Prelims
a) R1C89 = {16/25/34}, no 7,8,9
b) R23C5 = {19/28/37/46}, no 5
c) R2C89 = {18/27/36/45}, no 9
d) R4C67 = {49/58/67}, no 1,2,3
e) R5C23 = {29/38/47/56}, no 1
f) R5C89 = {39/48/57}, no 1,2,6
g) R78C3 = {89}
h) R7C56 = {14/23}
i) R89C1 = {19/28/37/46}, no 5
j) R89C2 = {17/26/35}, no 4,8,9
k) R89C5 = {49/58/67}, no 1,2,3
l) 10(3) cage at R6C2 = {127/136/145/235}, no 8,9
m) 10(3) cage at R8C4 = {127/136/145/235}, no 8,9
n) 19(3) cage at R8C6 = {289/379/469/478/568}, no 1
1a. Naked pair in R78C3, locked for C3 and N7, clean-up: no 2,3 in R5C2, no 1,2 in R89C1
1b. 45 rule on N7 3 innies R7C12 + R9C3 = 10 = {127/145/235} (cannot be {136} which clashes with R89C1), no 6
1c. 45 rule on C12 2 outies R15C3 = 9 = {27/36/45}, no 1
1d. 45 rule on N1 3 innies R2C3 + R3C23 = 10 = {127/136/145/235}, no 8,9
1e. 10(3) cage at R6C2 = {127/145/235} (cannot be {136} = 6{13} which clashes with R7C56), no 6
1f. {127} cannot be 7{12} which clashes with R7C56), no 7 in R6C2
1g. 10(3) cage at R8C4 = {127/136/145/235}
1h. 7 of {127} must be in R89C4 (R89C4 cannot be {12} which clashes with R7C56), no 7 in R9C3
[Alternatively 45 rule on N7 1 innie R9C3 = 1 outie R6C2]
1i. 45 rule on R12 2 outies R3C15 = 8 = {17/26}/[53], no 4,8,9, no 3 in R3C1, clean-up: no 1,2,6 in R2C5
1j. 45 rule on N9 2 innies R7C9 + R9C7 = 10 = [19]/{28/37/46}, no 5, no 9 in R7C9
1k. 45 rule on N5 3 innies R4C46 + R6C4 = 20 = {389/479/569/578}, no 1,2
1l. 45 rule on N5 2 innies R46C4 = 1 outie R4C7 + 7 -> no 7 in R6C4 (IOU)
1m. 45 rule on R89 3 innies R8C378 = 14
1n. R8C3 = {89} -> R8C78 = 5,6 = {14/15/23/24}
1o. 45 rule on R89 3 outies R7C378 = 22 = {589/679}, 9 locked for R7
2a. 45 rule on N6 3(2+1) outies R37C9 + R4C6 = 23
2b. Max R37C9 = 17 -> min R4C6 = 6, clean-up: no 8,9 in R4C7
2c. Max R3C9 + R4C6 = 18 -> R7C9 = {678} (5 already eliminated), R9C7 = {234} (step 1i)
2d. R7C378 = 22 (step 1o), R7C9 = {678} -> R7C3789 = 28,29,30 = {5689/5789/6789}, 8 locked for R7
2e. 45 rule on N3 1 innie R3C9 = 2 outies R23C6 + 4
2f. Min R23C6 = 3 -> min R3C9 = 7
2g. Max R3C9 = 9 -> max R23C6 = 5 = {12/13/14/23}
2h. 45 rule on N23 2 innies R3C49 = 1 outie R2C3 + 14
2i. Max R3C49 = 17 -> max R2C3 = 3
2j. Min R3C49 = 15 -> R3C4 = {6789}
2k. 19(3) cage at R8C6 = {289/379/469/478} (cannot be {568} because R9C7 only contains 2,3,4), no 5
2l. R9C7 = {234} -> no 2,3,4 in R89C6
3a. 45 rule on N69 4(1+3) outies R3C9 + R489C6 = 32
3b. Max R489C6 = 24 -> min R3C9 = 8
3c. R3C9 = {89} -> R4C489 = 23,24 = {689/789}, 8,9 locked for C6
4a. 45 rule on N6 2 outies R37C9 = 1 innie R4C7 + 10 -> R4C7 less than either of R37C9
4b. R3C9 ‘sees’ all except R7C7 in C7 (R89C7 only contain 1,2,3,4), R3C9 = {89} -> R7C7 = {89}
4c. Naked pair {89} in R7C37, 8 locked for R7, R7C8 = 5 (step 1o), clean-up: no 2 in R1C9, no 4 in R2C9, no 7 in R5C9, no 2 in R9C7 (step 1j)
4d. R7C78 = [85/95] = 13,14 -> R8C78 = 5,6 = {14/23/24}
4e. Killer pair 3,4 in R8C78 and R9C7, locked for N9
4f. 19(3) cage at R8C6 (step 2k) = {379/469/478}
4g. Hidden killer pair 8,9 in R4C6 and R89C6 for C6, R89C6 contains one of 8,9 -> R4C6 = {89}, R4C7 = {45}
4h. 19(3) cage = {379/469} (cannot be {478} = {78}4 which clashes with R4C67 = [85/94]), 9 locked for C6 and N8 -> R4C67 = [85], clean-up: no 7 in R5C8
4i. R5C23 = {47/56}/[92] (cannot be [83] which clashes with R5C89), no 3,8, clean-up: no 6 in R1C3 (step 1c)
4j. R89C5 = {58} (cannot be {67} which clashes with R89C6), locked for C5, 5 locked for N8, clean-up: no 2 in R3C5, no 6 in R3C1 (step 1i)
4k. R4C6 = 8 -> R46C4 = 12 (step 1k) = [39/75/93]
4l. 18(3) cage at R6C8 = {279/369/468} (cannot be {189} because R7C9 only contains 6,7, cannot be {378} which clashes with R5C89), no 1
4m. R7C9 = {67} -> no 6,7 in R6C89
4n. 1,6,7 in N6 only in 25(5) cage at R3C9 = {12679/13678}, no 4
4o. R3C9 = {89} -> no 8,9 in R4C89 + R56C7
[An unexpected breakthrough step.]
5a. 45 rule on N8 2 outies R9C37 = 1 innie R7C4 + 2
5b. 45 rule on N7 1 outie R6C2 = 1 innie R9C3
5c. Consider whether or not R6C2 and R9C3 contain 5
R6C2 = 5 => R6C34 cannot total 8 => no 6 in R7C4
or R6C2 and R9C3 don’t contain 5 => max R9C37 = 7 => max R7C4 = 5
-> no 6 in R7C4
5d. R7C9 = 6 (hidden single in R7), R9C7 = 4 (step 1j), 16(3) cage at R8C9 = {178} (only remaining combination), 1,8 locked for N9, clean-up: no 1 in R1C8, no 3 in R2C8, no 4 in R6C2, no 6 in R8C1
5e. R7C3 = 9, 8 in C7 only in R123C7, locked for N3 -> R3C9 = 9 (or step 4b in reverse), clean-up: no 1 in R2C89, no 3 in R5C8
5f. R7C9 = 6 -> R6C89 = 12 = [48/84/93]
5g. Naked quad {3489} in R56C89, 3 locked for C9 and N6, clean-up: no 4 in R1C8, no 6 in R2C8
5h. R1C89 = [34/61] (cannot be [25] which clashes with R2C89)
5i. R2C9 = 5 (hidden single in C9) -> R2C8 = 4, R1C89 = [61], clean-up: no 6 in R3C5, no 2 in R3C1 (step 1i)
5j. R9C7 = 4 -> R89C6 = 15 = {69}, 6 locked for C6 and N8
5k. 7 in N8 only in R789C4, locked for C4, clean-up: no 5 in R6C4 (step 4k)
5l. Naked pair {39} in R46C4, locked for C4 and N5
6a. 3 in N8 only in R7C56 = {23}, locked for R7, 2 locked for N8
6b. R3C9 = R23C6 + 4 (step 2e)
6c. R3C9 = 9 -> R23C6 = 5 = [14] (cannot be {23} which clashes with R7C6), clean-up: no 9 in R2C5, no 7 in R3C1 (step 1i)
6d. Naked pair {37} in R23C5, locked for C5 and N2 -> R7C56 = [23], R1C5 = 9
6e. R2C6 = 1 -> R12C7 = 11 = {38}, locked for N3, 3 locked for C7 -> R8C78 = [23], R3C78 = [72], R23C5 = [73], R3C1 = 5 (step 1i), clean-up: no 4 in R5C3 (step 1c), no 7 in R5C2, no 7 in R9C1, no 5,6 in R9C2
6f. Naked pair {16} in R3C23, 6 locked for R3 and N1 -> R3C4 = 8
6g. Naked pair {25} in R1C46, 2 locked for R1, N2 and 25(5) cage at R1C4 -> R2C34 = [36], clean-up: no 6,7 in R5C3, no 4,5 in R5C2
7a. Naked pair {78} in R89C9, locked for C9 and N9
7b. R9C8 = 1 -> R9C4 = 7, R8C4 + R9C3 = 3 = [12], R5C3 = 5 -> R5C2 = 6, R1C3 = 4 (step 1c), R6C2 = 2 (step 5b)
7c. R3C23 = [16], R4C8 = 7 -> R4C3 = 1, R4C4 = 9 (cage sum)
7d. R5C7 = 1, R5C5 = 4, R5C9 = 3 -> R5C8 = 9
and the rest is naked singles.