Prelims
a) R1C67 = {12}
b) R1C89 = {39/48/57}, no 1,2,6
c) R45C1 = {17/26/35}, no 4,8,9
d) R6C78 = {29/38/47/56}, no 1
e) R7C56 = {15/24}
f) 7(2) cage at R8C2 = {16/25/34}, no 7,8,9
g) R8C45 = {69/78}
h) 17(2) cage at R8C9 = {89}
i) 19(3) cage at R2C4 = {289/379/469/478/568}, no 1
j) 10(3) cage at R2C6 = {127/136/145/235}, no 8,9
k) 28(4) cage at R2C7 = {4789/5689}, no 1,2,3
l) 35(5) cage at R1C3 = {56789}
m) 17(5) cage at R5C9 = {12347/12356}, no 8,9
1a. Naked pair {12} in R1C67, locked for R1
1b. Naked pair {89} in 17(2) cage at R8C9, locked for N9
1c. Killer pair 8,9 in R8C45 and R8C9, locked for R8
1d. 10(3) cage at R2C6 = {136/145/235} (cannot be {127} which clashes with R1C6), no 7
1e. Killer pair 1,2 in R1C6 and 10(3) cage, locked for C6, clean-up: no 4,5 in R7C5
1f. 10(3) cage = {136/235} (cannot be {145} which clashes with R7C6), no 4, 3 locked for C6
1g. 4 in N2 only in R23C4, locked for C4
1h. 19(3) cage at R2C4 = {469/478}, no 2,3,5
1i. R123C6 = {123} (hidden triple in N2), locked for C6
1j. 5 in N2 only in R1C45 + R23C5, locked for 35(7) cage at R1C3, no 5 in R1C3
1k. 17(5) cage at R5C9 = {12347/12356}, CPE no 1,2,3 in R9C9
1l. 34(7) cage at R1C1 must contain 1, CPE no 1 in R2C2
2a. 45 rule on complete grid 4(1+3) innies R2C2 + R234C3 = 14
2b. Min R234C3 = 6 -> max R2C2 = 8
2c. Min R2C2 + R23C3 = 6 -> max R4C2 = 8
3a. 45 rule on N7 1 innie R7C1 = 2 outies R79C4 + 2
3b. Min R79C4 = 4 (cannot be {12} which clashes with R7C5) -> min R7C1 = 6
3c. Max R7C1 = 9 -> max R79C4 = 7, no 7,8,9 in R79C4
3d. Max R79C4 + R7C56 = 13 must contain 1, locked for N8
[Ed pointed out CPE no 1 in R7C3. He’s better at spotting those less obvious CPEs than I am!]
3e. 45 rule on N78 1 innie R7C1 = 1 outie R9C7 + 4, R7C1 = {6789} -> R9C7 = {2345}
3f. 45 rule on N9 2 outies R56C9 = 1 innie R9C7 + 1, max R9C7 = 5 -> max R56C9 = 6, no 6,7 in R56C9
3g. 8,9 in R7 only in R7C123, locked for N7
4a. 45 rule on C6789 2 innies R67C6 = 1 outie R9C5 + 6, IOU no 6 in R6C6
4b. Min R67C6 = 9 -> min R9C5 = 3
4c. Max R67C6 = 14 -> min R9C5 = 8
4d. R67C6 cannot total 10 -> no 4 in R9C5
4e. 4 in C5 only in R456C5, locked for N5
5a. 22(4) cage at R8C6 = {2569/2578/3469/3478} (cannot be {2389/2479/3568/4567} which clash with R8C45)
5b. 2 of {2569/2578} must be in R9C7 -> no 5 in R9C7, clean-up: no 9 in R7C1 (step 3e)
5c. Killer quad 6,7,8,9 in 22(4) cage and R8C45, locked for N8
5d. Max R9C7 = 4 -> max R56C9 (step 3f) = 5, no 5 in R56C9
6a. 45 rule on N2356789 2 innies R6C4 + R7C1 = 1 outie R1C3 + 2
[Or 45 rule on N14 using step 2a for the 4 zero cells.]
6b. Max R1C3 = 9 -> max R6C4 + R7C1 = 11, min R7C1 = 6 -> max R6C4 = 5
6c. 36(6) cage at R5C3 must contain 9, locked for N4
7a. 45 rule on N3 2 outies R4C79 = 2 innies R1C7 + R3C8 + 9
7b. Max R4C79 = 17 -> max R1C7 + R3C8 = 8 -> max R3C8 = 7
7c. Min R1C7 + R3C8 = 3 -> min R4C79 = 12, no 1,2 in R4C9
8a. 45 rule on N36 1 outie R5C6 = 3(1+2) innies R1C7 + R56C9 + 3
8b. Min R1C7 + R56C9 = 1 + 3 -> min R5C6 = 7
8c. 45 rule on N5 4 innies R4C46 + R5C6 + R6C4 = 24 = {1689/2589/2679/3579/3678} -> R6C4 = {123}
8d. R4C6 = {56} -> no 6 in R4C4
8e. 19(3) cage at R2C4 (step 1h) = {469/478}
8f. 9 of {469} only in R4C4 -> no 9 in R23C4
8g. 9 in N2 only in R1C45 + R23C5, locked for 35(7) cage at R1C3, no 9 in R1C3
9a. R7C1 = R9C7 + 4 (step 3e), R6C4 + R7C1 = R1C3 + 2 (step 6a), 22(4) cage at R8C6 (step 5a) = {2569/2578/3469/3478}, R4C46 + R5C6 + R6C4 = 24 (step 8c)
9b. Consider placements for R9C7 = {234}
R9C7 = 2 => R1C67 = [21], R23C6 = {13} => R4C6 = 6 (cage sum)
or R9C7 = 3 => 22(4) cage must contain 4, locked for N8 => R7C6 = 5, R4C6 = 6
or R9C7 = 4, 22(4) cage must contain 3, R7C6 = 4 (hidden single in N8) => R7C5 = 2, R79C4 = {15}, 1 locked for C4, R7C1 = 8 => R1C3 + R6C4 = [82] (only possibility) => R23C4 = {48} (hidden pair in N2), R4C4 = 7 (cage sum), R46C4 = [72] = 9 => R46C6 = 15 = [69]
-> R4C6 = 6, R23C6 = 4 = {13}, R1C67 = [21], clean-up: no 2 in R5C1
9c. 1 in C9 only in R567C9, locked for 17(5) cage at R5C9, no 1 in R7C8
9d. 14(3) cage at R2C9 = {239/248/257/347/356}
9e. R56C9 = R9C7 + 1 (step 3f)
9f. R9C7 = {234} -> R56C9 = 3,4,5 = {12/13/14} (cannot be {23} which clashes with 14(3) cage), 1 locked for C9 and N6
9g. R8C8 = 1 (hidden single in N9) -> R7C7 + R9C9 = 11 = {47/56}, no 1,2, clean-up: no 6 in R9C1
10a. R6C4 + R7C1 = R1C3 + 2 (step 6a), R7C1 = R79C4 + 2 (step 3a)
10b. Consider placements for R7C1 = {678}
R7C1 = 6 => R79C4 = 4 = {13}, 3 locked for C4
or R7C1 = 7 => R79C4 = 5 = {23}, 3 locked for C4
or R7C1 = 8 => R1C3 + R6C4 = [71/82]
-> R6C4 = {12}
[The final key step. The rest is long but fairly straightforward.]
10c. R4C46 + R5C6 + R6C4 (step 8c) = {1689/2589/2679}, 9 locked for N5
10d. 36(6) at R5C3 = {156789/246789} (cannot be {345789} because R6C4 only contains 1,2)
10e. R6C4 = {12} -> 1,2 in R5C3 + R6C123
10f. 36(6) = {156789/246789}, CPE no 6,7 in R45C1, clean-up: no 1,2 in R45C1
10g. Naked pair {35} in R45C1, locked for N4, clean-up: no 2,4 in R8C2
10h. 36(6) = {246789} -> R6C4 = 2, clean-up: no 9 in R6C78
10i. R4C46 + R5C6 + R6C4 = 24 (step 8c), R4C6 + R6C4 = [62] = 8 -> R4C4 + R5C6 = 16 = {79}, 7 locked for N5
10j. 19(3) cage at R2C4 = {469/478}, R4C4 = {79} -> no 7 in R23C4
10k. R7C5 = 2 (hidden single in N8) -> R7C6 = 4, clean-up: no 7 in R9C9 (step 9g)
10l. 1 in N8 only in R79C4, locked for C4
10m. R79C4 = {13/15} = 4,6 -> R7C1 = {68}, clean-up: no 3 in R9C7 (step 3e)
10n. R6C4 = 2 -> R1C3 = R7C1 = {68}
10o. 2 in N9 only in R89C7, locked for C7
10p. 3 in N9 only in R7C89 + R8C7, locked for 17(5) cage at R5C9, no 3 in R56C9
10q. 9 in R6 only in R6C123, locked for 36(6) cage, no 9 in R5C3
10r. 9 in R5 only in R5C567, locked for 28(5) cage at R3C8, no 9 in R4C8
10s. 14(3) cage at R23C9 = {239/257/347/356} (cannot be {248} which clashes with R56C9, ALS block), no 8
11a. R67C6 = R9C5 + 6 (step 4a), R7C6 = 4 -> R6C6 + R9C5 = [53/86]
11b. Hidden killer pair 8,9 in R8C45 and R9C6, R8C45 contains only of 8,9 -> R9C6 = {89}
11c. 7 in R9 only in R9C23, locked for N7
11d. 7 in R7 only in R7C789, locked for N9
12a. R4C79 = R1C7 + R3C8 + 9 (step 7a)
12b. R4C79 cannot be {79} which clashes with R4C4 -> R1C7 + R3C8 cannot total 7, no 6 in R3C8
12c. 28(5) cage at R3C8 contains 2 for C8 and 9 for R5 = {23689/24589/24679}
12d. {23689} must have one of 2,3 in R3C8, {24589/24679} must have one of 2,4 in R3C8 (both of 2,4 in R4C8 + R5C78 would clash with R56C9, ALS block) -> R3C8 = {234}
12e. R6C78 = {38/47} (cannot be {56} which clashes with 28(5) cage), no 5,6
12f. 6 in R6 only in R6C123, locked for 36(6) cage -> R7C1 = 8
12g. R6C78 = {38} (cannot be {47} which clashes with R6C123, ALS block), locked for R6 and N6
12h. R5689C6 = [9578], naked pair {69} in R8C45, locked for R8, 6 locked for N8, R9C5 = 3 -> R9C7 = 4 (cage sum), clean-up: no 7 in R7C7 (step 9g), no 3 in R8C2, no 1 in R9C1
12i. R8C7 = 2 (hidden single in N9), R7C89 = {37} (hidden pair in N9), naked pair {14} in R56C9, 4 locked for C9 and N6
12j. R4C4 = 7 -> naked pair {59} in R4C79, 5 locked for R4 and N6, R4C8 = 2
12k. R4C79 = {59} = 14 -> R1C7 + R3C8 = 5 = [14]
12l. Naked pair {67} in R5C78, locked for R5 -> R5C3 = 4
12m. R4C4 = 7 -> R23C4 = 12 = [48]
13a. R8C3 = 3 (hidden single in R8), R79C4 = {15} = 6 -> R79C3 = 13 = [67]
13b. R8C1 = 4, 7(2) cage at R8C2 = [52] -> R79C2 = [91], R45C2 = [82]
13c. R16C3 = [89], R4C3 = 1, R23C3 = {25} = 7 -> R2C2 = 6 (cage sum for zero cages, step 2a)
13d. Killer pair 7,9 in R1C1 and R1C89, locked for R1
13e. Naked pair {56} in R1C45, locked for N2, 5 locked for R1, clean-up: no 7 in R1C89
and the rest is naked singles.