Prelims
a) 10(3) cage at R1C1 = {127/136/145/235}, no 8,9
b) 19(3) cage at R2C2 = {289/379/469/478/568}, no 1
c) 10(3) cage at R2C6 = {127/136/145/235}, no 8,9
d) 9(3) cage at R3C1 = {126/135/234}, no 7,8,9
e) 23(3) cage at R3C3 = {689}
f) 19(3) cage at R4C4 = {289/379/469/478/568}, no 1
g) 20(3) cage at R4C6 = {389/479/569/578}, no 1,2
h) 10(3) cage at R6C4 = {127/136/145/235}, no 8,9
i) 11(3) cage at R6C6 = {128/137/146/236/245}, no 9
j) 13(4) cage at R3C8 = {1237/1246/1345}, no 8,9
1a. 45 rule on R89 2 innies R8C27 = 6 = {15/24}
1b. 45 rule on C89 2 innies R27C8 = 12 = {39/48/57}, no 1,2,6
1c. Max R7C8 + R8C7 = 14 -> min R7C7 = 3
1d. 45 rule on N9 2 innies R7C9 + R9C7 = 8 = {17/26/35}, no 4,8,9
1e. 45 rule on N5 3 innies R4C6 + R6C46 = 11 = {128/137/146/236/245}, no 9
1f. Min R4C6 + R6C4 = 4 -> max R6C6 = 7
1g. 45 rule on R6789 1 outie R5C8 = 1 innie R6C5 + 3 -> R5C8 = {456789}, R6C5 = {123456}
2a. 45 rule on C1 4 innies R6789C1 = 1 outie R1C2 + 26
2b. Max R6789C1 = 30 -> max R1C2 = 4
2c. Min R6789C1 = 27, no 1,2
2d. 8,9 in C1 only in R6789C1 -> no 8,9 in R7C2 (CPE)
3a. 45 rule on N47 2 innies R48C3 = 1 outie R3C1 + 2
3b. Max R3C1 = 6 -> max R48C3 = 8 -> R4C3 = 6
[Ed and wellbeback both used 45 rule on R123 4 innies R3C1348 = 28 = {4789/5689} with 8,9 only in R3C34 -> R4C3 = 6.]
3c. R3C1 = R8C3 + 4 -> R3C1 + R8C3 = [51/62]
3d. Naked pair {89} in R3C45, locked for R3
3e. 19(3) cage at R2C2 = {379/469/478/568} (cannot be {289} which clashes with R3C3), no 2
3f. 45 rule on C123 1 outie R3C4 = 2 innies R18C3 + 5
3g. R3C4 = {89} -> R18C3 = 3,4 = {12/13}, 1 locked for C3
3h. Killer pair 1,2 in R8C27 and R8C3, locked for R8
3i. 9(3) cage at R3C1 = {126/135} (cannot be {234} because R3C1 only contains 5,6), no 4
3j. R3C1 = {56} -> R45C1 = {12/13}, 1 locked for C1 and N4
3k. 10(3) cage at R1C1 = {127/136/145/235}
3l. 1 of {145} must be in R1C2 -> no 4 in R1C2
3m. 19(3) cage = {379/469/478} (cannot be {568} which clashes with R3C1), no 5
3n. 1 in N1 only in R1C23, locked for R1
3o. 5 in N1 only in R123C1, locked for C1
3p. 45 rule on R123 2 remaining innies R3C18 = 11 = {56}, locked for R3
3q. 13(4) cage at R3C8 = {1246/1345} (cannot be {1237} because R3C8 only contains 5,6), no 7, 1,4 locked for N6, clean-up: no 1 in R6C5 (step 1g)
3r. R3C8 = {56} -> no 5,6 in R4C89 + R5C9
3s. 1 in R6 only in R6C46, locked for N5
3t. R4C6 + R6C46 (step 1e) = {128/137/146}, no 5
3u. 4 of {146} must be in R4C6 -> no 4 in R6C46
3v. 20(3) cage at R4C6 = {389/479/578} (cannot be {569} because no 5,6,9 in R4C6), no 6
4a. Consider permutations for R3C1 + R8C3 (step 3c) = [51/62]
R3C1 + R8C3 = [51] => R45C1 = 4 = {13}, 3 locked for C1, R1C2 = 1 (hidden single in N1) => R12C1 = 9 = {27}
or R3C1 + R8C3 = [62] => R1C3 = 1, 10(3) cage at R1C1 = {235}
-> 10(3) cage = {127/235}, no 4,6, 2 locked for N1
4b. 4 in N1 only in 19(3) cage at R2C2 = {469/478}, no 3
4c. 6 of {469} must be in R2C2 -> no 9 in R2C2
4d. 9 in N1 only in R23C3, locked for C3
4e. 4 in C1 only in R6789C1, CPE no 4 in R78C2, clean-up: no 2 in R8C7 (step 1a)
4f. 2 in R8 only in R8C23, locked for N7
4g. 45 rule on N7 1 outie R6C1 = 2 innies R78C3
4h. Min R78C3 = 5 (cannot be [31] which clashes with R1C3) -> no 3,4 in R6C1
4i. R6C1 = {789}, R8C3 = {12} -> R7C3 = {578}
4j. 4 in C1 only in R789C1, locked for N7
4k. R6789C1 = R1C2 + 26 (step 2a) -> R6789C1 = {4689/4789}, no 3, R1C2 = {12}
4l. 18(3) cage at R6C2 = {279/378/459}
4m. 5 of {459} must be in R7C3 -> no 5 in R6C23
4n. 5 in N4 only in 16(3) cage at R4C2
4o. Consider combinations for 10(3) cage (step 4a)
10(3) cage = {127}, 2,7 locked for C1, R45C1 = {13}, 3 locked for N4, R6C1 = {89} => no 5 in R7C3 (step 4i) => 18(3) cage = {279}, 2 locked for N4
or 10(3) cage = {235}, 3 locked for C1, R45C1 = {12}, 2 locked for N4
-> 16(3) cage = {358/457}, no 2,9
4p. 9 in N4 only in R6C12, locked for R6
4q. 18(3) cage = {279/378/459}
4r. {378} = {38}7 (cannot be {37}8 which clashes with 16(3) cage) -> no 7 in R6C23, no 8 in R7C3
4s. 9 of {459} must be in R6C2 -> no 4 in R6C2
5a. 17(3) cage at R7C7 = {179/359/458/467} (cannot be {368} because R8C7 only contains 1,4,5)
5b. Consider placements for R7C3 = {57}
R7C3 = 5, naked pair {12} in R8C23, locked for R8 => 17(3) cage = {359/458/467}
or R7C3 = 7 => 17(3) cage = {359/458}
-> 17(3) cage = {359/458/467}, no 1, clean-up: no 5 in R8C2 (step 1a)
5c. Naked pair {12} in R8C23, 1 locked for N7
5d. Naked pair {12} in R18C2, locked for C2
[Taking that forcing chain a bit further …]
5e. R6C1 = {789} -> R78C3 = [52/71/72] (steps 4e and 4g), R8C27 = [15/24] (step 1a)
5f. R7C3 = 5 => R8C3 = 2, R8C27 = [15]
or R7C3 = 7
-> 17(3) cage = {359/458}, no 6,7, 5 locked for N9, clean-up: no 3 in R7C9 + R9C7 (step 1d)
6a. 45 rule on N3 2 innies R3C78 = 1 outie R1C6 + 2
6b. Min R3C78 = 6 -> min R1C6 = 4
6c. R3C78 cannot total 11 -> no 9 in R1C6
6d. Max R1C6 = 8 -> max R3C78 = 10, no 7 in R3C7
7a. Consider placements for 5 in R3C18
R3C1 = 5 => R45C1 = 4 = {13} => R6C3 = 2 (hidden single in N4)
or R3C8 = 5 => R4C89 + R5C9 = 8 = {134} => 2 in N6 only in R6C789
-> 2 in R6C3 or R6C789, locked for R6, clean-up: no 5 in R5C8 (step 1g)
7b. R4C6 + R6C46 (step 3t) = {137/146}, no 8
7c. 20(3) cage at R4C6 (step 3v) = {389/479/578}
7d. 3 of {389} must be in R4C6 -> no 3 in R45C7
8a. R6C1 = R78C3 (step 4g), R8C3 = {12}
8b. Consider placements for R7C3 = {57}
R7C3 = 5 => R6C1 = 7
or R7C3 = 7
-> 7 in R6C1 or R7C3, CPE no 7 in R5C3 + R789C1 + R7C2
8c. 10(3) cage (step 4a) = {127/235}, 18(3) cage at R6C2 (step 4q) = {279/378/459}
8d. Consider placements for 7 in C1
7 in 10(3) cage = {127}, 2 locked for C1 => R45C1 = {13}, 3 locked for N4
or R6C1 = 7 => R7C3 = 5 => 18(3) cage = {459}
-> 18(3) cage = {279/459} -> R6C2 = 9, R6C3 = {24}
9a. R5C8 = R6C5 + 3 (step 1g)
9b. 20(3) cage at R4C6 (step 3v) = {389/479/578}
9c. Consider placement for 9 in N6
9 in R45C7 => 20(3) cage = {389/479}
or R5C8 = 9 => R6C5 = 6 => 5 in R6 only in R6C789, locked for N6 => 20(3) cage = {389/479}
-> 20(3) cage = {389/479}, no 5, 9 locked for C7 and N6, clean-up: no 6 in R6C5
9d. 20(3) cage = {389/479} -> R4C6 = {34}
9e. 5 in N6 only in R6C789, locked for R6, clean-up: no 8 in R5C8
9f. Naked pair {34} in R4C6 + R6C5, locked for N5
9g. 19(3) cage at R4C4 = {289/568}, no 7, 8 locked for N5
9h. 6 of {568} must be in R5C4 -> no 5 in R5C4
10a. 17(3) cage at R7C7 (step 5f) = {359/458}
10b. 9 of {359} must be in R7C8 -> no 3 in R7C8
10c. R6C1 = R78C3 (step 4g), R8C27 (step 1a) = [15/24]
10d. Consider combinations for R4C6 + R6C46 (step 7b) = {137/146}
R4C6 + R6C46 = {137}, 7 locked for R6, R6C1 = 8 => R78C3 = 8 = [71] => R8C27 = [24] => 17(3) cage = {58}4
or R4C6 + R6C46 = {146} = [461] => R7C45 = 4 = {13}, 3 locked for R7=> 17(3) cage = {458}
or R4C6 + R6C46 = {146} = [416] => R6C7 + R7C6 = 5 = {23} (because no 1,4 in R6C7), CPE no 3 in R7C7 => 17(3) cage = {458}
-> 17(3) cage = {458}, 4,8 locked for N9, 8 locked for R7
10e. R7C1 = 9 (hidden single in R7)
10f. R6C1 + R78C2 = 13 = {238} (only possible combination, cannot be {157} = [751] which clashes with R6C1 + R78C3 = [752], cannot be {256} because R6C1 only contains 7,8) -> R6C1 = 8. R78C2 = [32], R78C3 = [71], R8C7 = 4, R89C1 = [64], R3C1 = 5 -> R45C1 = 4 = {13}, 3 locked for C1 and N4, R6C3 = 2 (cage sum)
11a. R2C2 = 6 (hidden single in N1) -> R2C3 + R3C2 = 13 = [94], R3C34 = [89]
11b. R3C8 = 6 -> R5C8 = 7, R5C2 = 5
11c. R6C5 = 4 (hidden single in R6) -> R5C56 = 11 = {29}, locked for R5 and N5
11d. R4C6 = 3 -> R45C1 = [13], R5C9 = 1 (hidden single in N6)
11e. R7C9 + R9C7 = 8 (step 1d) = {26}, locked for N9
11f. R45C7 = [98] -> R7C78 = [58], R2C8 = 4 (step 1b)
11g. 10(3) cage at R6C4 = {127} (only remaining combination) -> R6C4 = 7, R7C45 = {12}, locked for R7 and N8, R7C9 = 6 -> R9C7 = 2
11h. R7C6 = 4 -> R6C67 = 7 = [16]
11i. R9C7 = 2 -> R8C56 + R9C6 = 23 -> R9C6 = 6, R8C56 = {89}, locked for N8, 9 locked for R8, R8C89 = [37], R8C4 = 5, R9C45 = [37], R4C45 = [85]
11j. R1C3 = 3, R1C7 = 7, R12C1 = [27]
11k. R3C6 = 7 (hidden single in N2) -> R2C6 + R3C7 = 3 = [21]
and the rest is naked singles.