Prelims

a) R1C23 = {49/58/67}, no 1,2,3

b) R1C78 = {15/24}

c) R3C12 = {19/28/37/46}, no 5

d) R3C89 = {16/25/34}, no 7,8,9

e) R7C12 = {17/26/35}, no 4,8,9

f) R7C89 = {29/38/47/56}, no 1

g) R9C23 = {17/26/35}, no 4,8,9

h) R9C78 = {17/26/35}, no 4,8,9

i) 10(3) cage at R3C3 = {127/136/145/235}, no 8,9

j) 19(3) cage at R3C7 = {289/379/469/478/568}, no 1

k) 42(7) cage at R4C5 = {3456789}

1. R3C89 = {16/34} (cannot be {25} which clashes with R1C78), no 2,5

1a. Killer pair 1,4 in R1C78 and R3C89, locked for N3

1b. 45 rule on N3 2 innies R23C7 = 14 = {59/68}

1c. R23C7 = 14 -> R34C7 cannot be 14 (CCC) -> no 5 in R4C8

1d. 7 in N3 only in 18(3) cage at R1C9 = {279/378} (cannot be {567} which clashes with R23C7), no 5,6

1e. R3C12 = {19/28/37} (cannot be {46} which clashes with R3C89), no 4,6

1f. Min R2C7 = 5 -> max R234C6 = 11, no 9 in R234C6

2. 45 rule on N1 2 innies R23C3 = 4 = {13}, locked for C3 and N1, clean-up: no 7,9 in R3C12, no 5,7 in R9C2

2a. Naked pair {28} in R3C12, locked for R3 and N1, clean-up: no 5 in R1C23, no 6 in R2C7 (step 1b)

2b. Killer pair 1,3 in R3C3 and R3C89, locked for R3

2c. 7 in R3 only in R3C456, locked for N2

2d. 6 in N3 only in R3C789, locked for R3

2e. Min R2C7 + R3C6 = 9 -> max R24C6 = 7, no 7,8 in R24C6

2f. R34C3 cannot total 4 -> no 6 in R4C2

3. 45 rule on N7 2 innies R78C3 = 15 = {69/78}

3a. Killer triple 5,6,7 in R7C12, R78C3 and R9C23, locked for N7

3b. Min R7C3 = 6 -> max R6C23 = 9, no 8,9 in R6C2, no 9 in R6C3

3c. Min R8C3 = 6 -> max R678C4 = 10, no 8,9 in R678C4

4. 45 rule on N9 2 innies R78C7 = 11 = {29/38/47} (cannot be {56} which clashes with R23C7), no 1,5,6

4a. R78C7 = 11 -> R67C7 cannot be 11 (CCC) -> no 7 in R6C8

5. 45 rule on R1234 3 innies R4C159 = 20 = {389/479/569/578}, no 1,2

6. 45 rule on N6 2 outies R37C7 = 1 innie R5C7 + 8, IOU no 8 in R7C7, clean-up: no 3 in R8C7 (step 4)

6a. Max R37C7 = 16 -> max R5C7 = 8

7. 45 rule on N4 2 outies R37C3 = 1 innie R5C3 + 5, max R37C3 = 12 -> max R5C3 = 7

7a. 42(7) cage at R4C5 = {3456789}, 9 locked for N5

8. 45 rule on C789 3 innies R258C7 = 17

8a. R23C7 = 14 (step 1c) -> R28C7 cannot be 14 (CCC) -> no 3 in R5C7

8b. R78C7 = 11 (step 4) -> R28C7 cannot be 11 (CCC) -> no 6 in R5C7

8b. R258C7 = {278/458}, no 9, 8 locked for C7, clean-up: no 5 in R3C7 (step 1b), no 2 in R7C7 (step 4)

8c. 2 of {278} must be in R8C7 -> no 7 in R8C7, clean-up: no 4 in R7C7 (step 4)

8d. 5 in R3 only in R3C456, locked for N2

8e. 42(7) cage at R4C5 = {3456789}, 3 locked for N5

8f. Max R34C7 = 16 -> min R4C8 = 3

9. 16(4) cage at R2C6 = {1258/1348/1357/1456} (cannot be {1267/2347} because R2C7 only contains 5,8, cannot be {2356} because 2,3,6 only in R24C6), 1 locked for C6

9a. 5 of {1258} must be in R3C6, 5 of {1357/1456} must be in R2C7 -> no 5 in R4C6

10. 45 rule on N2 4(2+2) outies R2C37 + R4C46 = 14, min R2C37 = 6 -> max R4C46 = 8, no 8 in R4C4

10a. R2C3 = {13}, R2C7 = {58}, R23C7 (step 1b) = [59/86], 16(4) cage at R2C6 (step 9) = {1258/1348/1357/1456}

10b. R2C37 + R4C46 = [1571/1841] (both because 16(4) cage must contain 1)/[3551/3812/3821] (cannot be [3524/3542] because no valid combinations for 16(4) cage with R3C6 = {457}) -> R4C4 = {12457}, R4C6 = {12}, 1 in R4C46, locked for R4 and N5

10c. 3,6 of 16(3) cage = {1348/1456} must be in R2C6 -> no 4 in R2C6

[Now analysing further which permutations of 19(4) cage at R2C3 and 16(4) cage at R2C6 fit with the possible permutations for R2C37 + R4C46]

10d. R2C37 + R4C46 = [1571/1841/3812] (cannot be [3551] because 19(4) cage = [3295] clashes with R23C7 = [59] and 19(4) cage = [3475] clashes with 16(4) cage = [6541], cannot be [3821] because 19(4) cage = [3952] clashes with 16(4) cage = [2851]), no 2,5 in R4C4

10e. R2C37 + R2C46 + R3C46 + R4C46 = [15635771/18935441/38619512] (cannot be other permutations which clash with R23C7 or the 19(4) cage and 16(4) cage clash with each other) -> R2C4 = {69}, R2C6 = {13}, R3C4 = {59}

10f. R2C37 + R2C46 + R3C46 + R4C46 = [15635771/38619512] (cannot be [18935441] because R23C7 = [86] and then R3C67 = [46] clashes with R3C89) -> R2C4 = 6, R3C6 = {57}, R4C4 = {17}, 5 in R3C46, locked for R3

10g. Naked pair {13} in R2C36, locked for R2

10h. 18(3) cage at R1C9 (step 1d) = {279/378}

10i. 3 of {378} must be in R1C9 -> no 8 in R1C9

10j. 8 in R1 only in R1C456, locked for N2

10k. 10(3) cage at R3C3 = {136/145/235} (cannot be {127} which clashes with R4C46, ALS block), no 7

11. 45 rule on N4 5 innies R46C2 + R456C3 = 20 = {12458/23456} (cannot be {12368} because 15(3) cage at R6C2 = [186] clashes with 6 in R45C3, cannot be {12467} because 15(3) cage cannot be [177] and 15(3) cage = [168], R78C3 = [87] clashes with 7 in R5C3, cannot be {13457} because R46C2 = [31] doesn’t provide any combinations for 10(3) cage at R3C3), no 7, 2,4,5 locked for N4

11a. R3C3 = {13} and 15(3) cage at R6C2 -> {12458/23456} can only be {25}[418]/{25}[634]/[364]{25} (cannot be [365]{24} because R78C3 = [96] clashes with R4C3 = 6, other combinations aren’t compatible with 10(3) cage at R3C3 or 15(3) cage at R6C2) -> 4 in R56C3, locked for C3 and N4, R5C3 = {46}, clean-up: no 9 in R1C2

11b. R6C23 = [18/25/34/52] -> R7C3 = {68}, clean-up: no 6,8 in R8C3 (step 3)

11c. R1C3 = {679}, R78C3 = [69/78] -> combined half cage R178C3 must contain 7, locked for C3, clean-up: no 1 in R9C2

[Cracked. The rest is fairly straightforward. No clean-ups in the remaining steps.]

12. 19(4) cage at R2C3 = [1657/3691]

12a. R8C3 = {79} -> 16(4) cage at R6C4 = {234}7/{124}9 (cannot be {135}7 which clashes with 19(4) cage) -> R678C4 = {124/234}, 2,4 locked for C4

12b. 45 rule on N8 4(2+2) outies R6C46 + R8C37 = 20

12c. R8C3 is odd, R6C4 and R8C7 are both even -> R6C6 must be odd = {57}

12d. Naked pair {57} in R36C6, locked for C6

12e. 6,8 in N5 only in 42(7) cage at R4C5 -> no 6 in R5C3, no 8 in R5C7

12f. R5C3 = 4, R6C4 = 4 (hidden single in N5), clean-up: no 3 in R6C2 (step 11b)

12g. R6C46 + R8C37 = 20, min R6C46 + R8C4 = 16 -> max R8C7 = 4

13. R2C7 = 8 (hidden single in C7) -> R234C6 = 8 = [152], R6C6 = 7, R4C4 = 1, R23C3 = [31], R3C47 = [96]

13a. Naked pair {23} in R78C4, locked for C4 and N8, R159C4 = [857], R5C7 = 7

13b. R678C4 = 4{23} = 9 -> R8C3 = 7 (cage sum), R7C3 = 8 (step 3)

13c. R3C3 = 1 -> R4C23 = 9 = [36]

13d. R7C3 = 8 -> R6C23 = 7 = {25}, locked for R6

13e. R1C3 = 9 -> R1C2 = 4, R1C56 = [23], R23C5 = [47], R1C9 = 7

13f. R4C1 = 7 (hidden single in N4)

13g. R7C8 = 7 (hidden single in N9) -> R7C9 = 4, R8C7 = 2 -> R7C7 = 9 (step 4)

13h. R67C6 = [76], R8C7 = 2 -> R8C6 = 8 (cage sum)

13i. R3C7 = 6, R4C7 = 4 (hidden single in C7) -> R4C8 = 9 (cage sum)

13j. R9C78 = {35} (only remaining combination), locked for R9 and N9

13k. R78C4 = [23], R7C1 = 3 (hidden single in R7) -> R7C2 = 5

13l. R5C6 = 9, naked pair {18} in R5C12, locked for R5 and N4

13m. R9C9 = 8 (hidden single in R9)

13n. R6C8 = 8 (hidden single in N6), R7C7 = 9 -> R6C7 = 1 (cage sum)

and the rest is naked singles.