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Ordered NC 4, 4H and 5 http://rcbroughton.co.uk/sudoku/forum/viewtopic.php?f=13&t=1548 |
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Author: | Andrew [ Mon Apr 06, 2020 2:54 am ] |
Post subject: | Re: Ordered NC 4, 4H and 5 |
Ordered NC Killer 4 is straightforward, so I won't post my walkthrough. Solution: 6 4 8 7 1 9 5 3 2 9 7 3 2 5 8 1 6 4 1 5 2 6 4 3 9 8 7 5 9 7 1 8 2 6 4 3 3 6 4 9 7 5 8 2 1 2 8 1 3 6 4 7 9 5 4 1 6 8 3 7 2 5 9 8 3 9 5 2 1 4 7 6 7 2 5 4 9 6 3 1 8 |
Author: | Andrew [ Mon Apr 06, 2020 11:08 pm ] |
Post subject: | Re: Ordered NC 4, 4H and 5 |
Thanks HATMAN for that amusing comment! While Vanilla Sudokus aren't my favourites, they're occasionally useful when learning how to solve new variants. Ordered NC Killer 3 was a fairly hard one, so I'd taken a break to do number 4 yesterday and came back to it today. Here is my walkthrough for Ordered NC Killer 3: NC increasing (ONC), 89 not allowed, 98 allowed. Prelims a) R23C2 = {16/25}/[43] (ONC) b) R23C8 = [98] (ONC) c) R45C3 = [98] (ONC) d) R67C5 = {49/58}/[76] (ONC) e) R8C12 = {19/28/37/46}, no 5 f) 13(4) cage at R4C1 = {1237/1246/1345}, no 8,9 g) 12(4) cage at R7C8 = {1236/1245}, no 7,8,9 1a. 13(4) cage at R4C1 = {1237/1246/1345}, 1 locked for N4 1b. 12(4) cage at R7C8 = {1236/1245}, 1,2 locked for N9 1c. R3C8 = 8 -> no 7 in R3C7 (ONC) 1d. R5C3 = 8 -> no 7 in R5C2, no 9 in R5C4 (ONC) 1e. 8,9 in C9 only in R456C9, locked for N6 1f. Hidden killer pair 8,9 for R45C9 and R6C9, R45C9 cannot be [89] (ONC) -> R45C9 must contain only one of 8,9 and R6C9 = {89} 1g. 18(3) cage at R3C3 = {279/369/459/567}, no 1 2a. 45 rule on N1 3 innies R123C3 = 14 = {167/257/347} (cannot be {356} which clashes with R23C2), 7 locked for C3 and N1 2b. 45 rule on C12345 2 innies R12C5 = 8 = {17/26/35} 2c. 45 rule on C12 2 innies R9C12 = 13 = {49/58}/[76] (ONC) 2d. R9C12 = 13 -> R89C3 = 8 = {26/35} 2e. 45 rule on N47 2 innies R67C3 = 6 = [24/42/51] 2f. R67C3 = 6 -> R67C4 = 9 = {18/27/36} (cannot be {45} which clashes with R67C3), no 4,5,9 2g. Killer pair 2,5 in R67C3 and R89C3, locked for C3 2h. Combined cage R67C3 + R89C3 = [2435/4253/5162] (ONC), no 2 in R8C3, no 6 in R9C3 2i. 45 rule on N4 3 innies R6C123 = 15 = {267/357/456} 2j. 2 of {267} must be in R6C3 -> no 2 in R6C12 2k. 5 of {357} must be in R6C3, {456} = [465/654] (ONC) -> no 5 in R6C1, no 4 in R6C2 2l. 45 rule on C67 2 innies R3C67 = 12 = [57/75/93] 2m. 18(3) cage at R3C3 (step 1g) = {279/369/459/567} 2n. 4 of {459} must be in R3C3 -> no 4 in R3C45 2o. 7 of {279} must be in R3C3, {567} = [657/765] (ONC) -> no 7 in R3C4 3a. 45 rule on N7 3 innies R7C123 = 14 -> max R7C12 = 13 3b. Hidden killer triple 7,8,9 in R7C12, R8C12 and R9C12 for N7, none of them can contain more than one of 7,8,9 -> each must contain one of 7,8,9 -> R8C12 = {19/28/37}, no 4,6 3c. R7C123 = {149/158/248/347} (cannot be {239/257/356} which clash with R89C3, cannot be {167} = [761] which clashes with R6C12), no 6 3d. R67C3 = 6 (step 2e), R6C123 (step 2i) = {267/357/456} = [762/375/735/465/654] -> R7C123 = {149/158/248} (cannot be {347} = {37}4 which clashes with R6C123 = [762]), no 3,7 3e. Consider placement for 6 in N7 R8C3 = 6 => R9C3 = 2 (step 2d) or R9C2 = 6 => R6C123 = [375/735/654] -> no 2 in R6C3 -> R67C3 + R89C3 (step 2h) = [4253/5162] 3f. R6C123 = {357/456}, 5 locked for R6 and N4, clean-up: no 8 in R7C5 3g. 2 in C3 only in R79C3, locked for N7, clean-up: no 8 in R8C12 3h. R7C123 = {149/158/248} 3i. 1 of {149/158} must be in R7C3 -> no 1 in R7C12 3j. R7C3 = {12} -> no 2 in R7C4 (ONC), clean-up: no 7 in R6C4 (step 2f) 3k. R8C3 = {56} -> no 6 in R8C4 (ONC) 3l. R9C3 = {23} -> no 3 in R9C4 (ONC) 4a. 45 rule on R123 4 innies R12C5 + R3C67 = 16, R12C5 = 8 (step 2b) -> R3C67 = 8 = {26/35}/[71], no 4,9, no 1 in R3C6 4b. R3C67 = 8 -> R4C67 = 11 = {47}/[65/83] (cannot be [56], ONC), no 1,2, no 3,5 in R4C6, no 6 in R4C7 4c. 18(3) cage at R3C3 (step 1g) = {279/369/459} (cannot be {567} which clashes with R3C67), 9 locked for R3 and N2 4d. 7 of {279} must be in R3C3 -> no 7 in R3C5 4e. 4 of {459} must be in R3C3 -> no 5 in R3C4 (ONC) 4f. 9 in R1 only in R1C12 -> no 8 in R1C1 (ONC) 4g. R23C2 = {16/25} (cannot be [43] because R23C2 = [43] + 18(3) cage = 7{29} clashes with R3C67), no 3,4 [With hindsight that was my key step. Looks like it’s now time for another forcing chain.] 4h. Consider combinations for R23C2 R23C2 = {16}, locked for N1 => R123C3 = {347} or R23C2 = {25}, locked for C2 => R6C123 (step 3d) = [375/735/465] => R6C3 = 5 => R789C3 (step 3e) = [162] -> R123C3 = {347}, 3,4 locked for C3 and N1, R6789C3 = [5162], clean-up: no 8 in R67C4 (step 2f), no 9 in R8C12 4i. Naked pair {37} in R8C12, locked for R8, 7 locked for N7 4j. R6C12 = {37} (cannot be [46] because 23(4) cage at R6C1 = [4685], ONC clashes with R23C2), locked for R6 and N4, clean-up: no 6 in R7C4 (step 2f), no 6 in R7C5 4k. 12(4) cage at R7C8 = {1236} (cannot be {1245} which clashes with R8C8), 3,6 locked for N9, 1 locked for C9 4l. R6C3 = 5 -> R6C4 = 2 (ONC), R7C4 = 7 (step 2f) 4m. 18(3) cage = {279/369/459} 4n. 3 of {369} must be in R3C3 -> no 3 in R3C45 4o. R6C4 = 2 -> no 1 in R5C4 (ONC) 5a. 45 rule on R89 4(2+2) innies R89C89 = 15 = [4173/5271] -> R9C8 = 7, R9C9 = {13} 5b. R89C9 = [13/21] -> R7C89 = {26/36} 5c. 45 rule on R67 4 (2+2) innies R67C89 = 19 = [19]{36} -> R6C89 = [19], R7C89 = {36}, locked for R7, 3 locked for N9 -> R89C9 = [21], R8C8 = 5, clean-up: no 4 in R7C5 [Cracked.] 5d. Naked triple R789C7 = {489}, 4 locked for C7 5e. R6C7 = 6, R7C6 = 2 (hidden single in N8) -> R6C6 + R7C7 = 12 = {48} 5f. Naked pair {48} in R6C56, locked for N5 5g. 45 rule on N5 2 remaining innies R45C6 = 15 = [69] -> R5C7 = 3 (step 2l), no 4 in R5C8 (ONC) 5h. R45C8 = [42], R4C9 = 8 (hidden single in N6) -> R5C9 = 7 (cage sum), R5C45 = [51], R4C45 = [37] 5i. R4C67 = [65] = 11 -> R3C67 = 8 = [71], no 6 in R3C5 (ONC), clean-up: no 6 in R2C2 5j. R12C7 = {27} = 9 -> R12C6 = 7 = [43] (ONC) -> R6C6 + R7C7 = [84], R89C6 = [15], R67C5 = [49], R8C45 = [48], R89C7 = [98], R9C45 = [63] 5k. R3C4 = 9 -> R3C35 = 9 = [45], R12C3 = [37], R12C4 = [81] (ONC), R12C5 = [26] (ONC) 5l. R2C7 = 2, R2C2 = 5 -> R3C2 = 2 5m. R7C12 = [58] -> R6C12 = [73] (ONC), R8C12 = [37] -> R9C12 = [94] (ONC) and the rest is naked singles without using ordered NC. Solution: 1 9 3 8 2 4 7 6 5 8 5 7 1 6 3 2 9 4 6 2 4 9 5 7 1 8 3 2 1 9 3 7 6 5 4 8 4 6 8 5 1 9 3 2 7 7 3 5 2 4 8 6 1 9 5 8 1 7 9 2 4 3 6 3 7 6 4 8 1 9 5 2 9 4 2 6 3 5 8 7 1 |
Author: | Andrew [ Sat Apr 11, 2020 2:24 am ] |
Post subject: | Re: Ordered NC 4, 4H and 5 |
I enjoyed Ordered NC Killer 4H; an excellent puzzle about moderate Assassin level. Same solution as Ordered NC Killer 4. I used: MinMax, which is something I rarely use for Assassins, and a short forcing chain to finally crack the puzzle. Here is my walkthrough for Ordered NC Killer 4H: NC increasing (ONC), 89 not allowed, 98 allowed. Prelims a) R9C89 = {18/27/36}/[54] (ONC), no 9 b) 26(4) cage at R1C1 = {2789/3689/4589/4679/5678}, no 1 c) 27(4) cage at R7C8 = {3789/4689/5679}, no 1,2 d) 14(4) cage at R8C6 = {1238/1247/1256/1346/2345}, no 9 1a. 45 rule on R12 2 innies R12C9 = 6 = {15/24} 1b. 45 rule on C89 using R12C9 = 6, 2 outies R12C7 = 6 = {15/24} 1c. Naked quad {1245} in R12C79, locked for N3 1d. R12C7 = 6 -> R12C8 = 9 = {36}, locked for C8 and N3, clean-up: no 3,6 in R9C9 1e. Naked triple {789} in R3C789, locked for R3 1f. 45 rule on N9 3 innies R789C7 = {135/234} (cannot be {126} which clashes with R12C7), 3 locked for N9 1g. Naked quint {12345} in R12789C7, locked for C7 1h. 27(4) cage at R7C8 = {4689/5679} 1i. R9C89 = {18/27} (cannot be [54] which clashes with 27(4) cage), no 4,5 1h. 6 in C7 only in R456C7, locked for N6 1i. Min R3C89 = 15 -> max R4C89 = 7, no 7,8,9 1j. Min R34C7 = 13 = [76] -> max R34C6 = 7 but cannot be {16}, no 6,7,8,9 2a. 45 rule of C6789 using R12C9 = 6, 3 outies R127C5 = 9 = {126/135/234} 2b. 45 rule of C5 using R127C5 = 9, 2 innies R89C5 = 11 = {29/38/47}/[65] (ONC), no 1 2c. R89C5 = 11 -> R89C4 = 9 = {18/27/36}/[54] (ONC), no 9 2d. 45 rule of C6789 using R12C9 = 6, 2 innies R12C6 = 1 outie R7C5 + 14 2e. Min R12C6 = 15 = {69/79}/[87/98] (ONC) 2f. Max R12C6 = 17 -> max R7C5 = 3 2g. R127C5 = {126} can only be [621] (because [261] clashes with R12C6 = [87], ONC), no 6 in R2C5 2h. R12C6 = {79}/[87/98] (cannot be {69} because R7C5 = 1, R12C5 = {35} and 23(4) cage at R1C5 = [3659/5936] (ONC) clashes with one of R12C8), no 6 2i. 25(4) cage at R3C5 = {1789/2689/3589/3679/4579/4678} 2j. 1 of {1789} must be in R3C5 -> no 1 in R456C5 2k. Max R7C57 = [35] = 8 -> min R7C6 = 5 (because {345} must be [354], ONC) 3a. 45 rule on R89 4(3+1) innies R8C789 + R9C1 = 28, max R8C789 = 24 -> min R9C1 = 4 3b. 45 rule on R89 4(2+2) innies R89C1 + R8C89 = 28, max R8C89 = 17 -> min R89C1 = 11, no 1 in R8C1 [I ought to have spotted this earlier; I never used it when solving the simpler version.] 4a. 45 rule on N89 1 innie R8C4 = 8, clean-up: no 1 in R89C4 (step 2c), no 3 in R89C5 (step 2b) 4b. R8C4 = 8 = R7C34 + R8C3 = 10 (127/136/145/235}, no 9, no 7 in R7C4 + R8C3 (ONC) 4c. 8 in N2 only in R12C6 = [87/98] (ONC) = 15,17 -> R7C5 = {13} (step 2d) 4d. 12(3) cage at R7C5 = {129/147/156/237}/[354] (cannot be {246} because R7C5 only contains 1,3) 4e. R7C5 = {13} -> no 1,3 in R7C7 4f. 3 in N9 only in R89C7, locked for 14(4) cage at R8C6 4g. R89C4 + R89C5 cannot be {27}[65] which clashes with R127C5 = {35}1/[621] and with 12(3) cage at R7C5 = [354/372] -> no 2,7 in R89C4 4h. R89C5 = {29/47} (cannot be [65] which clashes with R89C4), no 5,6 4i. 12(3) cage = {129/147/156/237} (cannot be [354] which clashes with R89C4) 4j. 14(4) cage at R8C6 contains 3 = {1346/2345}, no 7 4k. 1 of {1346} must be in R89C6 (R89C6 cannot be {46} which clashes with R89C4), no 1 in R89C7 4l. 1 in C7 only in R12C7 (step 1b) = {15}, locked for N3, 5 locked for C7 4m. Naked pair {24} in R12C9, locked for C9 4n. Naked triple {234} in R789C7, 2,4 locked for N9, clean-up: no 7 in R9C89 4o. Naked pair {18} in R9C89, locked for R9, 8 locked for N9 4p. 12(3) cage = {129/147/237} (cannot be {156} because 5,6 only in R7C6) -> R7C6 = {79} 4q. Killer pair 7,9 in R12C6 and R7C6, locked for C6 4r. R12C6 + R7C5 (step 2d) = [871/983] -> 12(3) cage = [192/372] (cannot be [174] which clashes with R12C6 + R7C5) -> R7C7 = 2, R89C7 = [43] (ONC), no 2 in R9C6 (ONC) -> R89C6 = 7 = [16/25], clean-up: no 6 in R8C4 (step 2c), no 7 in R9C5 (step 2b), no 5 in R8C7 (ONC) 4s. 7 in R9 only in R9C123, locked for N7 4t. R127C5 (step 2a) = {126/135} (cannot be {234} which clashes with R89C5), no 4, 1 locked for C5 4u. 2 of {126} must be in R2C5 -> no 2 in R1C5 4v. 4,6 in N8 only in R9C456, locked for R9 5a. 45 rule on C34 using R89C4 = 9 (step 2c) 4(2+2) innies R5C34 + R89C3 = 27, max R5C34 = 16 (cannot be [89], ONC) -> min R89C3 = 11, no 1 5b. Min R89C3 = 11 -> max R89C2 = 8, no 8,9 5c. Min R89C2 = 5 (cannot be [12], ONC) -> max R89C3 = 14 -> min R5C34 = 13, no 1,2,3 6a. R89C1 + R8C89 = 28 (step 3b), min R8C89 = {79} = 16 -> max R89C1 = 12, no 2 6b. 2 in N7 only in 19(4) cage at R8C2 = {1279/2359} (cannot be {2368} because 3,6,8 only in R8C23), no 6,8, 9 locked for C3 and N7 6c. R8C1 = 8 (hidden single for N7) 6d. R8C9 = 6 (hidden single in R8) -> no 5 in R7C9 (ONC) 6e. R7C8 = 5 (hidden single in N9) -> no 4 in R6C8 (ONC) [With hindsight, if I’d spotted step 9a next, step 8 and particularly 8a would have been simpler.] 7a. Variable hidden killer pair 4,6 in R7C12 and R7C3 for R7 -> R7C12 must contain at least one of 4,6 7b. 15(4) cage at R6C1 = {1248/1347/2346} (cannot be {1239/1257/1356} which don’t contain 4 or 6), no 5,9 8a. 22(4) cage at R3C8 = [7915/8725/8743] (cannot be {79}{24} because 2,4 only in R4C8, cannot be [9823] (ONC), cannot be [9715] because R89C8 cannot be [78] (ONC), cannot be [9841] which clashes with R9C9), no 7 in R3C7, no 9 in R3C8, no 8 in R3C9, no 1 in R4C9 8b. 7 in C7 only in R456C7, locked for N7 8c. R3C89 = [79] or R3C89 = [87] => R7C9 = 9 -> no 9 in R56C9 8d. 17(4) cage at R5C8 = {1259/1349/2348} (cannot be {1358} which clashes with R4C9) 8e. 1,3,5 of {1259/1349} must be in R56C9, 3,8 of {2348} must be in R56C9 -> no 1,8 in R56C8 8f. {2348} must be [4328] (ONC) -> no 8 in R5C9 8g. Min R34C6 = 4 (cannot be [21] which clashes with R8C6) -> max R34C7 = 16, no 8,9 in R4C7 8h. Min R34C7 = 14 -> max R34C6 = 6 = {13/14/24}/[32] (ONC) (cannot be {15} which clashes with R89C6), no 5 8i. Killer pair 1,2 in R34C6 and R8C6, locked for C6 9a. Consider combinations for 19(4) cage at R8C2 (step 6b) = {1279/2359} 19(4) cage = {1279} => R9C1 = 5 (hidden single in N7) or 19(4) cage = {2359}, 3 locked for R8 => R8C4 = 5 -> R9C6 = 6, R8C6 = 1 (cage sum), R7C5 = 3 -> R7C6 = 7 (cage sum) 9b. R12C6 = [98] = 17 -> R12C5 = 6 = {15}, locked for N2, 5 locked for C5 9c. R7C9 = 9, R8C8 = 7, R3C789 = [987], R9C89 = [18] 9d. R89C4 = [54] 9e. Naked pair {29} in R89C5, locked for C5 9f. 4 in N2 only in R3C56, 4 locked for R3 9g. 7 in N2 only in R12C4, locked for C4 and 20(4) cage at R1C3 9h. 19(4) cage = {2359}, 5 locked for N7 -> R9C1 = 7 9i. 1,5 in R3 only in R3C123, locked for N1 9j. 15(4) cage at R6C1 (step 7b) = {1248/1347/2346}, R7C12 = {14/46} -> R6C12 = [28/37/32] (ONC) -> R6C1 = {23}, R6C2 = {278} 10a. R5C34 + R89C3 (step 5a) = 27 10b. R5C4 = {69} -> R589C3 must contain 9 in R89C3 = 18,21 = {279/369/459/579} 10c. 4,6,7 only in R5C3 -> R5C3 = {467} 11a. 18(4) cage at R6C3 = {136}8/{145}8 (cannot be {235}8 because R7C3 only contains 1,4,6, cannot be [7218] which clashes with R6C12), no 2,7 11b. 18(4) cage = {13}[68]/[5148] (cannot be {36}[18] which clashes with 15(4) cage at R6C1), 1 locked for R6, no 1 in R7C3 11c. 1 in R7 only in 15(4) cage at R6C1 (step 7b) = {1248/1347} -> R7C12 = {14}, R7C3 = 6, R6C34 = {13}, 3 locked for R6, R6C12 = [28], R6C89 = [95] = 14 -> R5C89 = 3 = [21], R4C89 = [43] 11d. R6C6 = 4, R5C6 = 5 (hidden single in C6) -> R56C7 = 15 = [87] -> R3456C5 = [4876], 20(4) cage at R3C6 = [3926], R456C4 = [193], R56C3 = [41] 11e. R4C3 = 7 (hidden single in C3), R4C4 = 1 -> R3C34 = 8 = [26], R2C3 = 6, R2C8 = 6 -> R2C7 = 1 (ONC), R1C8 = 3 -> R1C9 = 2 (ONC) and the rest is naked singles without using ordered NC. |
Author: | Andrew [ Sun Apr 12, 2020 7:24 pm ] |
Post subject: | Re: Ordered NC 4, 4H and 5 |
Ordered NC Killer 5 was my least favourite of these four puzzles. After an easy start which gave quite a lot of placements: and reduced some of the square cages to single combinations, I had to use heavy interactions between the 23(4) cage and each of the 15(4) cages in R12. Here is my walkthrough for Ordered NC Killer 5: NC increasing (ONC), 89 not allowed, 98 allowed. Prelims a) R23C1 = {19/28/37/46}, no 5 b) R45C1 = {17/26/35}, no 4,8,9 c) R67C1 = {15/24} d) R9C34 = {18/27/36}/[54] (ONC), no 9 e) R9C56 = {69/78} f) R9C78 = {13} g) 14(4) cage at R7C6 = {1238/1247/1256/1346/2345}, no 9 h) 27(4) cage at R7C8 = {3789/4689/5679}, no 1,2 1a. Naked pair {13} in R9C78, locked for R9 and N9, clean-up: no 6,8 in R9C34 1b. 27(4) cage at R7C8 = {4689/5679}, 6,9 locked for N9 2a. 45 rule on C1 3 innies R189C1 = 21 = {489/579/678} 2b. Hidden killer triple 1,2,3 in R23C1, R45C1 and R67C1 for C1, R45C1 and R67C1 each contains one of 1,2,3 -> R23C1 must contain one of 1,2,3 = {19/28/37}, no 4,6 2c. Killer triple 7,8,9 in R189C1 and R23C1, locked for C1, clean-up: no 1 in R45C1 2d. Killer pair 2,5 in R45C1 and R67C1, locked for C1, clean-up: no 8 in R23C1 2e. 45 rule on R123456 2 innies R16C1 = 8 = [62/71], clean-up: no 1,2 in R7C1 2f. R189C1 = {678} (only remaining combination) 2g. R7C1 = 4 (hidden single in C1) -> R6C1 = 2 -> R1C1 = 6 2h. Naked pair {35} in R45C1, locked for N4, 3 locked for C1 2i. R89C1 = [87] (ONC), clean-up: no 2 in R9C34, no 8 in R9C56 2j. Naked pair {19} in R23C1, locked for N1 2k. R9C34 = [54] 2l. R9C29 = [28] (hidden pair in R9) 2m. 27(4) cage at R7C8 = {5679}, 5,7 locked for N9 2n. 45 rule on C45 1 remaining innie R9C5 = 9 -> R9C6 = 6 2o. 45 rule on C67 1 remaining innie R9C7 = 3 -> R9C8 = 1 2p. R7C78 = [24] = 6, no 3,5 in R8C6 (ONC) -> R78C6 = 8 = [71] (ONC) 2q. 45 rule on N1 2 remaining innies R3C23 = 7 = [43/52] (ONC) -> R4C23 = 16 = {79}, locked for R4 and N4 ONC clean-ups: no 7 in R1C2, no 2 in R2C3, no 3 in R3C4, no 1 in R6C7, no 9 in R8C2, no 3 in R8C4, no 5 in R8C8, no 7 in R8C9 2r. R8C8 = 7 (hidden single in N9), no 6 in R7C8 (ONC) 2s. R7C8 + R78C9 = [596/965], 9 locked for R7, 6 locked for C9 2t. R8C3 = 9 (hidden single in N7) -> R4C23 = [97], no 8 in R4C4, no 8 in R5C3 (ONC) 2u. 45 rule on R12 1 remaining innie R2C1 = 9 -> R3C1 = 1 2v. 45 rule on R34 1 remaining innies R4C1 = 5 -> R5C1 = 3, no 4 in R5C2 (ONC) 3a. R2C2 = 7 (hidden single in N1) -> no 8 in R2C3 (ONC) 3b. R3C23 (step 2q) = [52] (cannot be [43] which clashes with R2C3) 3c. 8 in N1 only in R1C23, locked for R1 3d. R8C45 = {25}/[53] (cannot be [23], ONC), 5 locked for R8 and N8 -> R8C9 = 6, R7C89 = [59] (ONC), R8C2 = 3, no 4 in R6C8 (ONC) 3e. R1C23 + R2C3 = {48}3 (cannot be [834] ONC) -> R1C23 = {48}, 4 locked for R1, R2C3 = 3 4a. 1 in N2 only in 15(4) cage at R1C4 -> 15(4) cage at R1C4 = {1239/1248/1257/1347/1356} 4b. 23(4) cage at R1C6 = {1589/2489/2579/2678/3578} (cannot be {1679} because 1,6,7 on in R12C7, cannot be {3479} because R2C7 only contains 1,5,6,8, cannot be {3569} = [3956] ONC, cannot be {4568} because 4,6,8 only in R2C67) 4c. 5 in N3 only in R12C7 and 15(4) cage at R1C8 4d. 23(4) cage = {1589/2489/2579/3578} (cannot be {2678} = [2786] which clashes with 15(4) cage in R1C8 = {1257/1356}), no 6 4e. 2 in N2 only in 15(4) cage at R1C4 and R12C6 4f. 23(4) cage = {1589/2489/2579} (cannot be {3578} = [3785] (not [3758] ONC) which clashes with 15(4) cage in R1C4 = {1239/1248/1257}), no 3, 9 locked for R1 4g. 23(4) cage = {1589/2579} (cannot be {2489} = [2948] + R1C8 = 3 clashes with 15(4) cage at R1C4), no 4 4h. {2579} must be [9725] -> no 2 in R1C6 4i. 15(4) cage at R1C4 = {1257/1356} (cannot be {1248} which clashes with 23(4) cage, cannot be {1347} which clashes with [9725] and because 15(4) cage must contain 2 when 23(4) cage = {1589}), no 4,8, 5 locked for N2 -> R1C6 = 9 4j. 23(4) cage = {1589/2579} = [9581/9725] (cannot be [9185] which clashes with 15(4) cage at R1C8 = {2346} = [3264] ONC) -> R1C7 = {57}, R2C7 = {15} 4k. R1C8 = {23} -> no 3 in R1C9 (ONC) 4l. 15(4) cage at R1C8 = {1248/2346} (cannot be {1257/1356} which clash with R2C7, cannot be {1347} = [3741] ONC), no 5,7 4m. 6,8 of {1248} only in R2C8 -> R2C8 = {68}, R2C9 = 4, R1C89 = [21/32], 2 locked for R1 4n. 5 in C6 only in R56C6, locked for N5 and 24(4) cage at R5C6 5a. 5 in N6 only in 17(4) cage at R5C8 = {1259/1358/2456} (cannot be {1457} because 1,5,7 only in R56C9, cannot be {2357} = [2537/2735] ONC), no 7 5b. 7 in N6 only in R56C7, locked for C7 -> R12C7 = [51], R2C6 = 8 (cage sum), R1C89 = [32], R2C8 = 6, R3C49 = [67] 5c. 17(4) cage = {1259} (cannot be {1358} which clashes with R4C9, cannot be {2456} because 2,4 only in R5C8) -> R56C8 = [29] 5d. R2C89 + R3C8 = [874] = 19 -> R3C9 = 3 (cage sum) 5e. R3C7 + R4C6 = [92] = 11 -> R3C6 + R4C7 = 9 = [36] and the rest is naked singles with using ordered NC. Solution: 6 4 8 7 1 9 5 3 2 9 7 3 2 5 8 1 6 4 1 5 2 6 4 3 9 8 7 5 9 7 1 8 2 6 4 3 3 6 4 9 7 5 8 2 1 2 8 1 3 6 4 7 9 5 4 1 6 8 3 7 2 5 9 8 3 9 5 2 1 4 7 6 7 2 5 4 9 6 3 1 8 |
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