Prelims
a) R23C6 = {16/25/34}, no 7,8,9
b) 13(2) cage at R2C8 = {49/58/67}, no 1,2,3
c) R45C2 = {29/38/47/56}, no 1
d) R4C45 = {39/48/57}, no 1,2,6
e) R4C78 = {49/58/67}, no 1,2,3
f) 16(2) cage at R5C1 = {79}
g) R56C6 = {17/26/35}, no 4,8,9
h) 10(2) cage at R7C3 = {19/28/37/46}, no 5
i) 4(2) cage at R8C4 = {13}
j) R8C56 = {18/27/36/45}, no 9
k) 7(3) cage at R1C1 = {124}
l) 19(3) cage at R2C3 = {289/379/469/478/568}, no 1
m) 11(3) cage at R6C7 = {128/137/146/236/245}, no 9
n) 10(3) cage at R7C1 = {127/136/145/235}, no 8,9
o) 20(3) cage at R8C3 = {389/479/569/578}, no 1,2
p) 23(3) cage at R8C8 = {689}
Steps resulting from Prelims
1a. 7(3) cage at R1C1 = {124}, locked for N1
1b. 16(2) cage at R5C1 = {79}, locked for N4, clean-up: no 2,4 in R45C2
1c. 4(2) cage at R8C4 = {13}, locked for N8, clean-up: no 6,8 in R8C56
1d. 23(3) cage at R8C8 = {689}, locked for N9
[Two easy placements which the original version didn’t have.]
2a. 45 rule on N3 1 innie R1C9 = 2
2b. 45 rule on N7 1 innie R9C1 = 5
2c. 7(3) cage at R1C1 = {124}, 2 locked for R2, clean-up: 5 in R3C6
2d. 10(3) cage at R7C1 = {127/136}, no 4, 1 locked for N7, clean-up: no 9 in 10(2) cage at R7C3
2e. 7 of {127} must be in R78C1 (R78C1 cannot be {12} which clashes with R12C1, ALS block), no 7 in R7C2
2f. 20(3) cage at R8C3 = {389/479}, no 6
2g. 10(2) cage at R7C3 = {28/46} (cannot be {37} which clashes with 20(3) cage), no 3,7
2h. 9 in R7 only in R7C456, locked for N8
2i. 2 in N9 only in R789C7 + R7C8, CPE no 2 in R6C7
3. 20(3) cage at R8C3 + 23(3) cage at R8C8 total 43
3a. Max R9C2389 = 30 -> min R8C38 = 13, no 3 in R8C3
3b. Max R8C38 = 17 -> min R9C2389 = 26 -> R9C2389 = {3689/4679/4689/4789/6789} (cannot be {3789} because R9C23 must total at least 11)
3c. R9C46 = {26/27/28} (other combinations clash with R8C2389 and/or with R8C56), no 4, 2 locked for R9 and N8, clean-up: no 7 in R8C56
3d. Naked pair {45} in R8C56, locked for R8 and N8, clean-up: no 6 in R7C3
3e. R9C2389 = {3689/4679/4689/4789} (cannot be {6789} which clashes with R9C46)
3f. Killer triple 6,7,8 in R9C2389 and R9C46, locked for R9
3g. Min R7C56 = 13 -> max R7C7 = 5
4. 45 rule on N2 1 outie R2C3 = 1 innie R3C5 + 4, no 3 in R2C3, no 6,7,8,9 in R4C2
4a. 23(4) cage at R1C4 = {1589/1679/3479/3578} (cannot be {3569/4568} which clash with R23C6)
4b. 23(4) = {1679/3479/3578} (cannot be {1589} because 19(3) cage at R2C3 cannot be 6{67}), 7 locked for N2
4c. Consider placement of 2 in N2
R3C4 = 2 => R2C34 = {89}
or R3C5 = 2 => R2C3 = 6
or R3C6 => R2C6 = 5
-> no 5 in R2C3, no 1 in R3C5
5. Combined cages R23C6 + R56C6 + R8C6 = {16}{35}4/{34}{17}5/{34}{26}5/[52]{17}4, 4,5 locked for C6
6. 7(3) cage at R1C1 = {124}, 10(3) cage at R7C1 (step 2d) = {127/136}, 10(2) cage at R7C3 = [28/46/82], 20(3) cage at R8C3 (step 2f) = {389/479}
6a. Consider placement for 4 in N4
4 in R46C1 => R12C1 = {12}, locked for C1 => 10(3) cage = {136} (cannot be {127}, ALS block)
or 4 in R456C3 => 10(2) cage = {28}, locked for N7 => 10(3) cage = {136}
-> 10(3) cage = {136}, locked for N7, 10(2) cage = {28}, 20(3) cage at R8C3 = {479}, 4 locked for R9
6b. R35C1 = {79} (hidden pair in C1)
6c. 8 in C1 only in R46C1, locked for N4, clean-up: no 3 in R45C2
6d. Naked pair {56} in R45C2, locked for C2 and N4
6e. Max R45C3 = 7 -> min R3C3 = 5
6f. 1 in C3 only in R456C3, locked for N4
7. 45 rule on N6 1 innie R5C7 = 1 outie R7C8 + 4, no 1,2,3,4 in R5C7, no 7 in R7C8
7a. 2 in C7 only R789C7, locked for N9, clean-up: no 6 in R5C7
7b. 17(4) cage at R4C9 = {1259/1268/1349/1358/1367/2348/2357} (cannot be {1457/2456} which clash with R4C78)
8. R4C2 = {56}, R4C45 = {39/48/57}, R4C78 = {49/58/67} -> combined cage R4C24578 = 5{39}{67}/5{48}{67}/6{39}{58}/6{57}{49}, 5,6 locked for R4
9. 12(3) cage at R3C3 = {129/138/237/246/345} (cannot be {147} which clashes with 20(3) cage at R8C3, ALS block, cannot be {156} because 5,6 only in R3C3)
9a. 12(3) cage = {129/138/237/345} (cannot be {246} because R3789C3 = 68{79} clash with R2C3), no 6 in R3C3
9b. Consider combinations for 12(3) cage
12(3) cage = {129/138/237} => R12C3 = [56] (hidden pair in C3)
or 12(3) cage = {345} => R67C3 = [12] (hidden pair in C3) => R12C3 = {68} (hidden pair in C3)
-> R12C3 = [56/68/86], clean-up: no 3,5 in R3C5 (step 4)
9c. 19(3) cage at R2C3 = {289/469/568}, no 3
9d. R2C3 + R3C5 (step 4) = [62/84]
9e. {568} must be 6{58} (cannot be 8{56} because R23C4 + R3C5 = {56}4 clashes with R23C6)
9f. 19(3) cage = 6{49}/6{58}/[892], no 6 in R23C4
9g. 19(3) cage + R3C5 = 6{49}2/6{58}2/[892]4 -> 2 in R3C45, locked for R3, clean-up: no 5 in R2C6
9h. R23C6 = {16/34}, R56C6 = {17/26/35} -> combined cage R2356C6 = {16}{35}/{34}{17}/{34}{26}, 3 locked for C6
9i. 3 in N1 only in R13C2, locked for C2 -> R7C2 = 1, R78C1 = {36}, locked for C1, clean-up: no 5 in R5C7 (step 7)
9j. 17(4) cage at R4C9 (step 7b) = {1259/1268/1349/1358/1367/2348/2357}
9k. Killer triple 7,8,9 in R4C78, 17(4) cage and R5C7, locked for N6
9l. 11(3) cage at R6C7 = {146/236/245} = [164/425/524/614/623], no 3 in R6C7, no 3,4,5 in R6C8
9m. 3 in N6 only in 17(4) cage = {1349/1358/1367/2348/2357}
9n. 17(4) cage = {1349/1358/1367/2357} (cannot be {2348} which combined with R4C78 = {67} clashes with 11(3) cage)
9o. 17(4) cage = {1349/1358/1367} (cannot be {2357} which combined with R7C9 = {3457} clashes with 11(3) cage), no 2, 1 locked for N6
9p. R6C8 = 2 (hidden single in N6) -> 11(3) cage = [425/524/623], clean-up: no 6 in R5C6
10. 18(3) cage at R7C5 = {279/378/468/567) (cannot be {369} which clashes with R7C1)
Consider combinations for 18(3) cage
18(3) cage = {279} => R7C3 = 8, R7C4 = 6
or 18(3) cage = {378/468/567} => R7C4 = 9 (hidden single in R7)
-> R7C4 = {69}
10a. Killer triple 3,6,9 in R7C1, R7C4 and 18(3) cage, locked for R7, clean-up: no 7 in R5C7 (step 7)
10b. 11(3) cage at R6C7 (step 9p) = [425/524], no 6, CPE no 4,5 in R45C8 + R7C7, clean-up: no 8,9 in R4C7
10c. 18(3) cage = {279/378}, no 6, 7 locked for R7 and N8
10d. R9C23 = {47} (hidden pair in R9) -> R8C3 = 9
10e. 12(3) cage at R3C3 (step 9a) = {138/237/345}, 3 locked for C3
11. 20(4) cage at R3C5 = {1289/1478/2378/2468} (cannot be {1379/1568} because R3C5 only contains 2,4, cannot be {2567/3467} because R5C7 only contains 8,9, cannot be {2369} because 3,6 only in R5C5, cannot be {2459} which clashes with R8C5, cannot be {3458} because 3,5 only in R5C5, cannot be {1469} which clashes with R23C6), no 5, CPE no 8 in R5C4
11a. R35C5 cannot be {24} because R4C6 + R5C7 cannot total 14, R3C5 = {24} -> no 2,4 in R5C5
11b. R3C5 = 2 (hidden single in C5) -> R2C3 = 6 (step 4), clean-up: no 1 in R3C6, no 7 in R3C7
11c. R3C5 = 2 -> 20(4) cage = {1289/2378}
11d. 3 of {2378} must be in R4C6 -> no 7 in R5C5
11e. 20(4) cage = {1289} (cannot be {2378} = [2738] which clashes with combined cage R4C45 + R4C78 = {48}{67}/{57}[49]) -> R4C6 + R5C57 = {189}, 1 locked for N5, CPE no 9 in R5C4, clean-up: no 7 in R56C6
11f. Killer pair 3,6 in R23C6 and R56C6, locked for C6
11g. 2 in N5 only in R5C46, locked for R5
11h. 1 in R3 only in R3C89, locked for N3
11i. 13(3) cage at R2C9 = {139/148/157}, no 6
11j. 1 in C7 only in R89C7, locked for N9
12. Consider combinations for R23C6 = [16]/{34}
R23C6 = {16} => R56C6 = {35}, locked for N5 => R4C45 = {48}, locked for N5 => R4C6 = 9
or R23C6 = {34} => R56C6 = [26] => R9C6 = 8 => R4C6 = {19}
-> R4C6 = {19}
12a. R4C6 + R5C57 (step 11e) = {189}, 8 locked for R5
13. 45 rule on R1 2 outies R2C57 = 3 innies R1C123 + 3
13a. Max R2C57 = 17 -> max R123C3 = 14, min R1C13 = 6 -> max R1C2 = 8
13b. 9 in N1 only in R3C12, locked for R3, clean-up: no 4 in R2C4 (step 9g), no 4 in R2C8
13c. 13(3) cage at R2C9 (step 11h) = {139/148/157}
13d. 9 of {139} must be in R2C9 -> no 3 in R2C9
14. R5C7 + R7C8 (step 7) = [84/95]
14a. R35C7 cannot be [89] because R3C7 + R2C8 = [85] would clash with R5C7 + R7C8 = [95], R5C7 = {89} -> no 8 in R3C7, clean-up: no 5 in R2C8
14b. Hidden killer pair 8,9 in 17(3) cage at R1C7 and R5C7 for C7, R5C7 = {89} -> 17(3) cage must contain one of 8,9 in R12C7 = {359/368/458} (cannot be {467} which doesn’t contain one of 8,9), no 7
14c. One of 8,9 must be in R12C7 -> no 8,9 in R1C8
14d. 4 of {458} must be in R1C78 (R1C78 cannot be {58} which clashes with R1C3), no 4 in R2C7
14e. 13(2) cage at R2C8 = [94/76] (cannot be [85] which clashes with 17(3) cage)
14f. Naked quad {6789} in R2489C8, locked for C8
14g. 7 in C8 only in R2C8 + R3C7 = [76] or R4C78 = [67] -> 6 in R34C7, locked for C7
14h. 17(3) cage = {359/458}, 5 locked for N3
14i. 13(3) cage at R2C9 (step 11i) = {139/148}, no 7
14j. R2C8 = 7 (hidden single in N3) -> R3C7 = 6, clean-up: no 1 in R2C5
15a. Naked pair {34} in R23C6, locked for C6 and N2, R8C56 = [45], R56C6 = [26], R9C6 = 8, R9C4 = 2 (hidden single in R9), clean-up: no 9 in R2C4 (step 9g), no 8 in R4C4
15b. Naked pair {58} in R23C4, locked for C4 and N2, clean-up: no 7 in R4C5
15c. 18(3) cage at R7C5 (step 10c) = {279} (only remaining combination) -> R7C7 = 2, R7C56 = {79}, locked for R7
15d. 10(2) cage at R7C3 = [82], R2C2 = 4, R23C6 = [34]
15e. 13(3) cage at R2C9 (step 14i) = {139} (only remaining combination) -> R2C9 = 9, R3C89 = {13}, locked for R3 and N3, 23(3) cage at R8C8 = [896], R4C8 = 6 -> R4C7 = 7, clean-up: no 5 in R4C5
15f. R139C3 = [574], R3C3 = 7 -> R45C3 = 5 = [23]
and the rest is naked singles.