Prelims
a) R12C5 = {12}
b) R45C9 = {79}
c) R6C45 = {18/27/36/45}, no 9
d) R6C78 = {17/26/35}, no 4,8,9
e) R67C9 = {18/27/36/45}, no 9
f) R89C5 = {39/48/57}, no 1,2,6
g) R89C9 = {17/26/35}, no 4,8,9
h) R9C34 = {15/24}
i) 19(3) cage at R4C3 = {289/379/469/478/568}, no 1
j) 20(3) cage at R7C4 = {389/479/569/578}, no 1,2
k) 28(4) cage at R1C6 = {4789/5689}, no 1,2,3
1a. Naked pair {12} in R12C5, locked for C5 and N2, clean-up: no 7,8 in R6C4
1b. Naked pair {79} in R45C9, locked for C9 and N6
1c. 45 rule on N6 1 innie R6C9 = 1 -> R7C9 = 8, clean-up: no 8 in R6C5
1d. 45 rule on N3 1 innie R1C7 = 9
1e. 45 rule on N7 1 innie R9C3 = 5 -> R9C4 = 1, clean-up: no 7 in R8C5, no 3 in R8C9
1f. 45 rule on N8 3 remaining innies R789C6 = 12, must contain 2 for N8 = {237/246}, no 5,8,9
1g. 45 rule on C6 using R123C6 = 19, 3 remaining innies R456C6 = 14, must contain 1 for C6 = {149/158} (cannot be {167} which clashes with R789C6), no 2,3,6,7
[Oops! I was overlooking that R123C6 must contain both of 1,9 for C6! Fortunately this didn’t make much difference and was OK after step 1n.]
1h. R6C45 = [27] (cannot be {36} which clashes with R6C78, cannot be {45} which clashes with R456C6), 2 placed for D/, clean-up: no 6 in R6C78, no 5 in R8C5
1i. Naked pair {35} in R6C78, locked for R6 and N6
1j. 4 in C9 only in 12(3) cage at R1C9, locked for N3
1k. 2,3 in C6 only in R789C6 = {237}, 3,7 locked for N8, 7 locked for C6, clean-up: no 9 in R89C5
1l. Naked pair {48} in R89C5, locked for C5 and N8
1m. R123C6 = 19 = {568} (only remaining combination), locked for N2, 5,8 locked for C6
1n. Naked triple {149} in R456C6, 4,9 locked for N5
1o. 45 rule on N58 1 outie R3C5 = 1 innie R4C4 + 1 -> R3C5 = 9, R4C4 = 8, placed for D\
1p. 6,8 in R6 only in R6C123 -> 20(4) cage at R5C1 = 2{468} (cannot be {1568} because 1,5 only in R5C1), locked for N4
1q. R6C6 = 9 (hidden single in R6), placed for D\
1r. 45 rule on N4 2 innies R4C12 = 6 = {15}, locked for R4 and 20(5) cage at R2C3, 5 locked for N4
1s. R45C6 = [41], 4 placed for D/
1t. Naked pair {26} in R4C78, 6 locked for R4 and N6 -> R4C5 = 3
1u. R4C4 = 8 -> R23C4 = 10 = {37}, locked for N2 -> R1C4 = 4
1v. 20(5) cage at R2C3 contains 4 for N1 = {248}{15}/{347}{15}, no 6,9
1w. R2C1 = 9 (hidden single in N1)
1x. 9 on D/ only in R7C3 + R8C2, locked for N7
1y. R9C8 = 9 (hidden single in R9) -> R8C8 + R9C7 = 7 = [16/34/43] (cannot be [52] which clashes with R89C9), no 2,5,7, no 6 in R8C8
1z. Killer pair 3,6 in R8C8 + R9C7 and R89C9, locked for N9
1aa. 12(3) cage at R1C9 = {246/345}
1ab. 6 of {246} must be in R1C9 -> no 6 in R23C9
1ac. Naked pair {56} in R57C5, CPE no 6 in R7C3 using D/, no 5 in R7C7 using D\
[17 placements after step 1 and a lot of cages reduced to one or two combinations, which may be the reason why the SS score is only 1.55.]
2a. 20(5) cage at R2C3 (step 1v) = {248}{15}/{347}{15}
2b. R2C3 + R3C23 = {248}/{37}4 (R23C3 cannot be {37} which clashes with R45C3 ALS block, R3C23 cannot be {37} which clashes with R3C4) -> no 3,7 in R3C3
2c. 13(3) cage at R7C7 = {157/247}
2d. Consider combinations for R8C8 + R9C7 = [16]/{34}
R8C8 + R9C7 = [16] => R4C7 = 2
or R8C8 + R9C7 = {34}, 4 locked for N9 => 13(3) cage = {157}
-> no 2 in R78C7
2e. 2 of {247} must be in R7C8 -> no 4 in R7C8
3a. 20(4) cages in N7 = {1289/1379/1469/1478/2369/2378/2468/3467}
3b. 20(4) cage at R7C1 = {1289/1379/1469/2369/3467} (cannot be {147}8 which clashes with R7C7, cannot be {237}8 which clashes with R7C6, cannot be {2468} because R7C3 only contains 1,3,7,9)
3c. 9 of {1289/1379/1469} must be in R7C3 -> no 1 in R7C3
3d. Naked triple {379} in R457C3, locked for C3
3e. R2C3 + R3C23 (step 2b) = {248} (only remaining combination), 2,8 locked for N1
3f. 20(4) cage at R8C2 = {1289/1478/2378/2468/3467} (cannot be {1379/1469/2369} which clash with 20(4) cage at R7C1)
3g. 9 of {1289} must be in R8C2 -> no 1 in R8C2
3h. 1 on D/ only in R2C8 + R3C7, locked for N3
3i. Variable hidden killer pair 1,4 in R7C12 and R7C78 for R7, R7C78 cannot contain both of 1,4 (step 2c) -> R7C12 must contain at least one of 1,4
3j. 20(4) cage at R7C1 = {1289/1379/1469/3467} (cannot be {2369} which doesn’t contain 1 or 4)
3k. 20(4) cage at R8C2 = {1289/2378/2468/3467} (cannot be {1478} which clashes with 20(4) cage at R7C1)
3l. Consider combinations for 13(3) cage at R7C7 (step 2c) = {157/247}
13(3) cage = {157} => 4 in R7 only in R7C12 => 20(4) cage at R7C1 = {1469/3467}
or 13(3) cage = {247} => R7C8 = 2 => R89C9 = {53}, R1C9 = 6, placed for D/ => 20(4) cage at R8C2 cannot be {2468} (because 2,4,6 then only in R8C3 + R9C2) => 20(4) cage at R7C1 cannot be {1379}
-> 20(4) cage at R7C1 = {1289/1469/3467}
3m. 8 of {1289} must be in R8C1, 6 of {1469} must be in R8C1 (R7C123 cannot be {169/469} which clash with R7C45, ALS block) -> no 1 in R8C1
3n. 20(4) cage at R8C2 = {1289/2378/3467} (cannot be {2468} which clashes with 20(4) cage at R7C1)
3o. Consider combinations for 20(4) cage at R7C1 = {1289/1469/3467}
20(4) cage at R7C1 = {1289}, 2 locked for R7 => 13(3) cage = {157}
or 20(4) cage at R7C1 = {1469} = {149}6, 4 locked for R7
or 20(4) cage at R7C1 = {3467} => 1 in R7 only in R7C78 => 13(3) cage = {157}
-> no 4 in R7C7, 13(3) cage = {157}/[724]
3p. 4 on D\ only in R3C3 + R8C8, CPE no 4 in R8C3
[At last I can use triangles.]
4a. 13(3) cage at R7C7 = {157}/[724], R8C8 + R9C7 = [16]{34}, R89C9 = {26}/[53]
4b. Consider placements for R1C1
R1C1 = 1, placed for D\ => R8C8 + R9C9 = {34}, 4 locked for N9
or R1C1 = 3, placed for D\ => R89C9 = {26}, 2 locked for N9
or R1C1 = {56} => naked pair {56} in R1C1 + R5C5 => R1C9 = 3 (CPE using diagonals) => R89C9 = {26}, 2 locked for N9
or R1C1 = 7, placed for D\ => R7C7 = 1
-> 13(3) cage = {157}, 1,5 locked for N9, clean-up: no 3 in R9C9
4c. Naked pair {26} in R89C9, locked for C9, 6 locked for N9
4d. Naked triple {345} in R123C9, 3,5 locked for N3
4e. 4 in R7 only in R7C12, locked for N7
4f. 4 in R7C12 -> 20(4) cage at R7C1 (step 3n) = {1469/3467}, no 2,8, 6 locked for N7
4g. 5 on D/ only in R1C9 + R5C5, CPE no 5 in R1C1 using diagonals
4h. 20(4) cage at R7C1 = {3467} (cannot be {1469} = {149}6 because of killer ALS clash with R7C45 = {56} and R7C78 = {157}), locked for N7
5a. 20(4) cage at R8C2 = {1289} = [9182], 8 placed for D/
5b. R9C9 = 6, placed for D\ -> R5C5 = 5, placed for both diagonals -> R1C9 = 3, placed for D/ -> R7C7 = 7, placed for D/ -> R7C7 = 1, place for D\
5c. R9C5 = 4 -> R9C7 = 3, R8C8 = 4, placed for D\
and the rest is naked singles, without using the diagonals.