Prelims
a) R1C12 = {19/28/37/46}, no 5
b) R1C89 = {17/26/35}, no 4,8,9
c) R67C9 = {89}
d) R89C1 = {49/58/67}, no 1,2,3
e) R8C23 = {19/28/37/46}, no 5
f) R9C89 = {14/23}
g) 20(3) cage at R1C6 = {389/479/569/578}, no 1,2
h) 9(3) cage at R4C3 = {126/135/234}, no 7,8,9
i) 10(3) cage at R7C1 = {127/136/145/235}, no 8,9
1. Naked pair {89} in R67C9, locked for C9
2. 45 rule on N7 2 innies R9C23 = 12 = {39/48/57}, no 1,2,6
3. 45 rule on N6 3 innies R4C79 + R6C9 = 14
3a. Min R6C9 = 8 -> max R4C79 = 6, no 6,7,8,9 in R4C79
3b. 18(3) cage at R4C8 = {279/369/378/459/468/567} (cannot be {189} which clashes with R6C9), no 1
4. 45 rule on N3 2(1+1) outies R2C6 + R4C9 = 4 = [13/22/31]
4a. R4C79 + R6C9 = 14 (step 3)
4b. Max R46C9 = 12 -> min R4C7 = 2
4c. 12(3) cage at R2C9 = {147/156/237/246/345}
4d. 1 of {147/156} must be in R4C9 -> no 1 in R23C9
5. 45 rule on C6789 3 outies R234C5 = 21 = {489/579/678}, no 1,2,3
6. 45 rule on R89 2 innies R8C46 = 1 outie R7C5 + 9 -> no 9 in R8C46 (IOU)
6a. Max R8C46 = 15 -> max R7C5 = 6
6b. Min R8C46 = 10, no 9 in R8C46 -> no 1 in R8C46
7. 24(5) cage at R8C7 = {14568/23568} (cannot be {12489/12579/12678/13479/13569/13578/23469/23478/24567} which clash with R9C89), no 7,9
[Alternatively, and simpler, 24(5) cage and R9C89 ‘see’ each other so form combined 29(7) cage = {1234568}, no 7,9.]
7a. 7,9 in N9 only in R7C789, locked for R7, 7 locked for 29(5) cage at R6C6
7b. Hidden killer pair 1,2 in 10(3) cage at R7C1 and R8C23, 10(3) cage contains one of 1,2 -> R8C23 must contain one of 1,2 = {19/28}, no 3,4,6,7
7c. 7 in N7 only in R89C1 = {67} or R9C23 = {57} -> R89C1 = {49/67} (cannot be {58}, locking-out pairs)
7d. 18(3) cage at R4C1 = {189/378/459/567} (cannot be {279/369/468} which clash with R89C1), no 2
7e. 9(3) cage at R4C3 = {126/234} (cannot be {135} which clashes with 18(3) cage at R4C1), no 5, 2 locked for C3 and N4, clean-up: no 8 in R8C2
7f. 18(3) cage at R4C2 = {189/378/459/567} (cannot be {369/468} which clash with 9(3) cage)
8. R1C89 = {17/26/35}, 12(3) cage at R2C9 (step 4c) = {147/156/237/246/345}
8a. Consider permutations for R2C6 + R4C9 (step 4) = [13/22/31]
R2C6 + R4C9 = [13] => 1 in N3 only in R1C89 = {17} => 12(3) cage = {45}3
or R2C6 + R4C9 = [22] => 2 in N3 only in R1C89 = [26] => 12(3) cage = {37}2
or R2C6 + R4C9 = [31] => 3 in N3 only in R1C89 = {35} => 12(3) cage = {47}1
-> R1C89 = {17/35}/[26], no 6 in R1C8, no 2 in R1C9
and R23C9 = {37/45/47}, no 2,6
8b. 12(3) cage = {147/237/345}
8c. 8,9 in N3 only in 29(6) cage at R1C7 = {123689/124589}, no 7
[I ought to have spotted this immediately after step 7. Step 6 is clearly a key step.]
9. 9 in N8 only in R8C5 + R9C56, locked for 29(6) cage at R7C5, clean-up: no 3 in R9C23 (step 2)
9a. Killer pair 4,7 in R89C1 and R9C23, 4 locked for N7
9b. 3 in N7 only in 10(3) cage at R7C1, locked for R7
9c. 3 in N9 only in R89C789, CPE no 3 in R9C6
9d. 29(6) cage at R7C5 contains 9 = {124589/134579} (cannot be {123689/124679/234569} because R9C23 only contains {48/57}), 1 locked for N8
9e. 29(6) cage = {124589/134579}, CPE no 4,5 in R9C6
9f. 29(6) cage = {124589/134579} -> R78C5 + R9C45 = {1259/1349}, no 6,7,8
9g. R8C4 = 7 (hidden single in N8), clean-up: no 6 in R9C1
9h. Combined 29(7) cage R8C789 + R9C6789 = {1234568}, 1,4,5 locked for N9
9i. 4 in R7 only in R7C456, locked for N8
9j. 6 in R9 only in R9C67, locked for 24(5) cage at R8C7
10. 8 in N7 only in R8C3 + R9C23 -> killer X-Wing for 8 in R8C3 + R9C23 and 24(5) cage at R8C7, no other 8 in R89
[Then I found that step wasn’t necessary because of …]
10a. R8C46 = R7C5 + 9 (step 6), R8C4 = 7 -> R8C6 = R7C5 + 2 -> R7C5 = {14}, R8C6 = {36}
10b. 29(5) cage at R6C6 contains 7 for R7C78 = {23789/25679/34679/35678} (cannot be {14789} because R8C6 only contains 3,6), no 1
11. Consider combinations for 29(6) cage at R7C5 (step 9f) = {124589/134579}
29(6) cage = {124589} => R9C23 = {48} => R8C23 = {19}, 9 locked for R8
or 29(6) cage = {134579} => R9C23 = {57}, R7C5 = 4, R8C5 + R9C45 = 1{39}/3{19} (cannot be 9{13} which clashes with R9C89)
-> no 9 in R8C5
11a. 9 in R8 only in R8C123, locked for N7, clean-up: no 4 in R8C1
11b. 4 in N7 only in R9C123, locked for R9, clean-up: no 1 in R9C89
11c. Naked pair {23} in R9C89, locked for R9 and N9
11d. 29(6) cage = {124589/134579}
11e. 2 of {124589} must be in R8C5, 5,7 of {134579} must be in R9C23 -> no 5 in R8C5
11f. 29(6) cage = {124589/134579}, 5 locked for R9
12. 12(3) cage at R2C9 (step 8b) = {147/345} (cannot be {237} which clashes with R9C9), no 2, 4 locked for C9 and N3
12a. 3 of {345} must be in R4C9 -> no 3 in R23C9
12b. Killer pair 1,5 in 12(3) cage at R8C9, locked for C9, clean-up: no 3,7 in R1C8
12c. R2C6 + R4C9 (step 4) = {13}, no 2
12d. Naked pair {13} in R2C6 + R4C9, CPE no 1,3 in R4C6
12e. 29(6) cage at R1C7 (step 8c) = {123689/124589}, 2 locked for N3, clean-up: no 6 in R1C9
[Reworked from here. First time through I carelessly overlooked {57} for R45C8.]
13a. R5C9 = 6 (hidden single in C9) -> R45C8 = 12 = {39/48/57}
13b. R9C9 = 2 (hidden single in C9) -> R9C8 = 3, clean-up: no 9 in R45C8
13c. Consider combinations for R45C8 = {48/57}
R45C8 = {48}, locked for C8 => R8C89 = {15}
or R45C8 = {57}, 5 locked for C8 => R1C89 = [17], R23C9 = {45}, 5 locked for C9 => R8C9 = 1
-> 1 in R8C89, 1 locked for R8 and N9, clean-up: no 9 in R8C23
13d. R8C23 = [28], R8C56 = [36], R8C1 = 9 -> R9C1 = 4, R9C67 = [86], R7C5 = 4 (step 10a), clean-up: no 8 in R1C1, no 1,6 in R1C2
13e. Naked pair {25} in R7C46, 5 locked for R7 and N8
13f. Killer triple 1,4,5 in R1C8, R45C8 and R8C8, locked for C8
14a. 18(3) cage at R4C1 (step 7d) = {378/567}, no 1, 7 locked for C1 and N4, clean-up: no 3 in R1C2
14b. Killer pair 3,6 in 18(3) cage at R4C1 and 9(3) cage at R4C3 (step 7e), locked for N4
14c. 18(3) cage at R4C2 (step 7f) = {189/459}, 9 locked for C2, clean-up: no 1 in R1C1
14d. R1C12 = [28/64] (cannot be [37] which clashes with R1C9)
14e. Killer pair 4,8 in R1C2 and 18(3) cage at R4C2, locked for C2
15a. R234C5 (step 5) = {579/678}, 7 locked for C5
15b. 7 in N25 only in 20(3) cage at R1C6 and 26(5) cage at R3C5 -> both must contain 7
15c. 20(3) cage at R1C6 = {479/578}, no 3,6, 7 locked for N2
15d. 8 of {578} must be in R2C5 -> no 5 in R2C5
16a. 18(3) cage at R4C1 (step 14a) = {378/567}
16b. Consider permutations for R1C12 = [28/64]
R1C12 = [28] => 8 in N4 only in 18(3) cage at R4C1 = {378}
or R1C12 = [64] => 18(3) cage at R4C1 = {378}
-> 18(3) cage at R4C1 = {378}, 3,8 locked for C1 and N4
16c. 18(3) cage at R4C2 (step 14c) = {459} (only remaining combination), 4,5 locked for C2, 4 locked for N4, R1C2 = 8 -> R1C1 = 2, R9C23 = [75]
16d. Naked triple {126} in 9(3) cage at R4C3, 1,6 locked for C3 -> R7C3 = 3
16e. 6 in R1 only in R1C45, locked for N2
17. 29(5) cage at R6C6 must contain 6,7 = {25679/35678} (cannot be {34679} because R7C6 only contains 2,5), no 4, 5 locked for C6
17a. 9 of {25679} must be in R7C78 -> no 9 in R6C6
17b. 20(3) cage at R1C6 = {479} (only remaining combination, cannot be {578} because 5,8 only in R2C5), 4,9 locked for N2, 4 locked for C6
18a. R4C79 + R6C9 = 14 (step 3)
18b. R4C79 + R6C9 = [239/419] (cannot be [518] which clashes with R45C8) -> R4C7 = {24}, R6C9 = 9 -> R7C9 = 8
18c. R7C78 = {79} = 16, R8C6 = 6 -> R67C6 = 7 = {25}, 2 locked for C6
18d. R25C6 = {13} (hidden pair in C6)
18e. 2 in N2 only in R23C4, locked for C4 -> R7C4 = 5, R67C6 = [52], R6C2 = 4
19a. 7 in N5 only in R4C56, locked for R4, clean-up: no 5 in R5C8 (step 13c)
19b. 26(5) cage at R3C5 must contain 7 = {14579/14678/23579/23678} (cannot be {13679} because 1,3 only in R5C6, cannot be {24578} because 2,4 only in R4C7)
19c. R4C7 = {24}, R5C6 = {13} -> R3C5 + R4C56 = {579/678}
19d. R3C5 = {58} -> no 8 in R4C5
20. R12C3 = {479} -> 22(4) cage at R1C3 R12C3 + R2C24 = {47}[38]/{49}[18]/{49}[63] (R12C3 cannot be {79} because R2C24 cannot total 6), 4 locked for N1, R2C4 = {38}
20a. Killer pair 1,3 in R2C24 and R2C6, locked for R2
20b. R3C4 = 2 (hidden single in N2)
20c. 1 in N2 only in R1C45 + R2C6, CPE no 1 in R1C7
[Repeat the forcing chain in step 13 now that I’ve reduced R4C79 + R6C9.]
21. R4C79 + R6C9 (step 18b) = [239/419]
21a. Consider combinations for R45C8 (step 13c) = {48}/[57]
R45C8 = {48}, locked for N4 => R4C79 = [23], R1C9 = 7 => R1C8 = 1
or R45C8 = [57] => R1C8 = 1
-> R1C89 = [17]
21b. Naked pair {45} in R23C9, 5 locked for C9 and N3 -> R48C9 = [31], R2C6 = 1 (step 4 or hidden single in N2), R5C6 = 3, R4C7 = 2 (step 18b)
21c. R1C457 = [653] (hidden triple in R1), R3C5 = 8 -> R4C56 = [67] (step 19c)
21d. R237C7 = [897] -> R6C7 = 1, R6C45 = [82], R6C8 = 7 -> R5C7 = 5 (cage sum)
and the rest is naked singles.