 does not contain the value 8 and A3 does not contain the value 3 (12), then A3 must contain the value 2 (only remaining possible value in the cell) (15) If A3 contains the value 2, then E3 cannot contain the value 2 (the value can occur only once in the row) (16) If A3 contains the value 2 (14), then D3 cannot contain the value 2 (the value can occur only once in the row) (17) If D3 does not contain the value 2 and E3 does not contain the value 2 (15), then F2 must contain the value 2 (only remaining possible position in the block) (18) If F2 contains the value 2, then F2 cannot contain the value 7 (the cell can contain only one value) (19) If G9 contains the value 7 (5), then F9 cannot contain the value 7 (the value can occur only once in the row) (20) If F9 does not contain the value 7 and F2 does not contain the value 7 (18), then F6 must contain the value 7 (only remaining possible position in the column) (21) If F6 contains the value 7, then E5 cannot contain the value 7 (the value can occur only once in the block) (22) If G9 contains the value 7 (5), then G5 cannot contain the value 7 (the value can occur only once in the column) (23) If G5 does not contain the value 7 and E5 does not contain the value 7 (21), then A5 must contain the value 7 (only remaining possible position in the row) (24) If A5 contains the value 7, then A7 cannot contain the value 7 (the value can occur only once in the column) (25) If A3 contains the value 2 (14), then A7 cannot contain the value 2 (the value can occur only once in the column) (26) If A8 contains the value 8 (initial assumption), then A7 cannot contain the value 8 (the value can occur only once in the block) (27) If A9 contains the value 3 (11), then B9 cannot contain the value 3 (the value can occur only once in the block) (28) If G9 contains the value 7 (5), then B9 cannot contain the value 7 (the value can occur only once in the row) (29) If A8 contains the value 8 (initial assumption), then B9 cannot contain the value 8 (the value can occur only once in the block) (30) If B9 does not contain the value 8, B9 does not contain the value 7 (28) and B9 does not contain the value 3 (27), then B9 must contain the value 9 (only remaining possible value in the cell) (31) If B9 contains the value 9, then A7 cannot contain the value 9 (the value can occur only once in the block) (32) If A7 does not contain the value 9, A7 does not contain the value 8 (26), A7 does not contain the value 2 (25) and A7 does not contain the value 7 (24), then A7 must contain the value 4 (only remaining possible value in the cell) (33) If A7 contains the value 4, then E7 cannot contain the value 4 (the value can occur only once in the row) Nested Contradiction Forcing Chain: I2.6 on ==> H4.6 both on & off Chain 26: If I2 contains the value 6, then H4 must contain the value 6 (View 26): (1) If I2 contains the value 6, then D2 cannot contain the value 6 (the value can occur only once in the row) (2) If I2 contains the value 6 (initial assumption), then E2 cannot contain the value 6 (the value can occur only once in the row) (3) If E2 does not contain the value 6 and D2 does not contain the value 6 (1), then F8 cannot contain the value 6 (Pointing: Cells F1,F3: 6 in block and column) (4) If I2 contains the value 6 (initial assumption), then I8 cannot contain the value 6 (the value can occur only once in the column) (5) If I8 does not contain the value 6, then E7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (6) If I8 does not contain the value 6 (4), then D7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (7) If E2 does not contain the value 6 (2) and D2 does not contain the value 6 (1), then F9 cannot contain the value 6 (Pointing: Cells F1,F3: 6 in block and column) (8) If F9 does not contain the value 6, F8 does not contain the value 6 (3), D7 does not contain the value 6 (6) and E7 does not contain the value 6 (5), then E9 must contain the value 6 (only remaining possible position in the block) (9) If E9 contains the value 6, then E9 cannot contain the value 1 (the cell can contain only one value) (10) If E9 does not contain the value 1, then C7 cannot contain the value 1 (Pointing: Cells D7,E7: 1 in block and row) (11) If I2 contains the value 6 (initial assumption), then I6 cannot contain the value 6 (the value can occur only once in the column) (12) If I6 does not contain the value 6, then A6 must contain the value 6 (only remaining possible position in the row) (13) If A6 contains the value 6, then A6 cannot contain the value 4 (the cell can contain only one value) (14) If A6 does not contain the value 4, then C7 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (15) If I8 does not contain the value 6 (4), then C7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (16) If C7 does not contain the value 6, C7 does not contain the value 4 (14) and C7 does not contain the value 1 (10), then C7 must contain the value 9 (only remaining possible value in the cell) (17) If C7 contains the value 9, then H7 cannot contain the value 9 (the value can occur only once in the row) (18) If C7 contains the value 9 (16), then G7 cannot contain the value 9 (the value can occur only once in the row) (19) If G7 does not contain the value 9 and H7 does not contain the value 9 (17), then G8 must contain the value 9 (only remaining possible position in the block) (20) If G8 contains the value 9, then F8 cannot contain the value 9 (the value can occur only once in the row) (21) If G8 contains the value 9 (19), then G8 cannot contain the value 7 (the cell can contain only one value) (22) If G8 does not contain the value 7, then F9 cannot contain the value 7 (Claiming: Cells D8,E8: 7 in row and block) (23) If C7 contains the value 9 (16), then C1 cannot contain the value 9 (the value can occur only once in the column) (24) If G8 contains the value 9 (19), then G1 cannot contain the value 9 (the value can occur only once in the column) (25) If G1 does not contain the value 9 and C1 does not contain the value 9 (23), then D3 cannot contain the value 9 (Claiming: Cells E1,F1: 9 in row and block) (26) If G8 contains the value 9 (19), then D8 cannot contain the value 9 (the value can occur only once in the row) (27) If D8 does not contain the value 9 and D3 does not contain the value 9 (25), then D9 must contain the value 9 (only remaining possible position in the column) (28) If D9 contains the value 9, then F9 cannot contain the value 9 (the value can occur only once in the block) (29) If F9 does not contain the value 9, F9 does not contain the value 7 (22), F9 does not contain the value 6 (7), F8 does not contain the value 9 (20) and F8 does not contain the value 6 (3), then F2 cannot contain the value 2 (Naked Pair: Cells F8,F9: 2,8 in column) (30) If I2 contains the value 6 (initial assumption), then I2 cannot contain the value 2 (the cell can contain only one value) (31) If I2 does not contain the value 2 and F2 does not contain the value 2 (29), then G2 must contain the value 2 (only remaining possible position in the row) (32) If G2 contains the value 2, then G2 cannot contain the value 4 (the cell can contain only one value) (33) If G8 contains the value 9 (19), then G8 cannot contain the value 4 (the cell can contain only one value) (34) If G8 does not contain the value 4 and G2 does not contain the value 4 (32), then G4 must contain the value 4 (only remaining possible position in the column) (35) If G4 contains the value 4, then G4 cannot contain the value 6 (the cell can contain only one value) (36) If I2 contains the value 6 (initial assumption), then I4 cannot contain the value 6 (the value can occur only once in the column) (37) If I6 does not contain the value 6 (11), I4 does not contain the value 6 and G4 does not contain the value 6 (35), then H4 must contain the value 6 (only remaining possible position in the block) Chain 27: If I2 contains the value 6, then H4 cannot contain the value 6 (View 27): (1) If I2 contains the value 6, then D2 cannot contain the value 6 (the value can occur only once in the row) (2) If I2 contains the value 6 (initial assumption), then E2 cannot contain the value 6 (the value can occur only once in the row) (3) If E2 does not contain the value 6 and D2 does not contain the value 6 (1), then F8 cannot contain the value 6 (Pointing: Cells F1,F3: 6 in block and column) (4) If I2 contains the value 6 (initial assumption), then I8 cannot contain the value 6 (the value can occur only once in the column) (5) If I8 does not contain the value 6, then E7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (6) If I8 does not contain the value 6 (4), then D7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (7) If E2 does not contain the value 6 (2) and D2 does not contain the value 6 (1), then F9 cannot contain the value 6 (Pointing: Cells F1,F3: 6 in block and column) (8) If F9 does not contain the value 6, F8 does not contain the value 6 (3), D7 does not contain the value 6 (6) and E7 does not contain the value 6 (5), then E9 must contain the value 6 (only remaining possible position in the block) (9) If E9 contains the value 6, then E9 cannot contain the value 1 (the cell can contain only one value) (10) If E9 does not contain the value 1, then C7 cannot contain the value 1 (Pointing: Cells D7,E7: 1 in block and row) (11) If I2 contains the value 6 (initial assumption), then I6 cannot contain the value 6 (the value can occur only once in the column) (12) If I6 does not contain the value 6, then A6 must contain the value 6 (only remaining possible position in the row) (13) If A6 contains the value 6, then A6 cannot contain the value 4 (the cell can contain only one value) (14) If A6 does not contain the value 4, then C7 cannot contain the value 4 (Claiming: Cells A7,A9: 4 in column and block) (15) If I8 does not contain the value 6 (4), then C7 cannot contain the value 6 (Pointing: Cells G7,H7: 6 in block and row) (16) If C7 does not contain the value 6, C7 does not contain the value 4 (14) and C7 does not contain the value 1 (10), then C7 must contain the value 9 (only remaining possible value in the cell) (17) If C7 contains the value 9, then H7 cannot contain the value 9 (the value can occur only once in the row) (18) If C7 contains the value 9 (16), then G7 cannot contain the value 9 (the value can occur only once in the row) (19) If G7 does not contain the value 9 and H7 does not contain the value 9 (17), then G8 must contain the value 9 (only remaining possible position in the block) (20) If G8 contains the value 9, then F8 cannot contain the value 9 (the value can occur only once in the row) (21) If G8 contains the value 9 (19), then G8 cannot contain the value 7 (the cell can contain only one value) (22) If G8 does not contain the value 7, then F9 cannot contain the value 7 (Claiming: Cells D8,E8: 7 in row and block) (23) If C7 contains the value 9 (16), then C1 cannot contain the value 9 (the value can occur only once in the column) (24) If G8 contains the value 9 (19), then G1 cannot contain the value 9 (the value can occur only once in the column) (25) If G1 does not contain the value 9 and C1 does not contain the value 9 (23), then D3 cannot contain the value 9 (Claiming: Cells E1,F1: 9 in row and block) (26) If G8 contains the value 9 (19), then D8 cannot contain the value 9 (the value can occur only once in the row) (27) If D8 does not contain the value 9 and D3 does not contain the value 9 (25), then D9 must contain the value 9 (only remaining possible position in the column) (28) If D9 contains the value 9, then F9 cannot contain the value 9 (the value can occur only once in the block) (29) If F9 does not contain the value 9, F9 does not contain the value 7 (22), F9 does not contain the value 6 (7), F8 does not contain the value 9 (20) and F8 does not contain the value 6 (3), then F2 cannot contain the value 2 (Naked Pair: Cells F8,F9: 2,8 in column) (30) If I2 contains the value 6 (initial assumption), then I2 cannot contain the value 2 (the cell can contain only one value) (31) If I2 does not contain the value 2 and F2 does not contain the value 2 (29), then G2 must contain the value 2 (only remaining possible position in the row) (32) If G2 contains the value 2, then G6 cannot contain the value 2 (the value can occur only once in the column) (33) If G2 contains the value 2 (31), then G4 cannot contain the value 2 (the value can occur only once in the column) (34) If G2 contains the value 2 (31), then G5 cannot contain the value 2 (the value can occur only once in the column) (35) If G5 does not contain the value 2, G4 does not contain the value 2 (33) and G6 does not contain the value 2 (32), then H4 must contain the value 2 (only remaining possible position in the block) (36) If H4 contains the value 2, then H4 cannot contain the value 6 (the cell can contain only one value) ```

1to9only commented 3 years ago

When I "Get all hints" at this point, it only returns +MFC chains. My question is whether there is a way to force it to also look for +DFC chains - obviously it will do so if there are no +MFC chains available, but in this case there's the 11.7 so that's all that shows up.

Other SudokuExplainers have implemented Shift or Ctrl keypress with the "Get all hints" button to return more hints - this extra processing can take a bit longer to return hints. See: https://sourceforge.net/projects/sudoku-explainer/ (use Shift + 'Get all hints') https://github.com/1to9only/SukakuExplainer (use Shift or Crtl + 'Get all hints')

newman-iknewit commented 3 years ago

Thought I would get a count of puzzles at each rating to see how much overlap there was in the DFC levels; was surprised to find that I've actually generated a few lowering the bounds again:

........1.....1....23.4..5...6..3.2...74...1.2...6.5....8..4.6..79....3.6...3...5|11.3|1.2|1.2|DLFC+DFC ........1.....1....12.3..4..2..5.4..5....6.2.7..3...1..5..6...48....3.5.9.7....6.|11.3|1.2|1.2|DLFC+DFC ........1.....1.....2.3..4..5...3.6..78....9.6...9...4.6...9.2..8.3...1.21..6.4..|11.3|1.2|1.2|DLFC+DFC

........1..1..2....3.45..2...3.4..5..5.6.7....8..9...2.1..7.2...4......53...14.8.|11.3|1.2|1.2|DLFC+DFC ........1..1..2....3.45..2...3.4..5..5.6.7....8..9...2.18...2...4......53...14.8.|11.3|1.2|1.2|DLFC+DFC

........1....23.....41...2...1..52.4.6.....7..7.4..1....62.......75...4.8....19..|10.6|1.2|1.2|DDFC+MFC

So +DFC is 11.3-11.9, and +MFC is 10.6-11.7

newman-iknewit commented 3 years ago

And confirming +FC goes all the way down to 9.9:

........1.....2.....3.4..56....6.....4.3.71...7....8...3..7..4.5.....9..6....5.2.|9.9|1.2|1.2|DRFC+FC

6 more of these at 22c. I have 9 grids with 11.1 +FC (one of which is posted above).

SudokuMonster commented 3 years ago

Thought I would get a count of puzzles at each rating to see how much overlap there was in the DFC levels; was surprised to find that I've actually generated a few lowering the bounds again:

........1.....1....23.4..5...6..3.2...74...1.2...6.5....8..4.6..79....3.6...3...5|11.3|1.2|1.2|DLFC+DFC ........1.....1....12.3..4..2..5.4..5....6.2.7..3...1..5..6...48....3.5.9.7....6.|11.3|1.2|1.2|DLFC+DFC ........1.....1.....2.3..4..5...3.6..78....9.6...9...4.6...9.2..8.3...1.21..6.4..|11.3|1.2|1.2|DLFC+DFC

........1..1..2....3.45..2...3.4..5..5.6.7....8..9...2.1..7.2...4......53...14.8.|11.3|1.2|1.2|DLFC+DFC ........1..1..2....3.45..2...3.4..5..5.6.7....8..9...2.18...2...4......53...14.8.|11.3|1.2|1.2|DLFC+DFC

........1....23.....41...2...1..52.4.6.....7..7.4..1....62.......75...4.8....19..|10.6|1.2|1.2|DDFC+MFC

So +DFC is 11.3-11.9, and +MFC is 10.6-11.7

Thanks. These are currently on Wiki quoted as:

10.7-11.5?: Dynamic Forcing Chains(+Multiple Forcing Chains) 11.4-11.9: Dynamic Forcing Chains(+Dynamic Forcing Chains)

So I will need to update them

newman-iknewit commented 2 years ago

Hadn't checked this in a while, but at some point I lowered the +DFC range again:

........1.....1....23.4..5...6..3.2..7.4...1.2...8.5....9..4.8..67....3.8...3...5 ED=11.2/1.2/1.2 DLFC+DFC

1to9only commented 2 years ago

SE Chains ratings are made up of a Base Rating + a Complexity Rating which depends on the total (including nested) chain length. In the SE code chain length is also referred to as chain complexity. And elsewhere I refer to nodes, which is just chain complexity minus 2.

BASE RATING

Technique               Chain   Nesting                         Base
                        Level   Level   Multi   Dynamic Nishio  Rating

ForcingChainCycle       0               false   false   false    6.6    7.0 if both X-/Y- chains

NishioForcingChain      0                        true    true    7.5
DynamicForcingChain     0                true    true            8.5
MultipleForcingChain    0                true                    8.0

DynamicForcingChainPlus 1       0        true    true            9.0  + nested sets!

NestedForcingChain      2       0        true    true            9.5  + nested ForcingChainCycle        (6.6)
NestedForcingChain      3       0        true    true           10.0  + nested MultipleForcingChain     (8.0)

NestedForcingChain      4       0        true    true           10.5  + nested DynamicForcingChain      (8.5)
NestedForcingChain      4       1        true    true           10.5  + nested DynamicForcingChainPlus  (9.0)
NestedForcingChain      4       2        true    true           10.5  + nested NestedForcingChain       (9.5)

COMPLEXITY RATING

+ 0.0 :     1 -     4 nodes
+ 0.1 :     5 -     6 nodes
+ 0.2 :     7 -     8 nodes
+ 0.3 :     9 -    12 nodes
+ 0.4 :    13 -    16 nodes
+ 0.5 :    17 -    24 nodes
+ 0.6 :    25 -    32 nodes
+ 0.7 :    33 -    48 nodes
+ 0.8 :    49 -    64 nodes
+ 0.9 :    65 -    96 nodes
+ 1.0 :    97 -   128 nodes
+ 1.1 :   129 -   192 nodes
+ 1.2 :   193 -   256 nodes
+ 1.3 :   257 -   384 nodes
+ 1.4 :   385 -   512 nodes
+ 1.5 :   513 -   768 nodes
+ 1.6 :   769 -  1024 nodes
+ 1.7 :  1025 -  1536 nodes
+ 1.8 :  1537 -  2048 nodes
+ 1.9 :  2049 -  3072 nodes
+ 2.0 :  3073 -  4096 nodes
+ 2.1 :  4097 -  6144 nodes
+ 2.2 :  6145 -  8192 nodes
+ 2.3 :  8193 - 12288 nodes
+ 2.4 : 12289 - 16384 nodes
+ 2.5 : 16385 - 24576 nodes
+ 2.6 : 24577 - 32768 nodes
+ 2.7 : 32769 - 49152 nodes
+ 2.8 : 49153 - 65536 nodes
+ 2.9 : 65537 - 98304 nodes

So it is possible to get similar rated puzzles with vastly different chains length:

11.6, Contradiction Forcing Chain (w/816 nodes)
11.6, Nested Forcing Chains (+ Multiple Forcing Chains), ED=10.0 + 1.6 (769-1024 nodes)
4.69......2...8...5..6..7..9.4...5..........8.1...7.3...74..6.......9..1....3..2. # 0414  12  11.6/11.6/9.3

11.6, Region Forcing Chains (w/158 nodes)
11.6, Nested Forcing Chains (+ Dynamic Forcing Chains), ED=10.5 + 1.1 (129-192 nodes)
.....3..7.7.....1...9.6.8.......1..5..8.2.9..4..6.......6.4.2...5.....3.2..8..... # 0427   2  11.6/11.6/9.9

The 2 puzzles are from the Patterns Game.

In Classic Sudoku, typically the highest rated puzzles have 257-384 nodes (ED=11.8) and 385-512 nodes (ED=11.9).

In 16x16 Sudoku, the highest rated puzzle I've seen has 24577-32768 nodes (ED=13.1).

SudokuMonster / SukakuExplainer

Rating ranges #97