Prelims

a) R12C5 = {18/27/36/45}, no 9

b) R67C5 = {39/48/57}, no 1,2,6

c) R89C1 = {59/68}

d) R89C9 = {69/78}

e) 24(3) cage at R1C4 = {789}

f) 22(3) cage at R1C6 = {589/679}

g) 10(3) cage at R5C3 = {127/136/145/235}, no 8,9

h) 11(3) cage at R8C8 = {128/137/146/236/245}, no 9

i) 14(4) cage at R2C1 = {1238/1247/1256/1346/2345}, no 9

1a. 45 rule on N2 2 outies R2C37 = 3 outies R3C123 + 10 -> R2C37 = 16,17 = {79/89}, 9 locked for R2, R3C123 = 6,7 = {123/124}, 1,2 locked for R3, N2 and 21(5) cage at R3C4, clean-up: no 7,8 in R12C5

1b. 24(3) cage at R1C4 and 22(3) cage at R1C6 must both contain 9 -> 9 in R1C46, locked for R1

1c. 45 rule on N2 4(2+2) outies R2C37 + R45C5 = 31, R2C37 = 16,17 -> R45C5 = 14,15 = {59/68/69/78}, no 3,4

[Killer pair 3,4 in R12C5 and R3C123 also make these eliminations)

1d. Naked triple {789} in 24(3) cage at R1C4, CPE no 7,8 in R2C6

1e. 22(3) cage at R1C6 = {589/679}

1f. R2C6 = {56} -> no 5,6 in R1C6

1g. Naked triple {789} in R2C347, locked for R2

1h. Killer half-pair in R12C5 and R67C5, 3 locked for C5

2a. 45 rule on C6789 3 innies R389C6 = 8 = {125/134}, 1 locked for C6

2b. 45 rule on C1234 3 innies R389C4 = 15, max R3C4 = 4 -> min R89C4 = 11, no 1

2c. 45 rule on N7 2(1+1) outies R6C1 + R7C4 = 12, no 1,2

2d. 45 rule on N9 2(1+1) outies R6C9 + R7C6 = 11, no 1 in R6C9

2e. 45 rule on R89 2 innies R8C37 = 12 = {39/48/57}, no 1,2,6

2f. 45 rule on C12 2 outies R19C3 = 7 = {16/25/34}, no 7,8,9

2g. 45 rule on C89 2 outies R19C7 = 5 = {14/23}

2h. 19(5) = {12349/12358/12367/12457/13456}, 1 locked for N3

3a. R12C5 = {36/45}, R45C5 (step 1c) = {59/68/69/78}

3b. Consider combinations for R12C5

R12C6 = {36}, locked for N2, 6 locked for C5 => R3C123 = {124} = 7, R45C5 = 14 = {59}

or R12C5 = {45}, locked for C5 and N2 => R3C123 = {123} = 6, R67C5 = {39}, 9 locked for C5, R45C5 = 15 = {78}

-> R45C5 = {59/78}, no 6

3c. 5 in R12C6 = {45} or R45C5 = {59}, locked for C5, clean-up: no 7 in R67C5

3d. Killer pair 3,4 in R12C5 and R67C5, locked for C5

3e. Killer pair 8,9 in R45C5 and R67C5, locked for C5

4a. 45 rule on N1 2 innies R23C3 = 2 outies R4C12 + 10

4b. Min R4C12 = 3 -> min R23C3 = 13, no 3 in R3C3

4c. Max R23C3 = 17 -> max R4C12 = 7, no 7,8 in R4C12

5a. 45 rule on N7 2 innies R7C12 = 1 outie R7C4 + 6 -> no 6 in R7C12 (IOU)

5b. 45 rule on N9 2 innies R7C89 = 1 outie R7C6 + 4 -> no 4 in R7C89 (IOU)

6a. 18(3) cage at R6C1 = {279/378/459/468} (cannot be {189/369/567} which clash with R89C1), no 1

6b. 15(3) cage at R6C9 = {159/249/258/348/357/456} (cannot be {168/267} which clash with R89C9)

7a. R389C4 = 15 (step 2b), R389C6 (step 2a) = {125/134}

7b. 24(3) cage at R1C4 and 22(3) cage at R1C6 -> R2C46 = [75/86]

7c. Consider combinations for R2C46

R2C46 = [75] => R389C6 = {134}

or R2C46 = [86], R12C5 = {45}, R3C4 = 3 => R89C4 = 12 = {57}, 5 locked for N8 => R389C6 = {134}

or R2C46 = [86], R12C5 = {45}, R3C6 = 3 => R389C6 = {134}

-> R389C6 = {134}, locked for C6, clean-up: no 7,8 in R6C9 (step 2d)

7d. 45 rule on N8 3 innies R7C456 = 19 = {289/379/469/478/568}

7e. Hidden killer pair 8,9 in R789C4 and R7C56 for N8, R7C56 cannot contain both of 8,9 (because no 2 in R7C4) -> R789C4 must contain one of 8,9 (cannot be both because of R12C4)

7f. Killer triple 7,8,9 in R12C4 and R789C4, locked for C4, 7 locked for C4, N2 and 24(3) cage at R1C4, no 7 in R2C3, clean-up: no 5 in R6C1 (step 2c)

8a. R389C4 = 15 (step 2b) = {149/158/249/258/348} (cannot be {456} because [456] which clashes with 14(3) cage at R8C4 = [563] and cannot be [465] because 14(3) cage = [671] clashes with R3C456 = [421]), no 6 in R89C4

8b. Killer triple 7,8,9 in R12C4 and R89C4, locked for C4, clean-up: no 3,4 in R6C1 (step 2c)

8c. 14(3) cage at R8C4 = {149/239/248/347/356} (cannot be {158} because 5,8 only in R8C4, cannot be {257} because no 2,5,7 in R8C6)

8d. 1 of {149} must be in R8C5 -> no 1 in R8C6

8e. 12(3) cage at R9C4 = {129/138/147/156/237/246} (cannot be {345} because no 3,4,5 in R9C5)

8f. 2,8 of {138/237} must be in R9C4 -> no 3 in R9C4

8g. R7C456 (step 7d) = {379/478/568} (cannot be {289} because no 2,8,9 in R7C4, cannot be {469} which clashes with 14(3) cage), no 2, clean-up: no 9 in R6C9 (step 2d)

8h. 12(3) cage = {129/147/156/237/246} (cannot be {138} which clashes with R7C456), no 8

8i. 12(3) cage = {129/147/156/237} (cannot be {246} which clashes with R7C456 = {478/568} and clashes with R7C456 = {379} + 14(3) cage then = {248})

8j. R7C456 = {379/478/568} -> R7C5 = {89}, R7C6 = {567}, R6C5 = {34}, clean-up: no 2,3 in R6C9 (step 2d)

8k. R7C456 = {379/568} (cannot be {478} = [487] because R67C5 = [48] clashes with R6C9 + R7C6 = [47], step 2d), no 4, clean-up: no 8 in R6C1 (step 2c)

[With hindsight, having glanced at the archive, 45 rule on R89 4 outies R7C3467 = 16, R7C456 = 19 -> R7C5 = R7C37 + 3 would have got to step 8j more quickly.]

9a. R12C5 = {36/45}, R2C46 (step 7b) = [75/86]

9b. Consider permutations for R67C5 = [39/48]

R67C5 = [39] => 8 in N8 only in R89C4, locked for C4 => R2C4 = 7

or R67C5 = [48] => R12C5 = {36}, 6 locked for N2 => R2C6 = 5

-> R2C46 = [75], clean-up: no 4 in R12C5

[It gets easier from here.]

9c. Naked pair {36} in R12C5, locked for C5, 3 locked for N2 -> R67C5 = [48], R6C9 + R7C6 (step 2d) = [56]

9d. 3 in C6 only in R89C6, locked for N8 -> R7C4 = 5, R6C1 = 7 (step 2c)

9e. R6C1 = 7 -> R7C12 = 11 = {29}, locked for R7 and N7, clean-up: no 5 in R1C3 (step 2f), no 5 in R89C1

9f. R6C9 = 5 -> R7C89 = 10 = {37}, locked for N9, 3 locked for R7, clean-up: no 2 in R1C7 (step 2g), no 8 in R89C9

9g. R19C7 (step 2g) = [32] (cannot be {14} which clashes with R7C7) -> R12C5 = [63], clean-up: no 4,6 in R9C3 (step 2f)

9h. 12(3) cage at R9C4 = {147} (only remaining combination) = [471], R3C6 = 4, 14(3) cage at R8C4 = [923], R1C46 = [89], R2C37 = [98], R89C9 = [69], R89C1 = [86], R3C9 = 7 -> R7C89 = [73], clean-up: no 1 in R1C3 (step 2f)

[Clean-ups for R8C37 (step 2e) were overlooked but that doesn’t matter]

9i. R9C23 = {35} -> R8C2 = 4 (cage sum), R78C3 = [17], R78C7 = [45], R89C8 = [18]

9j. 45 rule on N3 2 remaining innies R2C9 + R3C7 = 11 = [29]

9k. Naked triple {456} in R123C8, 4,6 locked for C8, 4 locked for N3 -> R1C9 = 1

9l. R23C9 = [27] -> R4C89 = 11 = [38], R5C9 = 4

9m. R3C3 = 8 (hidden single in C3)

9n. 45 rule on N1 2 remaining innies R23C1 = 7 = [43] -> R1C123 = [572], R23C2 = [16], R4C12 = 7 = [25]

9o. R56C3 = {36} -> R6C4 = 1 (cage sum)

9p. R4C4 = 6, R4C67 = [71], R3C7 = 9 -> R5C6 = 8 (cage sum)

and the rest is naked singles.