Prelims
a) 10(2) cage at R1C1 = {19/28/37/46}, no 5
b) 7(2) cage at R1C9 = {16/25/34}, no 7,8,9
c) R6C23 = {18/27/36/45}, no 9
d) R67C4 = {16/25/34}, no 7,8,9
e) R67C5 = {39/48/57}, no 1,2,6
f) R67C6 = {39/48/57}, no 1,2,6
g) R6C78 = {14/23}
h) R9C12 = {19/28/37/46}, no 5
i) R9C89 = {18/27/36/45}, no 9
j) 22(3) cage at R1C4 = {589/679}
k) 19(3) cage at R4C8 = {289/379/469/478/568}, no 1
l) 7(3) cage at R5C4 = {124}
m) 19(3) cage at R7C2 = {289/379/469/478/568}, no 1
n) 11(3) cage at R7C3 = {128/137/146/236/245}, no 9
o) 26(4) cage at R4C9 = {2789/3689/4589/4679/5678}, no 1
Steps resulting from Prelims
1a. 22(3) cage at R1C4 = {589/679}, 9 locked for R1 and NR1C4, clean-up: no 1 in R2C2
1b. 7(3) cage at R5C4 = {124}, locked for R5
2. 45 rule on NR1C4 3 innies R3C46 + R5C5 = 9 = {126/135/234}, no 7,8
3. 45 rule on NR3C1 1 outie R7C1 = 1 innie R3C1 + 4, R3C1 = {12345}, R7C1 = {56789}
4. 45 rule on NR3C9 1 outie R7C9 = 1 innie R3C9 + 5, R3C1 = {1234}, R7C9 = {6789}
5. 45 rule on C12 1 outie R5C3 = 2 innies R16C2 + 1
5a. Max R16C2 = 8, no 8 in R16C2, clean-up: no 1 in R6C3
5b. Min R16C2 = 3 -> no 3 in R5C3
6. 1 in NR3C9 only in R3C9 or R6C78 = {14} -> no 4 in R3C9 (locking-out cages), clean-up: no 9 in R7C9 (step 4)
7. 45 rule on NR1C7 1 innie R3C7 = 1 outie R3C9 + 3, R3C9 = {123} -> R3C7 = {456}
8. 45 rule on NR1C7 3 innies R2C9 + R3C78 = 20 = {479/569/578} (cannot be {389} because R3C7 only contains 4,5,6), no 1,2,3
8a. 4 of {479} must be in R3C7 -> no 4 in R2C9 + R3C8
9. 45 rule on NR1C1 1 innie R3C3 = 1 outie R3C1 + 6, R3C1 = {123}, R3C3 = {789}, clean-up: R7C1 = {567} (step 3)
10. Law of Leftovers (LoL) for R123 two outies R45C5 must exactly equal two innies R3C19, R3C19 = {123} -> R4C5 = {123}, R5C5 = {12}
10a. Naked triple {124} in 7(3) cage at R5C4, 4 locked for NR4C4, clean-up: no 3 in R7C4, no 8 in R7C5, no 8 in R7C6
11. 14(3) cage at R2C5 = {158/167/248} (cannot be {257/356} which clash with 22(3) cage at R1C4, cannot be {347} which clashes with R67C5), no 3
11a. Naked pair {12} in R45C5, locked for C5 and NR1C4
11b. 3 in NR1C4 only in R3C46, locked for R3, clean-up: no 9 in R3C3 (step 9), no 6 in R3C7 (step 7), no 7 in R7C1 (step 3), no 8 in R7C9 (step 4)
11c. Naked pair {12} in R3C19, locked for R3
11d. Killer pair 1,2 in R3C9 and R6C78, locked for NR3C9
12. R2C9 + R3C78 (step 8) = {479/569/578}
12a. R3C7 = {45} -> no 5 in R2C9 + R3C8
13. 12(3) cage at R2C1 = {129/138/147/156/237/246} (cannot be {345} because R3C1 only contains 1,2)
13a. 12(3) cage = {129/138/156/237/246} (cannot be {147} which clashes with R3C13 = [17], step 9, killer combo blocker using NR1C1)
13b. 7,8,9 of {129/138/237} must be in R3C2 -> no 7,8,9 in R2C1
14. 45 rule on NR3C1 3(2+1) outies R2C1 + R3C2 + R7C1 = 16
14a. Max R26C1 = 11 -> min R3C2 = 5
14b. R2C1 + R3C2 + R7C1 cannot be [565] -> no 5 in R2C1
15. 45 rule on NR3C9 3(2+1) outies R2C9 + R3C8 + R7C9 = 22
15a. R27C9 cannot total 14 (cannot be [77] and [86] clashes with R3C8 = 8) -> no 8 in R3C8
16. 26(4) cage at R4C9 = {4679/5678} (cannot be {3689} which clashes with R6C78 = {23} because R37C9 = [16], step 4, cannot be {4589} because R7C9 only contains 6,7), no 3, 6,7 locked for C9, clean-up: no 1 in R2C8, no 2,3 in R9C8
[Alternatively consider combinations for R6C78 = {14/23}
R6C78 = {14} => R3C9 = 2, R7C9 = 7 (step 4) => 26(4) cage cannot be {3689}
or R6C78 = {23}, locked for NR3C9 …]
17. 45 rule on R9 2 innies R9C37 = 10 = [19]/{28/37/46}, no 5, no 1 in R9C7
[I ought to have spotted this after step 11; then analysing the 26(4) cage at R4C9 would have been simpler than in step 16 …]
18. R3C19 = [12] (cannot be [21] because R7C19 cannot be [66] using steps 3 and 4), 1 placed for NR3C1, 2 placed for NR3C9, R3C3 = 7 (step 9), placed for NR1C1, R3C7 = 5 (step 7), placed for NR1C7, R7C1 = 5 (step 3), placed for NR7C1, R7C9 = 7 (step 4), placed for NR6C6
[More formally R7C1 = R3C1 + 4 (step 3), R7C9 = R3C9 + 5 (step 4) -> R3C1 cannot be 1 more than R3C9 because R7C19 cannot be equal …]
[manu found R3C19 = {12} = 3 -> R7C19 = 12 (using steps 3 and 4) = [57] … which is a better way to get this result.]
18a. Clean-up: no 3 in R1C1, no 3,9 in R2C2, no 2 in R2C8, no 2 in R6C2, no 8 in R6C3, no 2 in R6C4, no 5,7 in R6C5, no 5,7 in R6C6, no 3 in R6C78, no 9 in R9C2
[Cracked, the rest is straightforward.]
19. Naked pair {14} in R6C78, locked for R6 and NR3C9, clean-up: no 5 in R6C23, no 6 in R7C4
19a. 2 in R6 only in R6C13, locked for NR3C1
20. R7C9 = 7 -> 26(4) cage at R4C9 = {5678} (only remaining combination), 5,6,8 locked for C9 and NR3C9 -> R2C9 = 9, R3C8 = 6, placed for NR1C7, clean-up: no 1 in R1C9, no 1,4 in R9C8, no 3 in R9C9
20a. Naked pair {34} in 7(2) cage at R1C9, locked for NR1C7
20b. 1 in C9 only in R89C9, locked for NR7C6
21. Naked pair {34} in R3C46, locked for NR1C4 -> R3C5 = 8, placed for NR1C4, R3C2 = 9, R2C1 = 2 (cage sum), placed for NR1C1, clean-up: no 8 in 10(2) cage at R1C1, no 5 in 22(3) cage at R1C4, no 4 in R7C5, no 8 in R9C2
21a. Naked pair {46} in 10(2) cage at R1C1, locked for NR1C1
22. Naked triple {679} in 22(3) cage at R1C4, locked for R1 and NR1C4 -> R2C5 = 5, R4C5 = 1 (cage sum), R5C5 = 2
22a. Naked pair {39} in R67C5, locked for C5
22b. 4 in C5 only in R89C5, locked for NR7C5
23. 45 rule on NR7C1 1 remaining innie R7C4 = 1 outie R9C3 = {12}, clean-up: no 3 in R6C4
24. 45 rule on NR7C6 1 remaining innie R7C6 = 1 outie R9C7 = {39}, clean-up: no 8 in R6C6
24a. Naked pair {39} in R7C56, locked for R7
24b. Naked pair {39} in R67C6, locked for C6
24c. Naked pair {39} in R6C56, locked for R6 and NR4C3, clean-up: no 6 in R6C23
24d. R6C23 = [72], 7 placed for NR3C1, R9C3 = 1, R7C4 = 1 (step 23), R6C4 = 6, R5C46 = [41], R3C46 = [34], R2C34 = [38], R1C23 = [15], 7(2) cage at R1C9 = [34], 10(2) cage at R1C1 = [46], R2C67 = [71], R6C1 = 8, placed for NR3C1, R6C9 = 5, 22(3) cage at R1C4 = [976], R6C78 = [41], R9C9 = 4, R9C8 = 5, R9C5 = 6, R8C5 = 4, clean-up: no 3,9 in R9C1, no 2 in R9C2
25. R9C3 = 1 -> R78C3 = 10 = [46] -> R4C3 = 8, R4C4 = 5 (cage sum)
26. R9C7 = 9 (hidden single in R9), R7C6 = 9 (step 24), placed for NR7C6
26a. R8C9 = 1 -> R78C8 = 11 = [83], both placed for NR7C6
and the rest is naked singles, without using the nonets.
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would not have made a more frightening one : thanks Ed !
Worth a pic.