Prelims

a) 11(3) cage at R1C8 = {128/137/146/236/245}, no 9

b) 20(3) cage at R2C1 = {389/479/569/578}, no 1,2

c) 24(3) cage at R2C7 = {789}

d) 6(3) cage at R5C1 = {123}

e) 7(3) cage at R5C4 = {124}

f) 11(3) cage at R7C8 = {128/137/146/236/245}, no 9

g) 19(3) disjoint cage at R1C1 = {289/379/469/478/568}, no 1

Steps resulting from the Prelims

1a. Naked triple {789} in 24(3) cage at R2C7, locked for N3

1b. Naked triple {123} in 6(3) cage at R5C1, locked for N4

1c. Naked triple {124} in 7(3) cage at R5C4, locked for N5

2. 1,2 in R4 only in R4C789, locked for N6

2a. Hidden killer pair 1,2 in R4C7 and R4C89 for R4, R4C89 cannot contain both of 1,2 -> R4C7 = {12}, R4C89 must contain one of 1,2 -> 13(3) cage at R4C8 = {139/148/157/238/247/256} (cannot be {346} which doesn’t contain 1 or 2)

2b. 15(3) cage at R5C7 = {348/357/456}, no 9

2c. 13(3) cage = {139/148/157/238/256} (cannot be {247} which clashes with 15(3) cage)

3. 45 rule on N1 3 innies R1C1 + R2C2 + R3C3 = 7 = {124}, locked for N1 and D\

4. 45 rule on N4 3 innies R4C1 + R5C2 + R6C3 = 23 = {689}, locked for N4 and 45(9) disjoint cage at R1C7

5. 7 in 45(9) disjoint cage at R1C7 only in R7C4 + R8C5 + R9C6, locked for N8

5a. 45 rule on N8 3 innies R7C4 + R8C5 + R9C6 = 12 = {147/237}, no 5

6. 6 in N3 only in 11(3) cage at R1C8 = {146/236}, no 5

7. 45 rule on N2 3 innies R1C4 + R2C5 + R3C6 = 10 = {136/145/235} (cannot be {127} which clashes with R4C7), no 7,8,9

7a. Killer pair 1,2 in R1C4 + R2C5 + R3C6 and R4C7, locked for 45(9) disjoint cage at R1C4

8. R1C1 = {24} -> 19(3) disjoint cage at R1C1 = {289/469/478}, no 3,5

9. 45 rule on N6 3 innies R4C7 + R5C8 + R6C9 = 17 = {179/269/278} (other combinations don’t contain 1 or 2 for R4C7), no 3,4,5

10. 45 rule on N7 3 innies R7C1 + R8C2 + R9C3 = 18 = {369/378/459/468} (cannot be {567} which clashes with R4C7 + R5C8 + R6C9)

11. Hidden killer pair 8,9 in 18(3) cage at R1C2 and 17(3) cage at R1C5 for R1, neither can have both of 8,9 -> 18(3) cage and 17(3) cage must each contain one of 8,9 in R1 -> no 8,9 in R2C3 and R2C6

11a. 18(3) cage = {369/378} (cannot be {567} which doesn’t contain 8 or 9), no 5, 3 locked for N1

11b. 17(3) cage = {179/269/278/368/458} (cannot be {359} which clashes with R1C4 + R2C5 + R3C6, cannot be {467} which doesn’t contain 8 or 9)

11c. Hidden killer pair 7,9 in 18(3) cage at R1C2 and 17(3) cage at R1C5 for R1, 18(3) cage cannot contain both of 7,9 -> 17(3) cage must contain at least one of 7,9 in R1 = {179/269/278} (cannot be {368/458} which don’t contain 7 or 9), no 3,4,5

11d. 18(3) cage at R2C4 = {369/378/459/468} (cannot be {189/279/567} which clash with 17(3) cage), no 1,2

12. Hidden killer pair 1,2 in R3C3 and 13(3) cage at R7C2 for C3, 13(3) cage cannot contain both of 1,2 -> R3C3 = {12}, no 1,2 in R7C2

13. 45 rule on R3 3 outies R2C147 = 3 innies R3C369 + 17 -> R2C147 = 23,24, R3C369 = 6,7

13a. R2C147 = 23,24 = {689/789}, no 3,4,5

13b. R3C369 = 6,7 = {123/124}, no 5,6

13c. 5 in N1 only in R3C12, locked for R3

14. 18(3) cage at R2C4 (step 11d) = {369/378/468}

14a. R1C4 + R2C5 + R3C6 (step 7) = {145/235} (cannot be {136} which clashes with 18(3) cage), no 6, 5 locked for 45(9) cage at R1C4)

15. 45 rule on R6 3 innies R6C369 = 3 outies R5C147 + 17 -> R5C147 = 6,7, R6C369 = 23,24

15a. R5C147 = 6,7 -> R5C14 = {12}, R5C7 = {34}

15b. R6C369 = 23,24 = {689/789}, no 3,5, 8 locked for R6

15c. 15(3) cage at R5C7 (step 2b) = {357/456}, 5 locked for N6

15d. R5C7 = {34} -> no 3,4 in R6C78

15e. Killer pair 6,7 in R5C8 + R6C9 and 15(3) cage, locked for N6

15f. 5 in C9 only in R789C9, locked for N9

16. 45 rule on N5 3 innies R4C4 + R5C5 + R6C6 = 20 = {389/569/578}

16a. 3,5 only in R5C5 -> R5C5 = {35}

17. 45 rule on N9 3 innies R7C7 + R8C8 + R9C9 = 18 = {369/378/567}

17a. 11(3) cage at R7C8 = {128/146/245} (cannot be {137/236} which clash with R7C7 + R8C8 + R9C9), no 3,7

17b. 16(3) cage at R8C7 = {169/178/349} (cannot be {268/367} which clash with R7C7 + R8C8 + R9C9), no 2

17c. 2 in N9 only in 11(3) cage = {128/245}, no 6

18. 45 rule on C9 3 innies R369C9 = 3 outies R147C8 + 10

18a. Max R369C9 = 21 -> max R147C8 = 11, no 9 in R4C8

19. 45 rule on C7 3 innies R369C8 = 3 outies R147C7 + 10

19a. Min R147C7 = 9 -> min R369C8 = 19, no 1 in R9C8

[I ought to have spotted this sooner.]

20. 16(3) cage at R8C7 (step 17b) = {169/178/349}

20a. 9 of {349} must be in R89C7 (R89C7 cannot be {34} which clashes with R5C7)

20b. Killer quad 6,7,8,9 in R23C7, R7C7 and R89C7, locked for C7 -> R6C7 = 5

20c. R2C8 = 5 (hidden single in N3), placed for D/

20d. R1C4 = 5 (hidden single in N2)

21. R5C5 = 3, placed for both diagonals

21a. R9C9 = 5 (hidden single on D\) -> R7C7 + R8C8 + R9C9 (step 17) = {567}, 6,7 locked for N9 and D\

21b. Naked pair {89} in R4C4 + R6C6, locked for N5

21c. R5C7 = 4 -> R6C8 = 6 (cage sum), R8C8 = 7, R7C7 = 6

21d. 16(3) cage at R8C7 (step 20) = {349} (only remaining combination) -> R9C8 = 4, R89C7, = {39}, locked for C7

21e. R6C9 = 7 (hidden single in R6), placed for disjoint 45(9) cage at R1C4

21f. Naked triple {124} in R268C5, locked for C5

21g. R1C4 + R2C5 + R3C6 (step 7) = {145/235}

21h. 3 of {235} must be in R3C6 -> no 2 in R3C6

22. R3C8 = 9 (hidden single in N3)

23. R5C8 = 8, placed for 45(9) disjoint cage at R1C4

23a. R4C7 + R5C8 + R6C9 (step 9) = {278} (only remaining combination) -> R4C7 = 2, placed for 45(9) disjoint cage at R1C4, R5C9 = 9, R5C2 = 6

23b. R1C4 + R2C5 + R3C6 (step 7) = {145} (only remaining combination), 1,4 locked for N2 and 45(9) disjoint cage at R1C4 -> R8C2 = 9, R7C1 = 3, R9C3 = 6

24. Naked pair {12} in R56C1, locked for C1 and N4 -> R1C1 = 4, R6C2 = 3

25. 18(3) cage at R1C2 = {378} (only remaining combination), 7,8 locked for N1, 8 also locked for R1 -> R3C12 = [65], R2C1 = 9, R4C1 = 8, R89C1 = [57], 7 placed for D/, R9C2 = 2 (cage sum)

26. R2C2 = 1, R2C5 = 4

26a. R6C4 = 4 (hidden single in N5), R3C7 = 8, both placed for D/ -> R7C3 = 1

and the rest is naked singles, without using the diagonals.