Prelims
a) R12C9 = {17/26/35}, no 4,8,9
b) R2C78 = {39/48/57}, no 1,2,6
c) R3C67 = {89}
d) R7C34 = {19/28/37/45}, no 5
e) R89C1 = {17/26/35}, no 4,8,9
f) R8C23 = {17/26/35}, no 4,8,9
g) 7(3) cage at R2C2 = {124}
h) 21(3) cage at R5C8 = {489//579/678}, no 1,2,3
i) 6(3) cage at R6C6 = {123}
j) 22(3) cage at R7C5 = {589/679}
j) 26(4) cage at R4C6 = {2789/3689/4589/4679/5678}, no 1
Steps resulting from Prelims
1a. Naked triple {124} in 7(3) cage at R2C2, locked for R2, clean-up: no 6,7 in R1C9, no 8 in R2C78
1b. Naked pair {89} in R3C67, locked for R3
1c. 22(3) cage at R7C5 = {589/679}, 9 locked for N8, clean-up: no 1 in R7C3
1d. R12C9 = [17/26] (cannot be {35} which clashes with R2C78), no 3,5
1e. 1 in N6 only in R4C89 + R6C8, CPE no 1 in R3C8
2. 19(5) cage at R4C5 must contain 1, locked for N5
2a. 6(3) cage at R6C6 = {123}, 1 locked for R7, clean-up: no 9 in R7C3
3. 45 rule on N5 4 innies R46C46 = 26 = {2789/3689} (cannot be {4589/4679/5678} because R6C6 only contains 2,3), no 4,5, 8,9 locked for N5
3a. R6C6 = {23} -> no 2,3 in R4C46 + R6C4
3b. Min R4C6 = 6 -> max R3C34 = 6, no 6,7 in R3C34
4. 45 rule on R1 2 innies R1C19 = 9, R1C9 = {12} -> R1C1 = {78}
4a. 45 rule on R1 1 innie R1C1 = 1 outie R2C9 + 1
4b. R1C1 + R2C9 = [76/87], CPE no 7 in R2C1
5. 45 rule on R9 2 innies R9C19 = 7 = {16/25}/[34], no 7,8,9, no 3 in R9C9, clean-up: no 1 in R8C1
6. 18(3) cage at R2C5 = {369/378/468/567} (cannot be {189} which clashes with R3C6, cannot be {279/459} which clash with R2C78), no 1,2
7. 45 rule on R789 4 outies R6C1268 = 14 = {1238/1247/1256/1346/2345}, no 9
8. 45 rule on R12 2 innies R12C1 = 1 outie R3C5 + 9
8a. Min R3C5 = 3 -> min R12C1 = 12, no 3 in R2C1
[I should have spotted this earlier]
9. 45 rule on N6 1 outie R4C6 = 3 innies R4C89 + R6C8 + 2
9a. Min R4C89 + R6C8 = 6 -> min R4C6 = 8
9b. R4C6 = {89} -> R4C89 + R6C8 = 6,7 = {123/124}, 2 locked for N6
9c. R4C89 + R6C8 = {123/124}, CPE no 2 in R3C8
9d. Naked pair {89} in R34C6, locked for C6
10. 15(4) cage at R3C8 = {1257/1347/1356/2346}
10a. Killer pair 6,7 in R2C9 and 15(4) cage, locked for N3, clean-up: no 5 in R2C78
10b. Naked pair {39} in R2C78, locked for R2 and N3 -> R3C67 = [98], R4C6 = 8
10c. 9 in N5 only in R46C4, locked for C4
11. R4C6 = R4C89 + R6C8 + 2 (step 9)
11a. R4C6 = 8 -> R4C89 + R6C8 = 6 = {123}, locked for N6
12. 8 in N6 only in 21(3) cage at R5C8 = {489/678}, no 5
12a. 5 in N6 only in 26(4) cage at R4C6, locked for C7
13. 18(3) cage at R2C5 (step 6) = {378/468/567}
13a. 5 of {567} must be in R2C56 (R2C56 cannot be {67} which clashes with R2C9), no 5 in R3C5
14. 45 rule on N36 3(2+1) remaining innies R1C78 + R6C8 = 8 = [143/152/242/251/413], no 2 in R1C8
14a. R1C78 + R6C8 = [143/152/242/251/413] -> R1C789 = [142/152/241/251/412], 1,2 locked for R1 and N3
[With hindsight, it would have been simpler to use the 15(4) cage at R3C8 to eliminate 1,2 from R3C9.]
15. R23C4 = {12} (hidden pair in N2), locked for C4, clean-up: no 8 in R7C3
15a. 7(3) cage at R2C2 = {124}, 4 locked for N1
16. 45 rule on C89 4 innies R1289C8 = 24 = {1689/2589/3489/3579/4569} (cannot be {2679/3678} because R1C8 only contains 1,4,5, cannot be {4578} because R2C9 only contains 3,9), 9 locked for C8
17. 15(4) cage at R3C8 (step 10) = {1257/1347/1356/2346} -> R3C89 = {46/47/56/57} = 10,11,12
17a. 45 rule on R123 2 outies R4C24 = 2 innies R3C89 + 4
17b. R3C89 = 10,11,12 -> R4C24 = 14,15,16 = {59/69/79}, 9 locked for R4
18. 12(3) cage at R3C2 = {129/156/237}
18a. 29(5) cage at R1C1 = {15689/23789/35678} (cannot be {25679} which clashes with 12(3) cage), 8 locked for C1 and N1
18b. Combined cage 29(5) + 12(3) = {15689}[327]/{23789}[516]/{35678}{12}9, 3 locked for R3 and N1
19. 18(3) cage at R2C5 (step 13) = {468/567}, 6 locked for N2
20. 6 in R1 only in R1C23, locked for N1
20a. 29(5) cage at R1C1 (step 18a) = {23789/35678} (cannot be {15689} because 6,9 only in R4C2), no 1, 3 locked for N1
20b. 6,9 only in R4C2 -> R4C2 = {69}
[Cracked. The 29(5) cage proved to be very important, but only after 4 had been eliminated in step 15a.
The rest is fairly straightforward.]
21. 12(3) cage at R3C2 (step 18) = {129/156}, no 7
21a. Naked pair {69} in R4C24, locked for R4
22. 45 rule on R123 R4C24 = R3C89 + 4 (step 17a)
22a. R4C24 = {69} = 15 -> R3C89 = 11 -> R4C89 = 4 = {13}, locked for R4 and N6 -> R6C8 = 2, R6C6 = 3, clean-up: no 5 in R9C1 (step 5), no 3 in R8C1
22b. Naked pair {12} in R7C67, locked for R7, clean-up: no 8 in R7C4
22c. R46C46 (step 3) = {3689} (only remaining combination), 6 locked for C4 and N5, clean-up: no 4 in R7C3
23. R1C9 = 2 (hidden single in C9), R2C9 = 6, clean-up: no 1 in R9C1 (step 5), no 7 in R8C1
23a. R3C89 = 11 (step 22a) = {47} (only remaining combination), locked for R3 and N3 -> R1C78 = [15], R7C67 = [12]
23b. R3C5 = 6 -> R2C56 = 12 = {57}, locked for R2 and N2 -> R1C6 = 4, R12C1 = [78]
23c. Naked pair {69} in R14C2, locked for C2, clean-up: no 2 in R8C3
24. R6C1268 = 14 (step 7)
24a. R6C68 = [32] = 5 -> R6C12 = 9 = [18]/{45}
24b. R6C12 = 9 -> R7C12 = 13 = [58/94] (cannot be [67] which clashes with R7C34)
24c. R6C12 = [18] (cannot be {45} which clashes with R7C12), R7C12 = [94], clean-up: no 6 in R7C3
24d. Naked pair {37} in R7C34, locked for R7
25. R2C3 = 4 (hidden single in R2)
25a. R8C5 = 9 (hidden single in N8)
26. 4 in C1 only in 14(3) cage at R4C1 = {347} (only remaining combination) -> R4C1 = 4, R5C1 = 3, R5C2 = 7, clean-up: no 5 in R8C1, no 1 in R8C3, no 4 in R9C9 (step 5)
26a. Naked pair {26} in R89C1, locked for C1 and N7 -> R3C1 = 5
26b. Naked pair {12} in R3C34, locked for R3 -> R3C2 = 3
26c. R3C34 = {12}, R4C4 = 9 (cage sum), R4C2 = 6, R6C4 = 6
27. Naked triple {259} in R456C3, locked for C3 -> R3C34 = [12]
28. R9C3 = 8 (hidden single in C3)
28a. 8 in N8 only in 22(3) cage at R7C5 = {589} (only remaining combination), locked for N8
29. 7 in N5 only in R46C5, locked for C5 -> R2C5 = 5, R7C5 = 8, R7C89 = [65], R9C9 = 1, R8C9 = 8 (cage sum), R9C1 = 6 (step 5), R9C2 = 5, R8C2 = 1, R8C3 = 7
and the rest is naked singles.
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