Prelims
a) R12C6 = {17/26/35}, no 4,8,9
b) R6C23 = {39/48/57}, no 1,2,6
c) R78C5 = {19/28/37/46}, no 5
d) R89C8 = {16/25/34}, no 7,8,9
e) R89C9 = {89}
f) R9C67 = {18/27/36/45}, no 9
g) 9(3) cage at R1C1 = {126/135/234}, no 7,8,9
h) 19(3) cage at R2C1 = {289/379/469/478/568}, no 1
i) 11(3) cage at R4C1 = {128/137/146/236/245}, no 9
j) 20(3) cage at R7C6 = {389/479/569/578}, no 1,2
k) 30(4) cage at R1C7 = {6789}
l) 13(4) cage at R1C9 = {1237/1246/1345}, no 8,9
Steps resulting from Prelims
1a. Naked pair {89} in R89C9, locked for C9 and N9, clean-up: no 1 in R9C6
1b. Naked quad {6789} in 30(4) cage at R1C7, locked for N3
1c. 13(4) cage at R1C9 = {1237/1246/1345}, 1 locked for C9
1d. 6,7 of {1237/1246} must be in R4C9 -> no 2 in R4C9
[Ed pointed out that I missed something else here; I found it at step 10.]
2. 45 rule on N12 2 innies R3C36 = 16 = {79}, locked for R3
2a. 45 rule on N4 1 outie R3C3 = 1 innie R6C1 + 1 -> R3C3 + R6C1 = [76/98]
2b. 19(3) cage at R2C1 = {289/469/478/568} (cannot be {379} because 7,9 only in R2C1), no 3
2c. 7,9 of {289/469/478} must be in R2C1 -> no 2,4 in R2C1
2d. 5 of {568} must be in R23C1 (R23C1 cannot be {68} which clashes with R6C1) -> no 5 in R3C2
2e. 4 of {469/478} must be in R3C1 (cannot be [964] which clashes with R3C3 + R6C1 = [76], cannot be [784] which clashes with R3C3 + R6C1 = [98]) -> no 4 in R3C2
2f. 12(3) cage at R6C1 = {138/156/246} (cannot be {129/147/237/345} because R6C1 only contains 6,8), no 7,9
2g. R6C1 = {68} -> R7C12 = {13/15/24}, no 6,8
2h. 37(7) cage at R3C6 must contain 8, locked for N6
3a. 45 rule on C6789 2 innies R4C67 = 5 = {14/23}
3b. 11(3) cage at R4C1 = {128/137/146/236/245} cannot be {12}8/{13}7/{24}5 (which clash with R4C67) -> no 5,7,8 in R5C1
3c. 45 rule on C6789 3 outies R456C5 = 21 = {489/579/678}, no 1,2,3
3d. Combined cage R456C5 + R78C5 = {489}{37}/{579}{28}/{579}{46}/{678}{19}, 7,9 locked for C5
4. 45 rule on C12 3 innies R568C2 = 21 = {489/579/678}, no 1,2,3, clean-up: no 9 in R6C3
5. 20(3) cage at R7C6 = {389/479/569/578}
5a. 3 of {389} must be in R8C7 -> no 3 in R78C6
5b. 4 of {479} must be in R78C6 (R78C6 cannot be {79} which clashes with R3C6) -> no 4 in R8C7
6. 45 rule on C1234 4 outies R1239C5 = 14 = {1238/1256/1346/2345}
6a. 8 of {1238} must be in R123C5 (R123C5 cannot be {123} which clashes with R12C6) -> no 8 in R9C5
7. 45 rule on R89 2 outies R7C56 = 2 innies R8C23 + 6
7a. Min R8C23 = 6 (cannot be [41] which clashes with R7C12) -> min R7C56 = 12, no 1,2 in R7C5, clean-up: no 8,9 in R8C5
8. R568C2 (step 4) = {489/579/678}
8a. Consider placements for R3C2 = {268}
R3C2 = 2 => R23C1 = [98], R6C1 = 6, R3C3 = 7 => 23(4) cage at R3C3 = {2579/3479} (cannot be {3578} = 7{358} which clashes with R6C23), no 6,7,8 in R5C2 => R568C2 = {489/579}
or R3C2 = 6 => R568C2 = {489/579}
or R3C2 = 8 => R568C2 = {579}
-> R568C2 = {489/579}, no 6, 9 locked for C2
[First key step.]
9. 45 rule on R6789 7 (6+1) outies R4C4567 + R5C456 = 37
9a. R4C67 (step 3a) = 5 -> R4C45 + R5C456 = 32 = {26789/35789/45689}, no 1, 8,9 locked for N5
[Second key step.]
9b. R456C5 (step 3c) = {489/579/678}
9c. 4 of {489} must be in R6C5 -> no 4 in R45C5
9d. 8 on R6 only in R6C123, locked for N4
9e. 11(3) cage at R4C1 (step 3b) = {137/146/236/245}
9f. Hidden killer pair 1,2 in 11(3) cage and 23(4) cage at R3C3 for N4, 11(3) cage contains one of 1,2 -> 23(4) cage must contain one of 1,2 = {1679/2579}, no 3,4
[Alternatively 23(4) = {1679/2579} (cannot be {3479/3569} which clashes with 11(3) cage).
First time through I made a careless mistake here by either overlooking a digit in N4 or carelessly deleting it from my worksheet, so reworked from here.]
9g. 11(3) cage = {137/146/245} (cannot be {236} which clashes with 23(4) cage)
9h. Hidden killer pair 3,4 in 11(3) cage and R6C23 for N4, 11(3) contains one of 3,4 -> R6C23 must contain one of 3,4 = {48}/[93], no 5,7
9i. Consider combinations for R6C23
R6C23 = {48}, 4 locked for N4 => 11(3) cage = {137}
or R6C23 = [93] => R3C3 = 9, 19(3) cage at R2C1 = {568}, 9(3) cage at R1C1 = {234} => R12C3 = {17}, locked for C3 => 23(4) cage = {2579} => 11(3) cage = {146}
-> 11(3) cage = {137/146}, no 2,5, 1 locked for N4
9j. 23(4) cage = {2579}, no 6, 2 locked for C3
9k. R4C12 = {14/16/17/37/46}, R4C67 (step 3a) = {14/23} -> combined cage R4C1267 must contain 3, locked for R4
[Just spotted.]
10. R12C9 + R3C789 = {12345}, CPE no 2,3,4,5 in R5C9
11. 37(7) cage at R4C4 must contain 9
11a. Consider placements for 9 in N2
9 in R12C4, locked for C4 => 9 in 37(7) cage in R7C3 + R8C23, locked for N7 => R2C1 = 9 (hidden single in C1) => R3C3 = 7
or 9 in R3C6 => R3C3 = 7
-> R3C3 = 7, R3C6 = 9
[Final key step. With hindsight this step was available after step 6, but I only saw it while doing the rework.
Back to my original steps, slightly modified where appropriate.]
11b. 23(4) cage at R3C3 (step 9j) = {2579}, 9 locked for N4, clean-up: no 3 in R6C3
11c. Naked pair {48} in R6C23, locked for N4, 4 locked for R6 -> R6C1 = 6
11d. Naked triple {137} in 11(3) cage, 7 locked for R4
11e. R456C5 (step 3c) = {579/678}, 7 locked for C5 and N5, clean-up: no 3 in R78C5
11f. 20(3) cage at R7C6 = {578} (only remaining combination), 8 locked for C6 and N8, clean-up: no 2 in R8C5, no 1 in R9C7
11g. 20(3) cage at R7C6 = {578}, CPE no 5,7 in R8C4
11h. R1239C5 (step 6) = {1238/2345} (cannot be {1256} which clashes with R456C5, cannot be {1346} which clashes with R78C5), no 6
11i. 9 in N6 only in R6C78 -> 17(4) cage at R5C6 = {1259/1349}, no 6,7, 1 locked for R6
11j. 4 of {1349} must be in R5C6 -> no 3 in R5C6
11k. R4C45 + R5C456 (step 9a) = {26789/45689} (cannot be {35789} which clashes with R6C5), no 3
11l. R6C1 = 6 -> R7C12 = 6 = {15/24}, no 3
11m 45 rule on N1 2 innies R12C3 = 10 = {19/46}, no 3,5,8
11n. 3 in N1 only in 9(3) cage at R1C1 = {135/234}, no 6
11o. Killer pair 1,4 in 9(3) cage and R12C3, locked for N1
11p. R568C2 (step 8a) = {489} (only remaining combination, cannot be {579} because R6C2 only contains 4,8) -> R5C2 = 9, R68C2 = {48}, locked for C2
11q. Naked pair {25} in R45C3, 5 locked for C3
11r. 8 in N1 only in R23C1, locked for C1
11s. 19(3) cage at R2C1 (step 2b) = {289/568}
11t. R3C2 = {26} -> no 2 in R3C1
11u. 13(4) cage at R1C9 = {1246/1345}, 4 locked for C9
11v. 17(4) cage at R6C9 = {1367/1457/2357} (cannot be {2456} = 2{456}/5{246} which clashes with R89C8)
11w. 17(4) cage = {1367/2357} (cannot be {1457} which clashes with R7C12), no 4
12. 45 rule on N9 2 innies R89C7 = 1 outie R6C9 + 4
12a. Min R89C9 = 7 -> min R6C9 = 3
12b. R6C9 = {357} and R8C7 = {57} are both odd -> R9C7 must be even = {246} -> R9C6 = {357}
12c. 4 in C6 only in R45C6, locked for N5
13. 17(4) cage at R5C6 (step 11i) = {1259/1349}
13a. Consider combinations for 17(4) cage
17(4) cage = {1259}, CPE no 2,5 in R6C4 => R6C4 = 3
or 17(4) cage = {1349}, 3 locked for R6
-> R6C9 = {57}
13b. Naked pair {57} in R6C59, 5 locked for R6
13c. 17(4) cage at R6C9 (step 11w) = {1367/2357}, 3 locked for R7 and N9, clean-up: no 4 in R89C8
13d. R9C7 = 4 (hidden single in N9) -> R9C6 = 5, clean-up: no 3 in R12C6, no 1 in R4C6 (step 3a)
13e. Naked pair {78} in R78C6, 7 locked for C6, N8 and 20(3) cage at R7C6 -> R8C7 = 5
13f. R89C8 = {16} (only remaining combination), locked for C8 and N9
13g. Naked triple {237} in R7C789, 2,7 locked for R7, 7 locked for 17(4) -> R6C9 = 5, R6C5 = 7, R78C6 = [87]
13h. 17(4) cage at R5C6 = {1349} (only remaining combination) -> R5C6 = 4, R6C4 = 2 (hidden single in R6), R4C6 = 3 -> R4C7 = 2 (step 3a), R6C6 = 1, R45C3 = [52]
13i. Naked pair {17} in R4C12, 1 locked for R4 and N4 -> R5C1 = 3
13j. R5C7 = 1 (hidden single in R5) -> R3C7 = 3, R6C78 = [93], R7C789 = [723]
13k. Naked triple {124} in R123C9, 4 locked for C9 and N3 -> R3C8 = 5, R4C9 = 6, R5C89 = [87], R4C8 = 4, R3C1 = 8
13l. Naked pair {26} in R12C6, locked for N2
13m. Naked pair {14} in R3C45, locked for R3 and N2, R2C4 = 7 (cage sum), R12C8 = [79], R2C1 = 5, R3C29 = [62]
13n. 9(3) cage (step 11n) = {234} (only remaining combination) -> R1C1 = 4, R12C2 = {23}, locked for C2, R12C3 = [91], R7C12 = [15], R9C2 = 7
13o. R9C5 = 2 (hidden single in C5) -> R89C1 = [29], R89C9 = [98]
13p. 15(4) cage at R8C4 = {2346} (only remaining combination) -> R8C4 = 4, R78C5 = [91]
and the rest is naked singles.
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