Prelims
a) R2C12 = {14/23}
b) R34C1 = {29/38/47/56}, no 1
c) R67C9 = {18/27/36/45}, no 9
d) R8C89 = {29/38/47/56}, no 1
e) 11(3) cage at R3C4 = {128/137/146/236/245}, no 9
f) 20(3) cage at R6C6 = {389/479/569/578}, no 1,2
g) 27(4) cage at R2C5 = {3789/4689/5679}, no 1,2
h) 26(4) cage at R2C8 = {2789/3689/4589/4679/5678}, no 1
i) 14(4) cage at R6C2 = {1238/1247/1256/1346/2345}, no 9
j) 18(5) cage at R4C7 = {12348/12357/12456}, no 9
1a. 45 rule on R1 2 outies R2C46 = 7 = {16/25} (cannot be {34} which clashes with R2C12)
1b. Killer pair 1,2 in R2C12 and R2C46, locked for R2
2a. 45 rule on R1234 2 innies R4C57 = 7 = {16/25/34}, no 7,8,9
2b. 45 rule on R6789 2 innies R6C35 = 6 = {15/24}
2c. 45 rule on N7 2 outies R6C12 = 9 = {18/27/36} (cannot be {45} which clashes with R6C35), no 4,5,9
2d. 45 rule on N9 2 outies R6C89 = 9 = {18/27/36} (cannot be {45} which clashes with R6C35), no 4,5,9, clean-up: no 4,5 in R7C9
2e. Killer triple 1,2,3 in R6C12, R6C35 and R6C89, locked for R6
[Unnecessary, in view of step 2f, but step 2e follows logically from steps 2b, 2c and 2d so I’ll keep it.]
2f. 45 rule on N8 3 outies R6C467 = 21 = {489/579} (cannot be {678} which clashes with R6C12), no 6
2g. 45 rule on N8 1 outie R6C4 = 1 innie R7C6 + 1, R6C4 = {45789} -> R7C6 = {34678}, no 5,9
2h. Max R46C5 = 11 -> min R6C5 = 2
3a. 45 rule on C123 1 outie R5C4 = 1 innie R4C3 + 1, no 1 in R5C4
3b. 45 rule on C789 1 innie R6C7 = 1 outie R5C6 + 3, R6C7 = {45789} -> R5C6 = {12456}
[The first interesting steps …]
4a. R4C7 ‘sees’ both R4C5 and R5C6, R4C57 (step 2a) = 7 -> R5C6 + R4C7 cannot total 7 (combination crossover clash, CCC) -> R5C789 cannot total 11
Note. This doesn’t eliminate 8 from R5C789, which can still total more than 11
4b. R6C5 ‘sees’ both R5C4 and R6C3, R6C35 (step 2b) = 6 -> R5C4 + R6C3 cannot total 6 (CCC) -> R5C123 cannot total 21
5a. 45 rule on N1 2 outies R4C12 = 10 = {28/37/46}/[91], no 5, no 9 in R4C2, clean-up: no 6 in R3C1
5b. 45 rule on N3 2 outies R4C89 = 15 = {69/78}
5c. Min R2C7 + R4C8 = 9 -> max R3C78 = 9, no 9 in R3C78
6a. R6C35 (step 2b) = {15/24}, R6C7 = R5C6 + 3 (step 3b)
6b. Consider position of 1 in N6
1 in R4C7 + R5C789, locked for 18(4) cage at R4C7, no 1 in R5C6
or 1 in R6C89, locked for R6 => R6C35 = {24}, 4 locked for R6, no 4 in R6C7 => no 1 in R5C6
-> no 1 in R5C6, clean-up: no 4 in R6C7
6c. 18(5) cage at R4C7 = {12348/12357/12456}, 1 locked for N6, clean-up: no 8 in R6C89 (step 2d), clean-up: no 1,8 in R7C9
6d. Killer pair 6,7 in R4C89 and R6C89, locked for N6, clean-up: no 1 in R4C5 (step 2a), no 4 in R5C6
6e. 18(5) cage at R4C7 = {12348/12456}
6f. 6 of {12456} must be in R5C6 -> no 5 in R5C6, clean-up: no 8 in R6C7
6g. 45 rule on C789 3 outies R567C6 = 17 = {269/278/368/467} (cannot be {359/458} because R5C6 only contains 2,6), no 5
7a. 18(5) cage at R4C7 (step 6e) = {12348/12456}, R4C89 = 15 (step 5b), R6C89 = 9 (step 2d), R6C7 = R5C6 + 3 (step 3b)
7b. Consider combinations for R6C467 (step 2f) = {489/579}
R6C467 = {489} => R6C7 = 9 => R5C6 = 6 => 18(5) cage = {12456}
or R6C467 = {579}, 7 locked for R6 => R6C89 = {36}, 3 locked for N6 => 18(5) cage = {12456}
-> 18(5) cage = {12456}, no 3,8 -> R5C6 = 6, R6C7 = 9, clean-up: no 1 in R2C4 (step 1a), no 4 in R4C5, no 1 in R4C7 (both step 2a), no 6 in R4C89, no 5 in R4C3 (step 3a), no 7 in R6C4, no 8 in R7C6 (both step 2g)
7c. Naked pair {78} in R4C89, locked for R4, 7 locked for N6, clean-up: no 2,3 in R4C12 (step 5a), no 3,4,8,9 in R3C1, no 8,9 in R5C4 (step 3a), no 2 in R6C89, no 2,7 in R7C9
7d. Naked pair {78} in R4C89, CPE no 7,8 in R2C8
7e. Naked pair {36} in R6C89, locked for R6
7f. Naked pair {36} in R67C9, locked for C9, clean-up: no 5,8 in R8C8
7g. 1 in N6 only in R5C789, locked for R5
7h. 13(3) cage at R4C5, without 6, must contain one of 7,8,9 -> R5C5 = {789}
8a. 45 rule on N2 1 outie R4C6 = 1 innie R3C4 + 2, R4C6 = {3459} -> R3C4 = {1237}
8b. 11(3) cage at R3C4 = {137/146/236} (cannot be {245} = [245] which clashes with R4C57), no 5
8c. 6 of {146/236} must be in R4C3 -> no 2,4 in R4C3, clean-up: no 3,5 in R5C4 (step 3a)
8d. 45 rule on C123 3 outies R345C4 = 12 = {147/237}, 7 locked for C4
8e. 3 in N5 only in R4C456, locked for R4, clean-up: no 4 in R5C4 (step 3a)
8f. 9 in N5 only in R4C6 or 13(3) cage at R4C5 = [391] -> no 3 in R4C6 (locking-out cages), clean-up: no 1 in R3C4
[With hindsight I missed 9 in R4C6 + R5C5, CPE no 9 in R23C5]
8g. 11(3) cage = {137/236} (cannot be {146} because R3C4 only contains 2,3,7), no 4
8h. R345C4 = {237}, 2,3 locked for C4, clean-up: no 5 in R2C6 (step 1a)
8i. 1 in R4 only in R4C23, locked for N4, clean-up: no 8 in R6C12 (step 2d), no 5 in R6C5 (step 2b)
8j. 1 in R4 only in R4C23, CPE no 1 in R3C3
8k. Naked pair {27} in R6C12, locked for R6 and N4, clean-up: no 4 in R6C35 (step 2b)
8l. R6C35 = [51] -> R45C5 = 12 = [39/57], clean-up: no 5 in R4C7 (step 2a), no 4 in R7C6 (step 2g)
8m. 5 in R4 only in R4C56, CPE no 5 in R23C5
8n. Naked pair {48} in R6C46, 4 locked for N5, clean-up: no 2 in R3C4
8o. Naked pair {27} in R6C12, CPE no 2,7 in R8C2
8p. 45 rule on R9 2 outies R8C46 = 7 = [43/52/61]
9a. R4C12 (step 5a) = 10
9b. Consider permutations for 11(3) cage at R3C4 (step 8g) = {137/236} = [362/713]
11(3) cage = [362] => R4C12 = [91]
or 11(3) cage = [713] => R4C12 = [64]
-> R4C12 = [64/91], clean-up: no 7 in R3C1
9c. Max R4C2 = 4 -> min R2C3 + R3C23 = 19, no 1 in R3C2
9d. 1 in R3 only R3C78, locked for N3
9e. 18(4) cage at R2C7 contains 1 = {1278/1368/1458/1467}
9f. 1,2 of {1278} must be in R3C78, 7,8 of {1368/1458/1467} must be in R4C8 -> no 7,8 in R3C78
10a. 27(4) cage at R2C5 = {4689/5679} (cannot be {3789} = {378}9 which clashes with R3C3), no 3, 6 locked for C5 and N2, clean-up: no 1 in R2C6 (step 1a)
10b. R2C46 = [52], clean-up: no 3 in R2C12
10c. Naked pair {14} in R2C12, locked for N1, 4 locked for R2
10d. R2C46 = 7 -> R1C456 = 13 containing 1 for N2 = {139/148}, no 7
10e. 16(3) cage at R1C1 = {268/367} (cannot be {259} which clashes with R3C1, cannot be {358} which clashes with R1C456), no 5,9, 6 locked for R1 and N1
10f. 5 in R1 only in 16(3) cage at R1C7, locked for N3
10g. 16(3) cage at R1C7 = {259/457} (cannot be {358} which clashes with 16(3) cage at R1C1), no 3,8
11a. 26(4) cage at R2C8 = {2789/3689/4679}, 9 locked for N3
11b. 16(3) cage at R1C7 (step 10g) = {457} (only remaining combination), 4,7 locked for R3 and N3
11c. 4 in N2 only in 27(4) cage at R2C5 = {4689} -> R4C6 = 9, R4C1 = 6 -> R3C1 = 5, R4C3 = 1 -> R34C4 = 10 = [73], R4C2 = 4, R2C12 = [41]
11d. 16(3) cage at R1C1 = {268} (only remaining combination), 2,8 locked for N1, 8 locked for R1
11e. 14(4) cage at R6C2 = {2345} -> R6C2 = 2, R7C2 = 5, R78C3 = {34}, locked for N7, 3 locked for C3
11f. R3C23 = [39], R2C3 = 7
11g. R9C2 = 7 (hidden single in C2) -> R9C13 = 8 = [26]
11h. R8C46 (step 8p) = 7 -> R9C456 = 17 = {359/458} -> R8C46 = [61] (cannot be [43] which clashes with R9C456), R1C456 = [193], R7C6 = 7 -> R6C6 = 4 (cage sum), R3C6 = 8, clean-up: no 5 in R8C9
11i. R1C1 = 8, R8C12 = [98], clean-up: no 2,3 in R8C89
11j. R3C9 = 2 -> 26(4) cage at R2C8 = {2789} -> R2C89 = [98]
and the rest is naked singles.