--before
%g: ..1...2....23....45........4.6....1..5......7..8..........96.....5.3.....9...8.3.
%i: ..1...2....23....45........4.6....1..5......7..8..........96.....5.3.....9...8.3.
%m: ..3..6789..34.678.1...........456789...45678....45.7.9.2...........56789..3.56.89.....6789.....678..2.........3......1...5678.1...5.7.91...56789....56789...4.........5......34.678...34..7.912.4.678912.4.678.12.4..7.91.3..6789.....67891.3..6.89...4......23...7.......6....2..5.789.2..5.78..23.5.7.9..3.5..891.........23.5..89123.....9....5......3.....91..4.6.8912.4...8.1234....9...4.6.8..2.4.6.89......7..123...7.9123...7.........8.12.4567.91..4567..12345.7....3456..9.2.456..9.2..56...123...78.1234..78...34..7..12.45.7..........9.....6...1..45.78..2.45.78.12..5..8.12...678.12.4.678.....5....12.4..7....3......12.4..7..1..4.6789.2.4.678912...6.8912...67..........9...4..7..12.45.7..12.45.7.........8.1..4567....3......12..56...
%M:
+-------------------------+-------------------------+-------------------------+
| 36789 34678 1 | 456789 45678 4579 | 2 56789 35689 |
| 6789 678 2 | 3 15678 1579 | 156789 56789 4 |
| 5 34678 3479 | 1246789 124678 12479 | 136789 6789 13689 |
+-------------------------+-------------------------+-------------------------+
| 4 237 6 | 25789 2578 23579 | 3589 1 23589 |
| 1239 5 39 | 14689 1248 12349 | 468 24689 7 |
| 12379 1237 8 | 1245679 14567 123457 | 34569 24569 256 |
+-------------------------+-------------------------+-------------------------+
| 12378 123478 347 | 12457 9 6 | 14578 24578 1258 |
| 12678 124678 5 | 1247 3 1247 | 146789 246789 12689 |
| 1267 9 47 | 12457 12457 8 | 14567 3 1256 |
+-------------------------+-------------------------+-------------------------+
--- end of before section ---
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%h:
<html>
<body>
<h2>Dynamic Contradiction Forcing Chains</h2>
<p>
With this solving technique, we will prove the two following assertions:
<ul>
<li>If r7c2 contains the value 4, then r8c6 must contain the value 4
<li>If r7c2 contains the value 4, then r8c6 cannot contain the value 4
</ul>
Because the same assumption yields to contradictory results, we can conclude that the
assumption is false, that is, <b><font color="#00AAAA">r7c2 cannot contain the value 4</font></b>.
</p><p>
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.
</p><p>
The details of each chain are given below. Use the view selector below the grid
to switch between the <font color="#E08000">graphical illustrations</font> of the two different chains.
</p><p>
Chain 1: <b>If r7c2 contains the value 4, then r8c6 cannot contain the value 4</b> (View 1):<br>
(1) If r7c2 contains the value 4, then r9c3 cannot contain the value 4 (the value can occur only once in the block)<br>(2) If r7c2 contains the value 4 (initial assumption), then r7c3 cannot contain the value 4 (the value can occur only once in the block)<br>(3) If r7c3 does not contain the value 4 and r9c3 does not contain the value 4 (1), then r3c3 must contain the value 4 (only remaining possible position in the column)<br>(4) If r3c3 contains the value 4, then r5c5 cannot contain the value 4 (the value can occur only once in the Main Diagonal)<br>(5) If r7c2 contains the value 4 (initial assumption), then r8c2 cannot contain the value 4 (the value can occur only once in the block)<br>(6) If r8c2 does not contain the value 4, r7c3 does not contain the value 4 (2) and r5c5 does not contain the value 4 (4), then r6c4 must contain the value 4 (only remaining possible position in the Anti Diagonal)<br>(7) If r6c4 contains the value 4, then r6c4 cannot contain the value 9 (the cell can contain only one value)<br>(8) If r3c3 contains the value 4 (3), then r3c3 cannot contain the value 9 (the cell can contain only one value)<br>(9) If r9c3 does not contain the value 4 (1), then r9c3 must contain the value 7 (only remaining possible value in the cell)<br>(10) If r9c3 contains the value 7, then r7c3 cannot contain the value 7 (the value can occur only once in the block)<br>(11) If r7c3 does not contain the value 4 (2) and r7c3 does not contain the value 7, then r7c3 must contain the value 3 (only remaining possible value in the cell)<br>(12) If r7c3 contains the value 3, then r3c7 cannot contain the value 3 (the value can occur only once in the Anti Diagonal)<br>(13) If r7c3 contains the value 3 (11), then r1c9 cannot contain the value 3 (the value can occur only once in the Anti Diagonal)<br>(14) If r1c9 does not contain the value 3 and r3c7 does not contain the value 3 (12), then r3c9 must contain the value 3 (only remaining possible position in the block)<br>(15) If r3c9 contains the value 3, then r3c9 cannot contain the value 9 (the cell can contain only one value)<br>(16) If r3c3 does not contain the value 9 (8), then r5c3 must contain the value 9 (only remaining possible position in the column)<br>(17) If r5c3 contains the value 9, then r5c4 cannot contain the value 9 (the value can occur only once in the row)<br>(18) If r5c3 contains the value 9 (16), then r5c6 cannot contain the value 9 (the value can occur only once in the row)<br>(19) If r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 and r5c4 does not contain the value 9 (17), then r4c9 cannot contain the value 9 (Cells r4c4,r4c6 on value 9 in block 5)<br>(20) If r4c9 does not contain the value 9, r3c9 does not contain the value 9 (15), r3c3 does not contain the value 9 (8) and r6c4 does not contain the value 9 (7), then r3c8 cannot contain the value 9 (Region Forcing Chains: 9 in Anti Diagonal ==> r3c8.9 off)<br>(21) If r6c4 does not contain the value 9 (7) and r3c3 does not contain the value 9 (8), then r1c8 cannot contain the value 9 (Contradiction Forcing Chain: r1c8.9 on ==> r4c4.9 both on & off)<br>(22) If r3c3 does not contain the value 9 (8), r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18) and r5c4 does not contain the value 9 (17), then r2c8 cannot contain the value 9 (Region Forcing Chains: 9 in block ==> r2c8.9 off)<br>(23) If r5c3 contains the value 9 (16), then r5c8 cannot contain the value 9 (the value can occur only once in the row)<br>(24) If r4c9 does not contain the value 9 (19), r3c9 does not contain the value 9 (15), r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18) and r5c4 does not contain the value 9 (17), then r8c8 cannot contain the value 9 (Region Forcing Chains: 9 in column ==> r8c8.9 off)<br>(25) If r8c8 does not contain the value 9, r5c8 does not contain the value 9 (23), r2c8 does not contain the value 9 (22), r1c8 does not contain the value 9 (21) and r3c8 does not contain the value 9 (20), then r6c8 must contain the value 9 (only remaining possible position in the column)<br>(26) If r6c8 contains the value 9, then r6c8 cannot contain the value 5 (the cell can contain only one value)<br>(27) If r6c4 contains the value 4 (6), then r6c4 cannot contain the value 2 (the cell can contain only one value)<br>(28) If r9c3 contains the value 7 (9), then r7c1 cannot contain the value 7 (the value can occur only once in the block)<br>(29) If r9c3 contains the value 7 (9), then r8c1 cannot contain the value 7 (the value can occur only once in the block)<br>(30) If r9c3 contains the value 7 (9), then r9c1 cannot contain the value 7 (the value can occur only once in the block)<br>(31) If r7c2 contains the value 4 (initial assumption), then r7c2 cannot contain the value 1 (the cell can contain only one value)<br>(32) If r3c3 contains the value 4 (3), then r3c3 cannot contain the value 3 (the cell can contain only one value)<br>(33) If r6c4 contains the value 4 (6), then r6c4 cannot contain the value 7 (the cell can contain only one value)<br>(34) If r9c3 contains the value 7 (9), then r8c2 cannot contain the value 7 (the value can occur only once in the block)<br>(35) If r3c9 contains the value 3 (14), then r3c9 cannot contain the value 1 (the cell can contain only one value)<br>(36) If r3c9 contains the value 3 (14), then r3c9 cannot contain the value 6 (the cell can contain only one value)<br>(37) If r6c4 contains the value 4 (6), then r6c4 cannot contain the value 5 (the cell can contain only one value)<br>(38) If r3c9 does not contain the value 1 (35), then r7c7 cannot contain the value 1 (Cells r2c7,r3c7 on value 1 in block 3)<br>(39) If r7c7 does not contain the value 1, r6c4 does not contain the value 5 (37), r3c9 does not contain the value 6 (36), r4c9 does not contain the value 9 (19), r3c9 does not contain the value 9 (15), r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18), r5c4 does not contain the value 9 (17), r3c9 does not contain the value 1 (35), r3c3 does not contain the value 9 (8), r8c2 does not contain the value 7 (34), r7c3 does not contain the value 7 (10), r6c4 does not contain the value 7 (33), r5c8 does not contain the value 9 (23), r3c3 does not contain the value 3 (32), r7c2 does not contain the value 1 (31), r9c1 does not contain the value 7 (30), r8c1 does not contain the value 7 (29), r7c1 does not contain the value 7 (28) and r6c4 does not contain the value 2 (27), then r4c4 cannot contain the value 5 (Contradiction Forcing Chain: r4c4.9 off ==> r6c9.2 both on & off)<br>(40) If r5c3 contains the value 9 (16), then r5c1 cannot contain the value 9 (the value can occur only once in the block)<br>(41) If r6c4 contains the value 4 (6), then r6c8 cannot contain the value 4 (the value can occur only once in the row)<br>(42) If r7c2 contains the value 4 (initial assumption), then r7c8 cannot contain the value 4 (the value can occur only once in the row)<br>(43) If r3c3 contains the value 4 (3), then r8c8 cannot contain the value 4 (the value can occur only once in the Main Diagonal)<br>(44) If r8c8 does not contain the value 4, r7c8 does not contain the value 4 (42) and r6c8 does not contain the value 4 (41), then r5c8 must contain the value 4 (only remaining possible position in the column)<br>(45) If r5c8 contains the value 4, then r5c8 cannot contain the value 2 (the cell can contain only one value)<br>(46) If r7c3 contains the value 3 (11), then r5c3 cannot contain the value 3 (the value can occur only once in the column)<br>(47) If r7c2 contains the value 4 (initial assumption), then r7c2 cannot contain the value 2 (the cell can contain only one value)<br>(48) If r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18) and r5c4 does not contain the value 9 (17), then r4c7 cannot contain the value 9 (Cells r4c4,r4c6 on value 9 in block 5)<br>(49) If r3c3 does not contain the value 3 (32), then r6c1 cannot contain the value 3 (Cells r1c1,r6c6 on value 3 in Main Diagonal)<br>(50) If r7c3 contains the value 3 (11), then r7c1 cannot contain the value 3 (the value can occur only once in the block)<br>(51) If r7c3 contains the value 3 (11), then r4c6 cannot contain the value 3 (the value can occur only once in the Anti Diagonal)<br>(52) If r4c6 does not contain the value 3, r7c1 does not contain the value 3 (50), r6c1 does not contain the value 3 (49), r5c8 does not contain the value 9 (23), r4c9 does not contain the value 9 (19), r4c7 does not contain the value 9 (48), r7c2 does not contain the value 2 (47), r3c3 does not contain the value 9 (8), r5c3 does not contain the value 3 (46), r5c8 does not contain the value 2 (45), r7c2 does not contain the value 1 (31), r5c1 does not contain the value 9 (40) and r3c9 does not contain the value 1 (35), then r6c6 cannot contain the value 5 (Cell Forcing Chains: r5c1 ==> r6c6.5 off)<br>(53) If r6c6 does not contain the value 5 and r4c4 does not contain the value 5 (39), then r7c8 cannot contain the value 5 (Cells r7c7,r9c9 on value 5 in Main Diagonal)<br>(54) If r7c8 does not contain the value 5 and r6c8 does not contain the value 5 (26), then r1c9 cannot contain the value 5 (Cells r1c8,r2c8 on value 5 in column 8)<br>(55) If r6c4 does not contain the value 5 (37) and r1c9 does not contain the value 5, then r2c6 cannot contain the value 5 (Cells r2c8,r4c6 on value 5 in Anti Diagonal)<br>(56) If r7c2 does not contain the value 1 (31), r7c7 does not contain the value 1 (38) and r3c9 does not contain the value 1 (35), then r2c6 cannot contain the value 1 (Region Forcing Chains: 1 in block ==> r2c6.1 off)<br>(57) If r3c3 does not contain the value 9 (8), r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18) and r5c4 does not contain the value 9 (17), then r2c6 cannot contain the value 9 (Region Forcing Chains: 9 in block ==> r2c6.9 off)<br>(58) If r2c6 does not contain the value 9, r2c6 does not contain the value 1 (56) and r2c6 does not contain the value 5 (55), then r2c6 must contain the value 7 (only remaining possible value in the cell)<br>(59) If r2c6 contains the value 7, then r4c6 cannot contain the value 7 (the value can occur only once in the column)<br>(60) If r2c6 contains the value 7 (58), then r2c8 cannot contain the value 7 (the value can occur only once in the row)<br>(61) If r9c1 does not contain the value 7 (30), r8c2 does not contain the value 7 (34), r7c3 does not contain the value 7 (10), r6c4 does not contain the value 7 (33), r2c8 does not contain the value 7 and r4c6 does not contain the value 7 (59), then r3c7 must contain the value 7 (only remaining possible position in the Anti Diagonal)<br>(62) If r3c7 contains the value 7, then r3c7 cannot contain the value 9 (the cell can contain only one value)<br>(63) If r7c7 does not contain the value 1 (38), r3c9 does not contain the value 9 (15), r6c4 does not contain the value 5 (37), r3c9 does not contain the value 6 (36), r6c4 does not contain the value 9 (7), r3c9 does not contain the value 1 (35), r3c3 does not contain the value 9 (8), r8c2 does not contain the value 7 (34), r7c3 does not contain the value 7 (10), r6c4 does not contain the value 7 (33), r5c8 does not contain the value 9 (23), r4c9 does not contain the value 9 (19), r4c7 does not contain the value 9 (48), r3c3 does not contain the value 3 (32), r7c2 does not contain the value 1 (31), r9c1 does not contain the value 7 (30), r8c1 does not contain the value 7 (29), r7c1 does not contain the value 7 (28) and r6c4 does not contain the value 2 (27), then r2c7 cannot contain the value 9 (Contradiction Forcing Chain: r2c7.9 on ==> r6c9.2 both on & off)<br>(64) If r6c8 contains the value 9 (25), then r6c7 cannot contain the value 9 (the value can occur only once in the block)<br>(65) If r6c7 does not contain the value 9, r4c7 does not contain the value 9 (48), r2c7 does not contain the value 9 (63) and r3c7 does not contain the value 9 (62), then r8c7 must contain the value 9 (only remaining possible position in the column)<br>(66) If r8c7 contains the value 9, then r8c7 cannot contain the value 4 (the cell can contain only one value)<br>(67) If r6c4 contains the value 4 (6), then r8c4 cannot contain the value 4 (the value can occur only once in the column)<br>(68) If r8c8 does not contain the value 4 (43), r8c4 does not contain the value 4, r8c2 does not contain the value 4 (5) and r8c7 does not contain the value 4 (66), then r8c6 must contain the value 4 (only remaining possible position in the row)<br>
</p><p>
Chain 2: <b>If r8c6 must contain the value 4, then r8c6 cannot contain the value 4</b> (View 2):<br>
(1) If r7c2 contains the value 4, then r9c3 cannot contain the value 4 (the value can occur only once in the block)<br>(2) If r7c2 contains the value 4 (initial assumption), then r7c3 cannot contain the value 4 (the value can occur only once in the block)<br>(3) If r7c3 does not contain the value 4 and r9c3 does not contain the value 4 (1), then r3c3 must contain the value 4 (only remaining possible position in the column)<br>(4) If r3c3 contains the value 4, then r5c5 cannot contain the value 4 (the value can occur only once in the Main Diagonal)<br>(5) If r7c2 contains the value 4 (initial assumption), then r8c2 cannot contain the value 4 (the value can occur only once in the block)<br>(6) If r8c2 does not contain the value 4, r7c3 does not contain the value 4 (2) and r5c5 does not contain the value 4 (4), then r6c4 must contain the value 4 (only remaining possible position in the Anti Diagonal)<br>(7) If r6c4 contains the value 4, then r6c4 cannot contain the value 9 (the cell can contain only one value)<br>(8) If r3c3 contains the value 4 (3), then r3c3 cannot contain the value 9 (the cell can contain only one value)<br>(9) If r9c3 does not contain the value 4 (1), then r9c3 must contain the value 7 (only remaining possible value in the cell)<br>(10) If r9c3 contains the value 7, then r7c3 cannot contain the value 7 (the value can occur only once in the block)<br>(11) If r7c3 does not contain the value 4 (2) and r7c3 does not contain the value 7, then r7c3 must contain the value 3 (only remaining possible value in the cell)<br>(12) If r7c3 contains the value 3, then r3c7 cannot contain the value 3 (the value can occur only once in the Anti Diagonal)<br>(13) If r7c3 contains the value 3 (11), then r1c9 cannot contain the value 3 (the value can occur only once in the Anti Diagonal)<br>(14) If r1c9 does not contain the value 3 and r3c7 does not contain the value 3 (12), then r3c9 must contain the value 3 (only remaining possible position in the block)<br>(15) If r3c9 contains the value 3, then r3c9 cannot contain the value 9 (the cell can contain only one value)<br>(16) If r3c3 does not contain the value 9 (8), then r5c3 must contain the value 9 (only remaining possible position in the column)<br>(17) If r5c3 contains the value 9, then r5c4 cannot contain the value 9 (the value can occur only once in the row)<br>(18) If r5c3 contains the value 9 (16), then r5c6 cannot contain the value 9 (the value can occur only once in the row)<br>(19) If r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 and r5c4 does not contain the value 9 (17), then r4c9 cannot contain the value 9 (Cells r4c4,r4c6 on value 9 in block 5)<br>(20) If r4c9 does not contain the value 9, r3c9 does not contain the value 9 (15), r3c3 does not contain the value 9 (8) and r6c4 does not contain the value 9 (7), then r3c8 cannot contain the value 9 (Region Forcing Chains: 9 in Anti Diagonal ==> r3c8.9 off)<br>(21) If r6c4 does not contain the value 9 (7) and r3c3 does not contain the value 9 (8), then r1c8 cannot contain the value 9 (Contradiction Forcing Chain: r1c8.9 on ==> r4c4.9 both on & off)<br>(22) If r3c3 does not contain the value 9 (8), r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18) and r5c4 does not contain the value 9 (17), then r2c8 cannot contain the value 9 (Region Forcing Chains: 9 in block ==> r2c8.9 off)<br>(23) If r5c3 contains the value 9 (16), then r5c8 cannot contain the value 9 (the value can occur only once in the row)<br>(24) If r4c9 does not contain the value 9 (19), r3c9 does not contain the value 9 (15), r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18) and r5c4 does not contain the value 9 (17), then r8c8 cannot contain the value 9 (Region Forcing Chains: 9 in column ==> r8c8.9 off)<br>(25) If r8c8 does not contain the value 9, r5c8 does not contain the value 9 (23), r2c8 does not contain the value 9 (22), r1c8 does not contain the value 9 (21) and r3c8 does not contain the value 9 (20), then r6c8 must contain the value 9 (only remaining possible position in the column)<br>(26) If r6c8 contains the value 9, then r6c8 cannot contain the value 5 (the cell can contain only one value)<br>(27) If r6c4 contains the value 4 (6), then r6c4 cannot contain the value 2 (the cell can contain only one value)<br>(28) If r9c3 contains the value 7 (9), then r7c1 cannot contain the value 7 (the value can occur only once in the block)<br>(29) If r9c3 contains the value 7 (9), then r8c1 cannot contain the value 7 (the value can occur only once in the block)<br>(30) If r9c3 contains the value 7 (9), then r9c1 cannot contain the value 7 (the value can occur only once in the block)<br>(31) If r7c2 contains the value 4 (initial assumption), then r7c2 cannot contain the value 1 (the cell can contain only one value)<br>(32) If r3c3 contains the value 4 (3), then r3c3 cannot contain the value 3 (the cell can contain only one value)<br>(33) If r6c4 contains the value 4 (6), then r6c4 cannot contain the value 7 (the cell can contain only one value)<br>(34) If r9c3 contains the value 7 (9), then r8c2 cannot contain the value 7 (the value can occur only once in the block)<br>(35) If r3c9 contains the value 3 (14), then r3c9 cannot contain the value 1 (the cell can contain only one value)<br>(36) If r3c9 contains the value 3 (14), then r3c9 cannot contain the value 6 (the cell can contain only one value)<br>(37) If r6c4 contains the value 4 (6), then r6c4 cannot contain the value 5 (the cell can contain only one value)<br>(38) If r3c9 does not contain the value 1 (35), then r7c7 cannot contain the value 1 (Cells r2c7,r3c7 on value 1 in block 3)<br>(39) If r7c7 does not contain the value 1, r6c4 does not contain the value 5 (37), r3c9 does not contain the value 6 (36), r4c9 does not contain the value 9 (19), r3c9 does not contain the value 9 (15), r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18), r5c4 does not contain the value 9 (17), r3c9 does not contain the value 1 (35), r3c3 does not contain the value 9 (8), r8c2 does not contain the value 7 (34), r7c3 does not contain the value 7 (10), r6c4 does not contain the value 7 (33), r5c8 does not contain the value 9 (23), r3c3 does not contain the value 3 (32), r7c2 does not contain the value 1 (31), r9c1 does not contain the value 7 (30), r8c1 does not contain the value 7 (29), r7c1 does not contain the value 7 (28) and r6c4 does not contain the value 2 (27), then r4c4 cannot contain the value 5 (Contradiction Forcing Chain: r4c4.9 off ==> r6c9.2 both on & off)<br>(40) If r5c3 contains the value 9 (16), then r5c1 cannot contain the value 9 (the value can occur only once in the block)<br>(41) If r6c4 contains the value 4 (6), then r6c8 cannot contain the value 4 (the value can occur only once in the row)<br>(42) If r7c2 contains the value 4 (initial assumption), then r7c8 cannot contain the value 4 (the value can occur only once in the row)<br>(43) If r3c3 contains the value 4 (3), then r8c8 cannot contain the value 4 (the value can occur only once in the Main Diagonal)<br>(44) If r8c8 does not contain the value 4, r7c8 does not contain the value 4 (42) and r6c8 does not contain the value 4 (41), then r5c8 must contain the value 4 (only remaining possible position in the column)<br>(45) If r5c8 contains the value 4, then r5c8 cannot contain the value 2 (the cell can contain only one value)<br>(46) If r7c3 contains the value 3 (11), then r5c3 cannot contain the value 3 (the value can occur only once in the column)<br>(47) If r7c2 contains the value 4 (initial assumption), then r7c2 cannot contain the value 2 (the cell can contain only one value)<br>(48) If r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18) and r5c4 does not contain the value 9 (17), then r4c7 cannot contain the value 9 (Cells r4c4,r4c6 on value 9 in block 5)<br>(49) If r3c3 does not contain the value 3 (32), then r6c1 cannot contain the value 3 (Cells r1c1,r6c6 on value 3 in Main Diagonal)<br>(50) If r7c3 contains the value 3 (11), then r7c1 cannot contain the value 3 (the value can occur only once in the block)<br>(51) If r7c3 contains the value 3 (11), then r4c6 cannot contain the value 3 (the value can occur only once in the Anti Diagonal)<br>(52) If r4c6 does not contain the value 3, r7c1 does not contain the value 3 (50), r6c1 does not contain the value 3 (49), r5c8 does not contain the value 9 (23), r4c9 does not contain the value 9 (19), r4c7 does not contain the value 9 (48), r7c2 does not contain the value 2 (47), r3c3 does not contain the value 9 (8), r5c3 does not contain the value 3 (46), r5c8 does not contain the value 2 (45), r7c2 does not contain the value 1 (31), r5c1 does not contain the value 9 (40) and r3c9 does not contain the value 1 (35), then r6c6 cannot contain the value 5 (Cell Forcing Chains: r5c1 ==> r6c6.5 off)<br>(53) If r6c6 does not contain the value 5 and r4c4 does not contain the value 5 (39), then r7c8 cannot contain the value 5 (Cells r7c7,r9c9 on value 5 in Main Diagonal)<br>(54) If r7c8 does not contain the value 5 and r6c8 does not contain the value 5 (26), then r1c9 cannot contain the value 5 (Cells r1c8,r2c8 on value 5 in column 8)<br>(55) If r6c4 does not contain the value 5 (37) and r1c9 does not contain the value 5, then r2c6 cannot contain the value 5 (Cells r2c8,r4c6 on value 5 in Anti Diagonal)<br>(56) If r7c2 does not contain the value 1 (31), r7c7 does not contain the value 1 (38) and r3c9 does not contain the value 1 (35), then r2c6 cannot contain the value 1 (Region Forcing Chains: 1 in block ==> r2c6.1 off)<br>(57) If r3c3 does not contain the value 9 (8), r6c4 does not contain the value 9 (7), r5c6 does not contain the value 9 (18) and r5c4 does not contain the value 9 (17), then r2c6 cannot contain the value 9 (Region Forcing Chains: 9 in block ==> r2c6.9 off)<br>(58) If r2c6 does not contain the value 9, r2c6 does not contain the value 1 (56) and r2c6 does not contain the value 5 (55), then r2c6 must contain the value 7 (only remaining possible value in the cell)<br>(59) If r2c6 contains the value 7, then r4c6 cannot contain the value 7 (the value can occur only once in the column)<br>(60) If r2c6 contains the value 7 (58), then r2c8 cannot contain the value 7 (the value can occur only once in the row)<br>(61) If r9c1 does not contain the value 7 (30), r8c2 does not contain the value 7 (34), r7c3 does not contain the value 7 (10), r6c4 does not contain the value 7 (33), r2c8 does not contain the value 7 and r4c6 does not contain the value 7 (59), then r3c7 must contain the value 7 (only remaining possible position in the Anti Diagonal)<br>(62) If r3c7 contains the value 7, then r3c7 cannot contain the value 9 (the cell can contain only one value)<br>(63) If r7c7 does not contain the value 1 (38), r3c9 does not contain the value 9 (15), r6c4 does not contain the value 5 (37), r3c9 does not contain the value 6 (36), r6c4 does not contain the value 9 (7), r3c9 does not contain the value 1 (35), r3c3 does not contain the value 9 (8), r8c2 does not contain the value 7 (34), r7c3 does not contain the value 7 (10), r6c4 does not contain the value 7 (33), r5c8 does not contain the value 9 (23), r4c9 does not contain the value 9 (19), r4c7 does not contain the value 9 (48), r3c3 does not contain the value 3 (32), r7c2 does not contain the value 1 (31), r9c1 does not contain the value 7 (30), r8c1 does not contain the value 7 (29), r7c1 does not contain the value 7 (28) and r6c4 does not contain the value 2 (27), then r2c7 cannot contain the value 9 (Contradiction Forcing Chain: r2c7.9 on ==> r6c9.2 both on & off)<br>(64) If r3c9 does not contain the value 9 (15), r3c8 does not contain the value 9 (20), r2c8 does not contain the value 9 (22), r2c7 does not contain the value 9, r1c8 does not contain the value 9 (21) and r3c7 does not contain the value 9 (62), then r1c9 must contain the value 9 (only remaining possible position in the block)<br>(65) If r1c9 contains the value 9, then r1c6 cannot contain the value 9 (the value can occur only once in the row)<br>(66) If r5c8 does not contain the value 2 (45) and r7c2 does not contain the value 2 (47), then r4c6 cannot contain the value 2 (Region Forcing Chains: 2 in column ==> r4c6.2 off)<br>(67) If r7c7 does not contain the value 1 (38), r6c4 does not contain the value 5 (37), r3c9 does not contain the value 6 (36), r4c9 does not contain the value 9 (19), r3c9 does not contain the value 9 (15), r3c9 does not contain the value 1 (35), r3c3 does not contain the value 9 (8), r8c2 does not contain the value 7 (34), r7c3 does not contain the value 7 (10), r6c4 does not contain the value 7 (33), r5c8 does not contain the value 9 (23), r3c3 does not contain the value 3 (32), r7c2 does not contain the value 1 (31), r9c1 does not contain the value 7 (30), r8c1 does not contain the value 7 (29), r7c1 does not contain the value 7 (28) and r6c4 does not contain the value 2 (27), then r4c6 cannot contain the value 9 (Contradiction Forcing Chain: r4c6.9 on ==> r6c9.2 both on & off)<br>(68) If r4c6 does not contain the value 9, r4c6 does not contain the value 3 (51), r4c6 does not contain the value 2 (66) and r4c6 does not contain the value 7 (59), then r4c6 must contain the value 5 (only remaining possible value in the cell)<br>(69) If r4c6 contains the value 5, then r1c6 cannot contain the value 5 (the value can occur only once in the column)<br>(70) If r2c6 contains the value 7 (58), then r1c6 cannot contain the value 7 (the value can occur only once in the block)<br>(71) If r1c6 does not contain the value 7, r1c6 does not contain the value 5 (69) and r1c6 does not contain the value 9 (65), then r1c6 must contain the value 4 (only remaining possible value in the cell)<br>(72) If r1c6 contains the value 4, then r8c6 cannot contain the value 4 (the value can occur only once in the column)<br>
</p>
<br><br>
<b>Nested Forcing Chains details</b> (Note that each Nested Forcing Chain relies on the fact that some <font color="blue">candidates</font> have been excluded by the main Forcing Chain): <br><br>
<i>Nested <b>Contradiction Forcing Chain: r2c7.9 on ==> r6c9.2 both on & off</b></i><br>
Chain 3: <b>If r2c7 contains the value 9, then r6c9 must contain the value 2</b> (View 3):<br>
(1) If r2c7 contains the value 9, then r3c7 cannot contain the value 9 (the value can occur only once in the block)<br>(2) If r2c7 contains the value 9 (initial assumption), then r1c9 cannot contain the value 9 (the value can occur only once in the block)<br>(3) If r2c7 contains the value 9 (initial assumption), then r2c8 cannot contain the value 9 (the value can occur only once in the block)<br>(4) If r2c8 does not contain the value 9, r1c9 does not contain the value 9 (2) and r3c7 does not contain the value 9 (1), then r4c6 must contain the value 9 (only remaining possible position in the Anti Diagonal)<br>(5) If r4c6 contains the value 9, then r4c6 cannot contain the value 7 (the cell can contain only one value)<br>(6) If r2c7 contains the value 9 (initial assumption), then r2c7 cannot contain the value 1 (the cell can contain only one value)<br>(7) If r2c7 does not contain the value 1, then r3c7 must contain the value 1 (only remaining possible position in the block)<br>(8) If r3c7 contains the value 1, then r3c7 cannot contain the value 7 (the cell can contain only one value)<br>(9) If r3c7 does not contain the value 7 and r4c6 does not contain the value 7 (5), then r2c8 must contain the value 7 (only remaining possible position in the Anti Diagonal)<br>(10) If r2c8 contains the value 7, then r2c1 cannot contain the value 7 (the value can occur only once in the row)<br>(11) If r2c7 contains the value 9 (initial assumption), then r2c1 cannot contain the value 9 (the value can occur only once in the row)<br>(12) If r2c1 does not contain the value 9, then r1c1 must contain the value 9 (only remaining possible position in the block)<br>(13) If r1c1 contains the value 9, then r1c1 cannot contain the value 7 (the cell can contain only one value)<br>(14) If r1c1 does not contain the value 7 and r2c1 does not contain the value 7 (10), then r6c1 must contain the value 7 (only remaining possible position in the column)<br>(15) If r6c1 contains the value 7, then r6c1 cannot contain the value 2 (the cell can contain only one value)<br>(16) If r3c7 contains the value 1 (7), then r8c2 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(17) If r8c2 does not contain the value 1, then r6c2 must contain the value 1 (only remaining possible position in the column)<br>(18) If r6c2 contains the value 1, then r6c2 cannot contain the value 2 (the cell can contain only one value)<br>(19) If r1c1 contains the value 9 (12), then r1c1 cannot contain the value 3 (the cell can contain only one value)<br>(20) If r1c1 does not contain the value 3, then r6c6 must contain the value 3 (only remaining possible position in the Main Diagonal)<br>(21) If r6c6 contains the value 3, then r6c6 cannot contain the value 2 (the cell can contain only one value)<br>(22) If r2c7 contains the value 9 (initial assumption), then r6c7 cannot contain the value 9 (the value can occur only once in the column)<br>(23) If r6c7 does not contain the value 9, then r6c8 must contain the value 9 (only remaining possible position in the block)<br>(24) If r6c8 contains the value 9, then r6c8 cannot contain the value 2 (the cell can contain only one value)<br>(25) If r6c8 does not contain the value 2, r6c6 does not contain the value 2 (21), r6c2 does not contain the value 2 (18) and r6c1 does not contain the value 2 (15), then r6c9 must contain the value 2 (only remaining possible position in the row)<br><br>
Chain 4: <b>If r2c7 contains the value 9, then r6c9 cannot contain the value 2</b> (View 4):<br>
(1) If r2c7 contains the value 9, then r3c7 cannot contain the value 9 (the value can occur only once in the block)<br>(2) If r2c7 contains the value 9 (initial assumption), then r1c9 cannot contain the value 9 (the value can occur only once in the block)<br>(3) If r2c7 contains the value 9 (initial assumption), then r2c8 cannot contain the value 9 (the value can occur only once in the block)<br>(4) If r2c8 does not contain the value 9, r1c9 does not contain the value 9 (2) and r3c7 does not contain the value 9 (1), then r4c6 must contain the value 9 (only remaining possible position in the Anti Diagonal)<br>(5) If r4c6 contains the value 9, then r4c6 cannot contain the value 7 (the cell can contain only one value)<br>(6) If r2c7 contains the value 9 (initial assumption), then r2c7 cannot contain the value 1 (the cell can contain only one value)<br>(7) If r2c7 does not contain the value 1, then r3c7 must contain the value 1 (only remaining possible position in the block)<br>(8) If r3c7 contains the value 1, then r3c7 cannot contain the value 7 (the cell can contain only one value)<br>(9) If r3c7 does not contain the value 7 and r4c6 does not contain the value 7 (5), then r2c8 must contain the value 7 (only remaining possible position in the Anti Diagonal)<br>(10) If r2c8 contains the value 7, then r2c8 cannot contain the value 5 (the cell can contain only one value)<br>(11) If r4c6 contains the value 9 (4), then r4c6 cannot contain the value 5 (the cell can contain only one value)<br>(12) If r4c6 does not contain the value 5 and r2c8 does not contain the value 5 (10), then r1c9 must contain the value 5 (only remaining possible position in the Anti Diagonal)<br>(13) If r1c9 contains the value 5, then r1c9 cannot contain the value 6 (the cell can contain only one value)<br>(14) If r1c9 does not contain the value 9 (2), then r8c9 must contain the value 9 (only remaining possible position in the column)<br>(15) If r8c9 contains the value 9, then r8c9 cannot contain the value 6 (the cell can contain only one value)<br>(16) If r3c7 contains the value 1 (7), then r8c2 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(17) If r8c2 does not contain the value 1, then r6c2 must contain the value 1 (only remaining possible position in the column)<br>(18) If r6c2 contains the value 1, then r6c6 cannot contain the value 1 (the value can occur only once in the row)<br>(19) If r3c7 contains the value 1 (7), then r5c5 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(20) If r5c5 does not contain the value 1 and r6c6 does not contain the value 1 (18), then r9c9 must contain the value 1 (only remaining possible position in the Main Diagonal)<br>(21) If r9c9 contains the value 1, then r9c9 cannot contain the value 6 (the cell can contain only one value)<br>(22) If r9c9 does not contain the value 6, r8c9 does not contain the value 6 (15) and r1c9 does not contain the value 6 (13), then r6c9 must contain the value 6 (only remaining possible position in the column)<br>(23) If r6c9 contains the value 6, then r6c9 cannot contain the value 2 (the cell can contain only one value)<br><br>
<i>Nested <b>Region Forcing Chains: 9 in Anti Diagonal ==> r3c8.9 off</b></i><br>
Chain 5: <b>If r1c9 contains the value 9, then r3c8 cannot contain the value 9</b> (View 5):<br>
(1) If r1c9 contains the value 9, then r3c8 cannot contain the value 9 (the value can occur only once in the block)<br><br>
Chain 6: <b>If r2c8 contains the value 9, then r3c8 cannot contain the value 9</b> (View 6):<br>
(1) If r2c8 contains the value 9, then r3c8 cannot contain the value 9 (the value can occur only once in the block)<br><br>
Chain 7: <b>If r3c7 contains the value 9, then r3c8 cannot contain the value 9</b> (View 7):<br>
(1) If r3c7 contains the value 9, then r3c8 cannot contain the value 9 (the value can occur only once in the block)<br><br>
Chain 8: <b>If r4c6 contains the value 9, then r3c8 cannot contain the value 9</b> (View 8):<br>
(1) If r4c6 contains the value 9, then r1c9 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(2) If r1c9 does not contain the value 9, then r8c9 must contain the value 9 (only remaining possible position in the column)<br>(3) If r8c9 contains the value 9, then r8c8 cannot contain the value 9 (the value can occur only once in the block)<br>(4) If r4c6 contains the value 9 (initial assumption), then r4c4 cannot contain the value 9 (the value can occur only once in the block)<br>(5) If r4c4 does not contain the value 9 and r8c8 does not contain the value 9 (3), then r1c1 must contain the value 9 (only remaining possible position in the Main Diagonal)<br>(6) If r1c1 contains the value 9, then r2c1 cannot contain the value 9 (the value can occur only once in the block)<br>(7) If r4c6 contains the value 9 (initial assumption), then r2c6 cannot contain the value 9 (the value can occur only once in the column)<br>(8) If r4c6 contains the value 9 (initial assumption), then r2c8 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(9) If r2c8 does not contain the value 9, r2c6 does not contain the value 9 (7) and r2c1 does not contain the value 9 (6), then r2c7 must contain the value 9 (only remaining possible position in the row)<br>(10) If r2c7 contains the value 9, then r3c8 cannot contain the value 9 (the value can occur only once in the block)<br><br>
<i>Nested <b>Contradiction Forcing Chain: r1c8.9 on ==> r4c4.9 both on & off</b></i><br>
Chain 9: <b>If r1c8 contains the value 9, then r4c4 must contain the value 9</b> (View 9):<br>
(1) If r1c8 contains the value 9, then r8c8 cannot contain the value 9 (the value can occur only once in the column)<br>(2) If r1c8 contains the value 9 (initial assumption), then r1c1 cannot contain the value 9 (the value can occur only once in the row)<br>(3) If r1c1 does not contain the value 9 and r8c8 does not contain the value 9 (1), then r4c4 must contain the value 9 (only remaining possible position in the Main Diagonal)<br><br>
Chain 10: <b>If r1c8 contains the value 9, then r4c4 cannot contain the value 9</b> (View 10):<br>
(1) If r1c8 contains the value 9, then r3c7 cannot contain the value 9 (the value can occur only once in the block)<br>(2) If r1c8 contains the value 9 (initial assumption), then r1c9 cannot contain the value 9 (the value can occur only once in the block)<br>(3) If r1c8 contains the value 9 (initial assumption), then r2c8 cannot contain the value 9 (the value can occur only once in the block)<br>(4) If r2c8 does not contain the value 9, r1c9 does not contain the value 9 (2) and r3c7 does not contain the value 9 (1), then r4c6 must contain the value 9 (only remaining possible position in the Anti Diagonal)<br>(5) If r4c6 contains the value 9, then r4c4 cannot contain the value 9 (the value can occur only once in the block)<br><br>
<i>Nested <b>Region Forcing Chains: 9 in block ==> r2c8.9 off</b></i><br>
Chain 11: <b>If r1c1 contains the value 9, then r2c8 cannot contain the value 9</b> (View 11):<br>
(1) If r1c1 contains the value 9, then r4c4 cannot contain the value 9 (the value can occur only once in the Main Diagonal)<br>(2) If r4c4 does not contain the value 9, then r4c6 must contain the value 9 (only remaining possible position in the block)<br>(3) If r4c6 contains the value 9, then r2c8 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br><br>
Chain 12: <b>If r2c1 contains the value 9, then r2c8 cannot contain the value 9</b> (View 12):<br>
(1) If r2c1 contains the value 9, then r2c8 cannot contain the value 9 (the value can occur only once in the row)<br><br>
<i>Nested <b>Region Forcing Chains: 9 in column ==> r8c8.9 off</b></i><br>
Chain 13: <b>If r1c9 contains the value 9, then r8c8 cannot contain the value 9</b> (View 13):<br>
(1) If r1c9 contains the value 9, then r4c6 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(2) If r4c6 does not contain the value 9, then r4c4 must contain the value 9 (only remaining possible position in the block)<br>(3) If r4c4 contains the value 9, then r8c8 cannot contain the value 9 (the value can occur only once in the Main Diagonal)<br><br>
Chain 14: <b>If r8c9 contains the value 9, then r8c8 cannot contain the value 9</b> (View 14):<br>
(1) If r8c9 contains the value 9, then r8c8 cannot contain the value 9 (the value can occur only once in the block)<br><br>
<i>Nested <b>Region Forcing Chains: 1 in block ==> r2c6.1 off</b></i><br>
Chain 15: <b>If r2c7 contains the value 1, then r2c6 cannot contain the value 1</b> (View 15):<br>
(1) If r2c7 contains the value 1, then r2c6 cannot contain the value 1 (the value can occur only once in the row)<br><br>
Chain 16: <b>If r3c7 contains the value 1, then r2c6 cannot contain the value 1</b> (View 16):<br>
(1) If r3c7 contains the value 1, then r8c2 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(2) If r8c2 does not contain the value 1, then r6c2 must contain the value 1 (only remaining possible position in the column)<br>(3) If r6c2 contains the value 1, then r6c6 cannot contain the value 1 (the value can occur only once in the row)<br>(4) If r3c7 contains the value 1 (initial assumption), then r5c5 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(5) If r5c5 does not contain the value 1 and r6c6 does not contain the value 1 (3), then r9c9 must contain the value 1 (only remaining possible position in the Main Diagonal)<br>(6) If r9c9 contains the value 1, then r9c5 cannot contain the value 1 (the value can occur only once in the row)<br>(7) If r3c7 contains the value 1 (initial assumption), then r3c5 cannot contain the value 1 (the value can occur only once in the row)<br>(8) If r6c2 contains the value 1 (2), then r6c5 cannot contain the value 1 (the value can occur only once in the row)<br>(9) If r6c5 does not contain the value 1, r5c5 does not contain the value 1 (4), r3c5 does not contain the value 1 (7) and r9c5 does not contain the value 1 (6), then r2c5 must contain the value 1 (only remaining possible position in the column)<br>(10) If r2c5 contains the value 1, then r2c6 cannot contain the value 1 (the value can occur only once in the block)<br><br>
<i>Nested <b>Region Forcing Chains: 9 in block ==> r2c6.9 off</b></i><br>
Chain 17: <b>If r1c1 contains the value 9, then r2c6 cannot contain the value 9</b> (View 17):<br>
(1) If r1c1 contains the value 9, then r4c4 cannot contain the value 9 (the value can occur only once in the Main Diagonal)<br>(2) If r4c4 does not contain the value 9, then r4c6 must contain the value 9 (only remaining possible position in the block)<br>(3) If r4c6 contains the value 9, then r2c6 cannot contain the value 9 (the value can occur only once in the column)<br><br>
Chain 18: <b>If r2c1 contains the value 9, then r2c6 cannot contain the value 9</b> (View 18):<br>
(1) If r2c1 contains the value 9, then r2c6 cannot contain the value 9 (the value can occur only once in the row)<br><br>
<i>Nested <b>Contradiction Forcing Chain: r4c4.9 off ==> r6c9.2 both on & off</b></i><br>
Chain 19: <b>If r4c4 does not contain the value 9, then r6c9 must contain the value 2</b> (View 19):<br>
(1) If r4c4 does not contain the value 9, then r4c6 must contain the value 9 (only remaining possible position in the block)<br>(2) If r4c6 contains the value 9, then r1c9 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(3) If r1c9 does not contain the value 9, then r8c9 must contain the value 9 (only remaining possible position in the column)<br>(4) If r8c9 contains the value 9, then r8c8 cannot contain the value 9 (the value can occur only once in the block)<br>(5) If r4c4 does not contain the value 9 (initial assumption) and r8c8 does not contain the value 9, then r1c1 must contain the value 9 (only remaining possible position in the Main Diagonal)<br>(6) If r1c1 contains the value 9, then r2c1 cannot contain the value 9 (the value can occur only once in the block)<br>(7) If r4c6 contains the value 9 (1), then r2c6 cannot contain the value 9 (the value can occur only once in the column)<br>(8) If r4c6 contains the value 9 (1), then r2c8 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(9) If r2c8 does not contain the value 9, r2c6 does not contain the value 9 (7) and r2c1 does not contain the value 9 (6), then r2c7 must contain the value 9 (only remaining possible position in the row)<br>(10) If r2c7 contains the value 9, then r2c7 cannot contain the value 1 (the cell can contain only one value)<br>(11) If r2c7 does not contain the value 1, then r3c7 must contain the value 1 (only remaining possible position in the block)<br>(12) If r3c7 contains the value 1, then r3c7 cannot contain the value 7 (the cell can contain only one value)<br>(13) If r4c6 contains the value 9 (1), then r4c6 cannot contain the value 7 (the cell can contain only one value)<br>(14) If r4c6 does not contain the value 7 and r3c7 does not contain the value 7 (12), then r2c8 must contain the value 7 (only remaining possible position in the Anti Diagonal)<br>(15) If r2c8 contains the value 7, then r2c1 cannot contain the value 7 (the value can occur only once in the row)<br>(16) If r1c1 contains the value 9 (5), then r1c1 cannot contain the value 7 (the cell can contain only one value)<br>(17) If r1c1 does not contain the value 7 and r2c1 does not contain the value 7 (15), then r6c1 must contain the value 7 (only remaining possible position in the column)<br>(18) If r6c1 contains the value 7, then r6c1 cannot contain the value 2 (the cell can contain only one value)<br>(19) If r3c7 contains the value 1 (11), then r8c2 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(20) If r8c2 does not contain the value 1, then r6c2 must contain the value 1 (only remaining possible position in the column)<br>(21) If r6c2 contains the value 1, then r6c2 cannot contain the value 2 (the cell can contain only one value)<br>(22) If r1c1 contains the value 9 (5), then r1c1 cannot contain the value 3 (the cell can contain only one value)<br>(23) If r1c1 does not contain the value 3, then r6c6 must contain the value 3 (only remaining possible position in the Main Diagonal)<br>(24) If r6c6 contains the value 3, then r6c6 cannot contain the value 2 (the cell can contain only one value)<br>(25) If r2c7 contains the value 9 (9), then r3c8 cannot contain the value 9 (the value can occur only once in the block)<br>(26) If r1c1 contains the value 9 (5), then r1c8 cannot contain the value 9 (the value can occur only once in the row)<br>(27) If r8c8 does not contain the value 9 (4), r2c8 does not contain the value 9 (8), r1c8 does not contain the value 9 and r3c8 does not contain the value 9 (25), then r6c8 must contain the value 9 (only remaining possible position in the column)<br>(28) If r6c8 contains the value 9, then r6c8 cannot contain the value 2 (the cell can contain only one value)<br>(29) If r6c8 does not contain the value 2, r6c6 does not contain the value 2 (24), r6c2 does not contain the value 2 (21) and r6c1 does not contain the value 2 (18), then r6c9 must contain the value 2 (only remaining possible position in the row)<br><br>
Chain 20: <b>If r4c4 does not contain the value 9, then r6c9 cannot contain the value 2</b> (View 20):<br>
(1) If r4c4 does not contain the value 9, then r4c6 must contain the value 9 (only remaining possible position in the block)<br>(2) If r4c6 contains the value 9, then r1c9 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(3) If r1c9 does not contain the value 9, then r8c9 must contain the value 9 (only remaining possible position in the column)<br>(4) If r8c9 contains the value 9, then r8c8 cannot contain the value 9 (the value can occur only once in the block)<br>(5) If r4c4 does not contain the value 9 (initial assumption) and r8c8 does not contain the value 9, then r1c1 must contain the value 9 (only remaining possible position in the Main Diagonal)<br>(6) If r1c1 contains the value 9, then r2c1 cannot contain the value 9 (the value can occur only once in the block)<br>(7) If r4c6 contains the value 9 (1), then r2c6 cannot contain the value 9 (the value can occur only once in the column)<br>(8) If r4c6 contains the value 9 (1), then r2c8 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(9) If r2c8 does not contain the value 9, r2c6 does not contain the value 9 (7) and r2c1 does not contain the value 9 (6), then r2c7 must contain the value 9 (only remaining possible position in the row)<br>(10) If r2c7 contains the value 9, then r2c7 cannot contain the value 1 (the cell can contain only one value)<br>(11) If r2c7 does not contain the value 1, then r3c7 must contain the value 1 (only remaining possible position in the block)<br>(12) If r3c7 contains the value 1, then r3c7 cannot contain the value 7 (the cell can contain only one value)<br>(13) If r4c6 contains the value 9 (1), then r4c6 cannot contain the value 7 (the cell can contain only one value)<br>(14) If r4c6 does not contain the value 7 and r3c7 does not contain the value 7 (12), then r2c8 must contain the value 7 (only remaining possible position in the Anti Diagonal)<br>(15) If r2c8 contains the value 7, then r2c8 cannot contain the value 5 (the cell can contain only one value)<br>(16) If r4c6 contains the value 9 (1), then r4c6 cannot contain the value 5 (the cell can contain only one value)<br>(17) If r4c6 does not contain the value 5 and r2c8 does not contain the value 5 (15), then r1c9 must contain the value 5 (only remaining possible position in the Anti Diagonal)<br>(18) If r1c9 contains the value 5, then r1c9 cannot contain the value 6 (the cell can contain only one value)<br>(19) If r8c9 contains the value 9 (3), then r8c9 cannot contain the value 6 (the cell can contain only one value)<br>(20) If r3c7 contains the value 1 (11), then r5c5 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(21) If r1c1 contains the value 9 (5), then r1c1 cannot contain the value 3 (the cell can contain only one value)<br>(22) If r1c1 does not contain the value 3, then r6c6 must contain the value 3 (only remaining possible position in the Main Diagonal)<br>(23) If r6c6 contains the value 3, then r6c6 cannot contain the value 1 (the cell can contain only one value)<br>(24) If r6c6 does not contain the value 1 and r5c5 does not contain the value 1 (20), then r9c9 must contain the value 1 (only remaining possible position in the Main Diagonal)<br>(25) If r9c9 contains the value 1, then r9c9 cannot contain the value 6 (the cell can contain only one value)<br>(26) If r9c9 does not contain the value 6, r8c9 does not contain the value 6 (19) and r1c9 does not contain the value 6 (18), then r6c9 must contain the value 6 (only remaining possible position in the column)<br>(27) If r6c9 contains the value 6, then r6c9 cannot contain the value 2 (the cell can contain only one value)<br><br>
<i>Nested <b>Cell Forcing Chains: r5c1 ==> r6c6.5 off</b></i><br>
Chain 21: <b>If r5c1 contains the value 1, then r6c6 cannot contain the value 5</b> (View 21):<br>
(1) If r5c1 contains the value 1, then r5c1 cannot contain the value 3 (the cell can contain only one value)<br>(2) If r5c1 does not contain the value 3, then r5c6 must contain the value 3 (only remaining possible position in the row)<br>(3) If r5c6 contains the value 3, then r5c6 cannot contain the value 2 (the cell can contain only one value)<br>(4) If r5c1 contains the value 1 (initial assumption), then r5c1 cannot contain the value 2 (the cell can contain only one value)<br>(5) If r5c1 does not contain the value 2 and r5c6 does not contain the value 2 (3), then r5c5 must contain the value 2 (only remaining possible position in the row)<br>(6) If r5c5 contains the value 2, then r9c1 cannot contain the value 2 (the value can occur only once in the Anti Diagonal)<br>(7) If r5c5 contains the value 2 (5), then r9c5 cannot contain the value 2 (the value can occur only once in the column)<br>(8) If r5c5 contains the value 2 (5), then r9c9 cannot contain the value 2 (the value can occur only once in the Main Diagonal)<br>(9) If r9c9 does not contain the value 2, r9c5 does not contain the value 2 (7) and r9c1 does not contain the value 2 (6), then r9c4 must contain the value 2 (only remaining possible position in the row)<br>(10) If r9c4 contains the value 2, then r8c6 cannot contain the value 2 (the value can occur only once in the block)<br>(11) If r5c5 contains the value 2 (5), then r4c6 cannot contain the value 2 (the value can occur only once in the block)<br>(12) If r5c5 contains the value 2 (5), then r6c6 cannot contain the value 2 (the value can occur only once in the block)<br>(13) If r6c6 does not contain the value 2, r5c6 does not contain the value 2 (3), r4c6 does not contain the value 2 (11) and r8c6 does not contain the value 2 (10), then r3c6 must contain the value 2 (only remaining possible position in the column)<br>(14) If r3c6 contains the value 2, then r3c6 cannot contain the value 1 (the cell can contain only one value)<br>(15) If r5c1 contains the value 1 (initial assumption), then r6c2 cannot contain the value 1 (the value can occur only once in the block)<br>(16) If r6c2 does not contain the value 1, then r8c2 must contain the value 1 (only remaining possible position in the column)<br>(17) If r8c2 contains the value 1, then r3c7 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(18) If r3c7 does not contain the value 1, then r2c7 must contain the value 1 (only remaining possible position in the block)<br>(19) If r2c7 contains the value 1, then r2c6 cannot contain the value 1 (the value can occur only once in the row)<br>(20) If r5c1 contains the value 1 (initial assumption), then r5c6 cannot contain the value 1 (the value can occur only once in the row)<br>(21) If r8c2 contains the value 1 (16), then r8c6 cannot contain the value 1 (the value can occur only once in the row)<br>(22) If r8c6 does not contain the value 1, r5c6 does not contain the value 1 (20), r2c6 does not contain the value 1 (19) and r3c6 does not contain the value 1 (14), then r6c6 must contain the value 1 (only remaining possible position in the column)<br>(23) If r6c6 contains the value 1, then r6c6 cannot contain the value 5 (the cell can contain only one value)<br><br>
Chain 22: <b>If r5c1 contains the value 2, then r6c6 cannot contain the value 5</b> (View 22):<br>
(1) If r5c1 contains the value 2, then r6c2 cannot contain the value 2 (the value can occur only once in the block)<br>(2) If r5c1 contains the value 2 (initial assumption), then r4c2 cannot contain the value 2 (the value can occur only once in the block)<br>(3) If r4c2 does not contain the value 2 and r6c2 does not contain the value 2 (1), then r8c2 must contain the value 2 (only remaining possible position in the column)<br>(4) If r8c2 contains the value 2, then r8c2 cannot contain the value 1 (the cell can contain only one value)<br>(5) If r8c2 does not contain the value 1, then r6c2 must contain the value 1 (only remaining possible position in the column)<br>(6) If r6c2 contains the value 1, then r6c2 cannot contain the value 3 (the cell can contain only one value)<br>(7) If r5c1 contains the value 2 (initial assumption), then r5c1 cannot contain the value 3 (the cell can contain only one value)<br>(8) If r5c1 does not contain the value 3, then r5c6 must contain the value 3 (only remaining possible position in the row)<br>(9) If r5c6 contains the value 3, then r6c6 cannot contain the value 3 (the value can occur only once in the block)<br>(10) If r6c6 does not contain the value 3 and r6c2 does not contain the value 3 (6), then r6c7 must contain the value 3 (only remaining possible position in the row)<br>(11) If r6c7 contains the value 3, then r6c7 cannot contain the value 9 (the cell can contain only one value)<br>(12) If r6c7 does not contain the value 9, then r6c8 must contain the value 9 (only remaining possible position in the block)<br>(13) If r6c8 contains the value 9, then r8c8 cannot contain the value 9 (the value can occur only once in the column)<br>(14) If r5c1 does not contain the value 3 (7), then r1c1 must contain the value 3 (only remaining possible position in the column)<br>(15) If r1c1 contains the value 3, then r1c1 cannot contain the value 9 (the cell can contain only one value)<br>(16) If r1c1 does not contain the value 9 and r8c8 does not contain the value 9 (13), then r4c4 must contain the value 9 (only remaining possible position in the Main Diagonal)<br>(17) If r4c4 contains the value 9, then r4c4 cannot contain the value 2 (the cell can contain only one value)<br>(18) If r5c1 contains the value 2 (initial assumption), then r5c5 cannot contain the value 2 (the value can occur only once in the row)<br>(19) If r8c2 contains the value 2 (3), then r8c8 cannot contain the value 2 (the value can occur only once in the row)<br>(20) If r6c8 contains the value 9 (12), then r6c8 cannot contain the value 2 (the cell can contain only one value)<br>(21) If r8c8 does not contain the value 2 (19) and r6c8 does not contain the value 2, then r7c8 must contain the value 2 (only remaining possible position in the column)<br>(22) If r7c8 contains the value 2, then r9c9 cannot contain the value 2 (the value can occur only once in the block)<br>(23) If r9c9 does not contain the value 2, r8c8 does not contain the value 2 (19), r5c5 does not contain the value 2 (18) and r4c4 does not contain the value 2 (17), then r6c6 must contain the value 2 (only remaining possible position in the Main Diagonal)<br>(24) If r6c6 contains the value 2, then r6c6 cannot contain the value 5 (the cell can contain only one value)<br><br>
Chain 23: <b>If r5c1 contains the value 3, then r6c6 cannot contain the value 5</b> (View 23):<br>
(1) If r5c1 contains the value 3, then r5c6 cannot contain the value 3 (the value can occur only once in the row)<br>(2) If r5c6 does not contain the value 3, then r6c6 must contain the value 3 (only remaining possible position in the block)<br>(3) If r6c6 contains the value 3, then r6c6 cannot contain the value 5 (the cell can contain only one value)<br><br>
<i>Nested <b>Region Forcing Chains: 2 in column ==> r4c6.2 off</b></i><br>
Chain 24: <b>If r4c2 contains the value 2, then r4c6 cannot contain the value 2</b> (View 24):<br>
(1) If r4c2 contains the value 2, then r4c6 cannot contain the value 2 (the value can occur only once in the row)<br><br>
Chain 25: <b>If r6c2 contains the value 2, then r4c6 cannot contain the value 2</b> (View 25):<br>
(1) If r6c2 contains the value 2, then r6c9 cannot contain the value 2 (the value can occur only once in the row)<br>(2) If r6c2 contains the value 2 (initial assumption), then r6c8 cannot contain the value 2 (the value can occur only once in the row)<br>(3) If r6c8 does not contain the value 2 and r6c9 does not contain the value 2 (1), then r4c9 must contain the value 2 (only remaining possible position in the block)<br>(4) If r4c9 contains the value 2, then r4c6 cannot contain the value 2 (the value can occur only once in the row)<br><br>
Chain 26: <b>If r8c2 contains the value 2, then r4c6 cannot contain the value 2</b> (View 26):<br>
(1) If r8c2 contains the value 2, then r4c6 cannot contain the value 2 (the value can occur only once in the Anti Diagonal)<br><br>
<i>Nested <b>Contradiction Forcing Chain: r4c6.9 on ==> r6c9.2 both on & off</b></i><br>
Chain 27: <b>If r4c6 contains the value 9, then r6c9 must contain the value 2</b> (View 27):<br>
(1) If r4c6 contains the value 9, then r1c9 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(2) If r1c9 does not contain the value 9, then r8c9 must contain the value 9 (only remaining possible position in the column)<br>(3) If r8c9 contains the value 9, then r8c8 cannot contain the value 9 (the value can occur only once in the block)<br>(4) If r4c6 contains the value 9 (initial assumption), then r4c4 cannot contain the value 9 (the value can occur only once in the block)<br>(5) If r4c4 does not contain the value 9 and r8c8 does not contain the value 9 (3), then r1c1 must contain the value 9 (only remaining possible position in the Main Diagonal)<br>(6) If r1c1 contains the value 9, then r2c1 cannot contain the value 9 (the value can occur only once in the block)<br>(7) If r4c6 contains the value 9 (initial assumption), then r2c6 cannot contain the value 9 (the value can occur only once in the column)<br>(8) If r4c6 contains the value 9 (initial assumption), then r2c8 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(9) If r2c8 does not contain the value 9, r2c6 does not contain the value 9 (7) and r2c1 does not contain the value 9 (6), then r2c7 must contain the value 9 (only remaining possible position in the row)<br>(10) If r2c7 contains the value 9, then r2c7 cannot contain the value 1 (the cell can contain only one value)<br>(11) If r2c7 does not contain the value 1, then r3c7 must contain the value 1 (only remaining possible position in the block)<br>(12) If r3c7 contains the value 1, then r3c7 cannot contain the value 7 (the cell can contain only one value)<br>(13) If r4c6 contains the value 9 (initial assumption), then r4c6 cannot contain the value 7 (the cell can contain only one value)<br>(14) If r4c6 does not contain the value 7 and r3c7 does not contain the value 7 (12), then r2c8 must contain the value 7 (only remaining possible position in the Anti Diagonal)<br>(15) If r2c8 contains the value 7, then r2c1 cannot contain the value 7 (the value can occur only once in the row)<br>(16) If r1c1 contains the value 9 (5), then r1c1 cannot contain the value 7 (the cell can contain only one value)<br>(17) If r1c1 does not contain the value 7 and r2c1 does not contain the value 7 (15), then r6c1 must contain the value 7 (only remaining possible position in the column)<br>(18) If r6c1 contains the value 7, then r6c1 cannot contain the value 2 (the cell can contain only one value)<br>(19) If r3c7 contains the value 1 (11), then r8c2 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(20) If r8c2 does not contain the value 1, then r6c2 must contain the value 1 (only remaining possible position in the column)<br>(21) If r6c2 contains the value 1, then r6c2 cannot contain the value 2 (the cell can contain only one value)<br>(22) If r1c1 contains the value 9 (5), then r1c1 cannot contain the value 3 (the cell can contain only one value)<br>(23) If r1c1 does not contain the value 3, then r6c6 must contain the value 3 (only remaining possible position in the Main Diagonal)<br>(24) If r6c6 contains the value 3, then r6c6 cannot contain the value 2 (the cell can contain only one value)<br>(25) If r2c7 contains the value 9 (9), then r3c8 cannot contain the value 9 (the value can occur only once in the block)<br>(26) If r1c1 contains the value 9 (5), then r1c8 cannot contain the value 9 (the value can occur only once in the row)<br>(27) If r8c8 does not contain the value 9 (3), r2c8 does not contain the value 9 (8), r1c8 does not contain the value 9 and r3c8 does not contain the value 9 (25), then r6c8 must contain the value 9 (only remaining possible position in the column)<br>(28) If r6c8 contains the value 9, then r6c8 cannot contain the value 2 (the cell can contain only one value)<br>(29) If r6c8 does not contain the value 2, r6c6 does not contain the value 2 (24), r6c2 does not contain the value 2 (21) and r6c1 does not contain the value 2 (18), then r6c9 must contain the value 2 (only remaining possible position in the row)<br><br>
Chain 28: <b>If r4c6 contains the value 9, then r6c9 cannot contain the value 2</b> (View 28):<br>
(1) If r4c6 contains the value 9, then r1c9 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(2) If r1c9 does not contain the value 9, then r8c9 must contain the value 9 (only remaining possible position in the column)<br>(3) If r8c9 contains the value 9, then r8c8 cannot contain the value 9 (the value can occur only once in the block)<br>(4) If r4c6 contains the value 9 (initial assumption), then r4c4 cannot contain the value 9 (the value can occur only once in the block)<br>(5) If r4c4 does not contain the value 9 and r8c8 does not contain the value 9 (3), then r1c1 must contain the value 9 (only remaining possible position in the Main Diagonal)<br>(6) If r1c1 contains the value 9, then r2c1 cannot contain the value 9 (the value can occur only once in the block)<br>(7) If r4c6 contains the value 9 (initial assumption), then r2c6 cannot contain the value 9 (the value can occur only once in the column)<br>(8) If r4c6 contains the value 9 (initial assumption), then r2c8 cannot contain the value 9 (the value can occur only once in the Anti Diagonal)<br>(9) If r2c8 does not contain the value 9, r2c6 does not contain the value 9 (7) and r2c1 does not contain the value 9 (6), then r2c7 must contain the value 9 (only remaining possible position in the row)<br>(10) If r2c7 contains the value 9, then r2c7 cannot contain the value 1 (the cell can contain only one value)<br>(11) If r2c7 does not contain the value 1, then r3c7 must contain the value 1 (only remaining possible position in the block)<br>(12) If r3c7 contains the value 1, then r3c7 cannot contain the value 7 (the cell can contain only one value)<br>(13) If r4c6 contains the value 9 (initial assumption), then r4c6 cannot contain the value 7 (the cell can contain only one value)<br>(14) If r4c6 does not contain the value 7 and r3c7 does not contain the value 7 (12), then r2c8 must contain the value 7 (only remaining possible position in the Anti Diagonal)<br>(15) If r2c8 contains the value 7, then r2c8 cannot contain the value 5 (the cell can contain only one value)<br>(16) If r4c6 contains the value 9 (initial assumption), then r4c6 cannot contain the value 5 (the cell can contain only one value)<br>(17) If r4c6 does not contain the value 5 and r2c8 does not contain the value 5 (15), then r1c9 must contain the value 5 (only remaining possible position in the Anti Diagonal)<br>(18) If r1c9 contains the value 5, then r1c9 cannot contain the value 6 (the cell can contain only one value)<br>(19) If r8c9 contains the value 9 (2), then r8c9 cannot contain the value 6 (the cell can contain only one value)<br>(20) If r3c7 contains the value 1 (11), then r5c5 cannot contain the value 1 (the value can occur only once in the Anti Diagonal)<br>(21) If r1c1 contains the value 9 (5), then r1c1 cannot contain the value 3 (the cell can contain only one value)<br>(22) If r1c1 does not contain the value 3, then r6c6 must contain the value 3 (only remaining possible position in the Main Diagonal)<br>(23) If r6c6 contains the value 3, then r6c6 cannot contain the value 1 (the cell can contain only one value)<br>(24) If r6c6 does not contain the value 1 and r5c5 does not contain the value 1 (20), then r9c9 must contain the value 1 (only remaining possible position in the Main Diagonal)<br>(25) If r9c9 contains the value 1, then r9c9 cannot contain the value 6 (the cell can contain only one value)<br>(26) If r9c9 does not contain the value 6, r8c9 does not contain the value 6 (19) and r1c9 does not contain the value 6 (18), then r6c9 must contain the value 6 (only remaining possible position in the column)<br>(27) If r6c9 contains the value 6, then r6c9 cannot contain the value 2 (the cell can contain only one value)<br><br>
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%g: ..1...2....23....45........4.6....1..5......7..8..........96.....5.3.....9...8.3.
%i: ..1...2....23....45........4.6....1..5......7..8..........96.....5.3.....9...8.3.
%m: ..3..6789..34.678.1...........456789...45678....45.7.9.2...........56789..3.56.89.....6789.....678..2.........3......1...5678.1...5.7.91...56789....56789...4.........5......34.678...34..7.912.4.678912.4.678.12.4..7.91.3..6789.....67891.3..6.89...4......23...7.......6....2..5.789.2..5.78..23.5.7.9..3.5..891.........23.5..89123.....9....5......3.....91..4.6.8912.4...8.1234....9...4.6.8..2.4.6.89......7..123...7.9123...7.........8.12.4567.91..4567..12345.7....3456..9.2.456..9.2..56...123...78.123...78...34..7..12.45.7..........9.....6...1..45.78..2.45.78.12..5..8.12...678.12.4.678.....5....12.4..7....3......12.4..7..1..4.6789.2.4.678912...6.8912...67..........9...4..7..12.45.7..12.45.7.........8.1..4567....3......12..56...
%M:
+-------------------------+-------------------------+-------------------------+
| 36789 34678 1 | 456789 45678 4579 | 2 56789 35689 |
| 6789 678 2 | 3 15678 1579 | 156789 56789 4 |
| 5 34678 3479 | 1246789 124678 12479 | 136789 6789 13689 |
+-------------------------+-------------------------+-------------------------+
| 4 237 6 | 25789 2578 23579 | 3589 1 23589 |
| 1239 5 39 | 14689 1248 12349 | 468 24689 7 |
| 12379 1237 8 | 1245679 14567 123457 | 34569 24569 256 |
+-------------------------+-------------------------+-------------------------+
| 12378 12378 347 | 12457 9 6 | 14578 24578 1258 |
| 12678 124678 5 | 1247 3 1247 | 146789 246789 12689 |
| 1267 9 47 | 12457 12457 8 | 14567 3 1256 |
+-------------------------+-------------------------+-------------------------+
%r: 11.9
%s: Contradiction Forcing Chain: r7c2.4 on ==> r8c6.4 both on & off
--- end of after section ---
by m_b_metcalf » Sun Dec 29, 2019 12:28 pm 
. I have a backlog of 400!
