Prelims
a) 21(3) cage at R3C3 = {489/579/678}, no 1,2,3
b) 22(3) cage at R5C4 = {589/679}
c) 22(3) cage at R9C4 = {589/679}
d) 12(4) cage at R1C1 = {1236/1245}, no 7,8,9
e) 13(4) cage at R1C3 = {1237/1246/1345}, no 8,9
f) 14(4) cage at R1C8 = {1238/1247/1256/1346/2345}, no 9
g) 13(4) cage at R3C9 = {1237/1246/1345}, no 8,9
h) 28(4) cage at R8C1 = {4789/5689}, no 1,2,3
i) 12(4) cage at R8C8 = {1236/1245}, no 7,8,9
Steps resulting from Prelims
1a. 22(3) cage at R5C4, 9 locked for N5
1b. 22(3) cage at R9C4 = {589/679}, 9 locked for R9 and N8
1c. 28(4) cage at R8C1 = {4789/5689}, 8,9 locked for N7, 9 also locked for R8
1d. 12(4) cage at R1C1 = {1236/1245}, 1,2 locked for N1
1e. 13(4) cage at R1C3 = {1237/1246/1345}, 1 locked for C4 and N2
1f. 12(4) cage at R8C8 = {1236/1245}, 1,2 locked for N9
2. 45 rule on N5 3 innies R4C56 + R5C6 = 6 = {123}, locked for N5
2a. 4 in N5 only in 17(3) cage at R4C4, locked for D\
2b. 23(4) cage at R1C5 can only contain one of 1,2,3, R4C5 = {123} -> no 2,3 in R123C5
2c. 23(4) cage at R2C7 can only contain one of 1,2,3, R4C6 = {123} -> no 1,2,3 in R2C7 + R3C78
2d. R478C5 = {123} (hidden triple in C5)
2e. 9 in R4 only in R4C123, locked for N4
2f. 9 on D\ only in R3C3 + R7C7. CPE no 9 in R3C7
3. 45 rule on N4 1 outie R3C3 = 1 innie R6C3 + 4 -> R6C3 = {12345}
4. 45 rule on N8 3 innies R7C456 = 10 = {127/136/145/235}, no 8
4a. 13(3) cage at R8C4 = {148/238/247/346} (cannot be {157/256} which clash with 22(3) cage at R9C4), no 5
4b. 8 in R7 only in R7C789, locked for N9
5. 45 rule on R89 4(2+2) innies R89C37 = 15
5a. Min R89C7 = 9 (cannot be {34/35} which clash with 12(4) cage at R8C8) -> max R89C3 = 6, no 6,7 in R89C3
6. 45 rule on N2 3(2+1) outies R1C37 + R4C2 = 14
6a. Max R1C3 + R4C2 = 10 -> min R1C7 = 4
6b. 1,2 in C7 only in R456C7, locked for N6
6c. Min R4C89 = 7 -> max R3C9 + R4C7 6, no 6,7 in R3C9 + R4C7
7. 45 rule on N2356 2(1+1) outies R1C3 + R7C9 = 1 = [38]/{47/56}, no 3,9 in R7C9
8. Hidden killer triple 7,8,9 in R3C3, 17(3) cage at R4C4 and R7C7 for D\, 17(3) cage can only contain one of 7,8 -> R3C3 + R7C7 = {789}, clean-up: no 1,2 in R6C3 (step 3)
8a. 1,2 in C3 only in R789C3, locked for N7
9. 45 rule on N4 3 innies R456C3 = 17 = {359/368/458/467}
9a. 12(3) cage at R4C1 = {129/138/147/237/246} (cannot be {156/345} which clash with R456C3), no 5
9b. 16(3) cage at R4C2 = {169/178/259/268/358/367/457} (cannot be {349} which clashes with R456C3)
10. 13(4) cage at R3C9 = {1246/1345} (cannot be {1237} which clashes with R4C56), no 7
10a. R4C789 can only contain one of 1,2,3 (otherwise clash with R4C56) -> R3C9 = {123}
10b. 13(4) cage = {1246/1345}, 4 locked for R4 and N6
10c. Killer triple 1,2,3 in R4C56 and 13(4) cage, locked for R4
10d. Min R4C1 = 6 -> max R56C1 = 6, no 6,7,8 in R56C1
11. 14(4) cage at R1C8 = {1247/1256/1346/2345} (cannot be {1238} which clashes with R3C9), no 8
12. 45 rule on N6 3(2+1) outies R37C9 + R5C6 = 11
12a. Max R3C9 + R5C6 = 6 -> min R7C9 = 5, clean-up: no 7 in R1C3 (step 7)
13. 45 rule on N3 4 outies R4C6789 = 1 innie R1C7 + 5
13a. Min R4C6789 = 12 (cannot be {1234/1235} which clash with R4C5) -> min R1C7 = 7
14. Hidden killer triple 1,2,3 in 13(4) cage at R1C3 and 23(4) cage at R1C6 for N2, 13(4) cage cannot contain all of 1,2,3 in R123C4 (because no 7 in R1C3), 23(4) cage cannot contain more than one of 2,3 (which would clash with R45C6) -> 13(4) cage contains two of 1,2,3 in R123C4 and 23(4) cage contains one of 2,3
14a. Killer triple 1,2,3 in 23(4) cage and R45C6, locked for C6
14b. 1 in C6 only in R45C6, locked for N5
15. R1C37 + R4C5 = 14 (step 6)
15a. Min R1C7 + R4C5 = 9 -> max R1C3 = 5, clean-up: no 5 in R7C9 (step 7)
[I could have omitted some earlier steps if I’d spotted this sooner…]
16. 45 rule on N23 6(1+5) outies R1C3 + R4C56789 = 19, min R4C56789 = 15 -> max R1C3 = 4, clean-up: no 6 in R7C9 (step 7)
16a. Killer pair 3,4 in 12(4) cage at R1C1 and R1C3, locked for N1
16b. Killer pair 3,4 in R1C3 and R456C3, locked for C3
16c. 3 in N7 only in R7C12, locked for R7
16d. 3 in N8 only in 13(3) cage at R8C4, locked for R8
16e. 13(3) cage (step 4a) = {238/346}, no 1,7
16f. 8 of {238} must be in R8C6 -> no 8 in R8C4
16g. R7C5 = 1 (hidden single in C5)
16h. R7C456 (step 4) = {127/145}, no 6
16i. 2 of {127} must be in R7C4 -> no 7 in R7C4
17. R3C469 = {123} (hidden triple in R3) -> R3C4 = {123}, R3C6 = {23}
17a. Naked triple {123} in R345C6, locked for C6
[At this stage I saw a nice forcing chain using the combinations for outies of N6, leading to R3C9 = R4C6 => R1C7 = 8 from 45 rule on N1.]
18. 20(5) cage at R6C3 = {13457/23456} (cannot be {12467} which clashes with 28(4) cage at R8C1)
18a. Killer pair 6,7 in 20(5) cage and 28(4) cage, locked for N7
18b. Hidden killer pair 1,2 in 20(5) cage and R9C3 for C3, 20(5) cage contains one of 1,2 -> R9C3 = {12}
18c. 20(5) cage must contain one of 1,2 in C3 -> no 2 in R7C4, clean-up: no 7 in R7C6 (step 16h)
18d. Naked pair {45} in R7C46, locked for R7 and N8 -> R7C1 = 3
18e. 20(5) cage contains 3 -> R6C3 = 3
18f. 22(3) cage at R9C4 = {679} (only remaining combination), locked for R9 and N8
18g. R8C6 = 8, naked pair {23} in R8C45, locked for R8
[The puzzle is cracked, the rest is fairly straightforward.]
19. R1C3 = 4 -> 12(4) cage at R1C1 = {1236}, locked for N1
19a. R7C4 = 4 (hidden single on C4), R7C6 = 5
19b. 20(5) cage at R6C3 (step 18) = {13457/23456} -> R8C3 = 5
19c. 28(4) cage at R8C1 = {4789} (only remaining combination), locked for N7 -> R7C23 = [62], 2 placed for D/
20. R6C3 = 3 -> R3C3 = 7 (step 3), placed for D\, R45C3 = 14 = {68}, locked for C3 and N3 -> R2C3 = 9
20a. R7C7 = 9 (hidden single on D\), R7C9 = 7 (step 7), R7C8 = 8
20b. 17(3) cage at R4C4 = {458} (only possible combination) -> R6C6 = 6, R4C4 + R5C5 = {58}, locked for N5 and D\
20c. Naked triple {679} in R569C4, locked for C4
21. R1C3 = 4 -> R123C4 = 9 = {135}, locked for C4 and N2 -> R3C6 = 2, R4C4 = 8, R5C5 = 5, placed for both diagonals, R45C3 = [68]
21a. R4C5 = 2 (hidden single in N5)
21b. Naked quad {1345} in 13(4) cage at R3C9, 5 locked for R4 and N6
22. 23(4) cage at R6C7 = {169}7/{268}7, 6 locked for R6 and N6
22a. Naked pair {79} in R6C45, locked for R6 and N5 -> R5C4 = 6, R6C789 = {268} = [268], R6C12 = [15]
22b. Naked pair {26} in R12C1, locked for C1 and N1 -> R5C1 = 4, R4C1 = 7 (cage sum)
23. R8C8 = 1, placed for D\ -> R2C2 = 3, placed for D\, R9C9 = 2, placed for D\, R1C1 = 6, R1C2 = 1
23a. R8C2 = 7 (hidden single in C2), placed for D/ -> R6C45 = [97]
23b. R3C7 + R4C6 = [61] (hidden pair on D/) -> R2C7 + R3C8 = 16 = [79]
23c. R8C79 = [46] -> R9C8 = 3 (cage sum)
and the rest is naked singles, without using the diagonals.
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