Prelims
a) R23C2 = {17/26/35}, no 4,8,9
b) R23C8 = {14/23}
c) 7(2) cage at R8C2 = {16/25/34}, no 7,8,9
d) 15(2) cage at R8C8 = {69/78}
e) 21(3) cage at R2C1 = {489/579/678}, no 1,2,3
f) 20(3) cage at R5C8 = {389/479/569/578}, no 1,2
g) 19(3) cage at R8C3 = {289/379/469/478/568}, no 1
h) 15(5) cage at R8C4 = {12345}
1a. Naked quint {12345} in 15(5) cage at R8C4, locked for N8
1b. 45 rule on R89 3 innies R8C159 = 22 = {589/679}, 9 locked for R8, clean-up: no 6 in R9C9
1c. Combined half cage R8C159 + R8C8 = {589}6/{589}7/{679}8, 8 locked for R8
1d. 45 rule on R1 2 outies R23C5 = 8 = {17/26/35}, no 4,8,9
1e. 45 rule on R123 2 outies R4C19 = 9 = {45}/[63/72/81], no 9, no 6,7,8 in R4C9
1f. 45 rule on N7 1 innie R7C3 = 1 outie R6C1 + 2, no 8,9 in R6C1, no 1,2 in R7C3
1g. 45 rule on N9 1 outie R6C9 = 1 innie R7C7 + 4 -> R6C9 = {56789}, R7C7 = {12345}
1h. 45 rule on N6 3 innies R4C9 + R6C79 = 9 = {126/135} cannot be {234} because no 2,3,4 in R6C9) -> R6C9 = {56}, R4C9 + R6C7 = {12/13}, 1 locked for N6, clean-up: no 4,5 in R4C1, no 3,4,5 in R7C7
1i. R4C9 + R6C79 + R7C7 = [1351/2162], no 3 in R4C9, no 2 in R6C7, clean-up: no 6 in R4C1
1j. R67C7 = [12/31], 1 locked for C7
1k. R67C7 = 3,4 -> R67C6 = 15,16 = {69/78/79}
1l. 45 rule on N58 4(2+2) outies R67C37 = 11, R67C7 = 3,4 -> R67C3 = 7,8, no 6,7,8,9 in R6C2, no 8,9 in R7C3, clean-up: no 6,7 in R6C1
1m. R67C3 = 7,8 -> R67C4 = 14,15 = [59]/{68/69/78}
1n. 45 rule on N4 3 innies R4C1 + R6C13 = 13, R4C1 = {78} -> R6C13 = 5,6 = {14/23/24} (cannot be {15} which clashes with R6C79), no 5, clean-up: no 7 in R7C3
1o. 45 rule on N1 2 innies R23C3 = 1 outie R4C1 + 4, min R4C1 = 7 -> min R23C3 = 11, no 1 in R23C3
1p. 45 rule on N3 2 innies R23C7 = 1 outie R4C9 + 12 -> min R23C7 = 13, no 2,3 in R23C7
2a. 45 rule on N9 4 innies R7C789 + R8C9 = 18 = {1269/1278/2349/2358/2457} (cannot be {1368/1467/2367} which clash with 15(2) cage at R8C8, cannot be {1359/1458} which clash with R6C9 + R7C7 = [51], cannot be {3456} because R7C7 only contains 1,2), 2 locked for R7 and N9
2b. 12(3) cage at R8C7 = {138/147/156/345}, no 9
2c. Min R89C7 = 7 -> max R9C8 = 5
2d. 45 rule on N7 4 innies R7C123 + R8C1 = 19 = {1369/1468/1567/3457} (cannot be {1459} which clashes with 7(2) cage at R8C2, cannot be {1378} which clashes with R6C1 + R7C3 = [13])
2e. 19(3) cage at R8C3 = {289/379/478} (cannot be {469/568} which clash with R7C123 + R8C1), no 5,6
2f. R7C123 + R8C1 contain at least one of 6,7
Consider it/their placement
At least one of 6,7 in R7C123, locked for R7 => R8C5 = {67}
and/or R8C1 = {67}
-> R8C159 (step 1b) = {679} (only possible combination), locked for R8 -> R8C8 = 8, R9C9 = 7, both placed for D\, clean-up: no 1 in R3C2, no 1 in R9C1
2g. 19(3) cage = {289} (only remaining combination) -> R8C3 = 2, R9C23 = {89}, locked for N7
2h. R7C123 + R8C1 = {1567/3457}, 5 locked for R7 and N7
3a. R4C1 + R6C13 (step 1n) = 13 = {148/238} (cannot be {247} = [724] which clashes with R6C1 + R7C3 = [24], step 1f) -> R4C1 = 8
[Cracked. Straightforward from here.]
3b. R4C1 = 8 -> R4C9 = 1 (step 1e), R6C7 = 3, R6C9 = 5 (step 1h), clean-up: no 5 in R7C3 (step 1f)
3c. 20(3) cage at R5C8 = {479} (only remaining combination), locked for N6, 7 locked for C8
3d. Naked pair {26} in R4C78, locked for R4 and N6 -> R5C7 = 8
3e. R7C7 = 1 (hidden single in C7), placed for D\, clean-up: no 7 in R3C2
3f. R8C2 = 1 (hidden single in N7) -> R9C1 = 6, both placed for D/, R8C1 = 7, clean-up: no 4 in R3C8, no 4 in R6C1 (step 1f)
3g. R4C1 + R6C13 = [814] (only remaining permutation) -> R7C3 = 3, placed for D/, clean-up: no 2 in R3C8
3h. Naked pair {45} in R89C7, locked for C7, 4 locked for N9, R9C8 = 3 (cage sum), clean-up: no 2 in R2C8
3i. R23C8 = [41], 4 placed for D/, clean-up: no 7 in R2C5 (step 1d)
3j. R67C7 = 4 -> R67C6 = 15 = {69} (cannot be {78} because 7,8 only in R7C6), locked for C6
3k. Naked pair {79} in R56C8, 9 locked for C8 and N6 -> R5C9 = 4
3l. 9 in N9 only in R78C9, locked for C9
3m. R1C8 = 5 (hidden single in C8) -> R1C79 = 11 = [92], 2 placed for D/
3n. R23C7 = [67], 7 placed for D/, clean-up: no 1 in R2C5, no 2 in R3C5 (both step 1d), no 2 in R3C2
3o. R23C7 = 13 -> R23C6 = 10 = {28}, locked for N2, 2 locked for C6, clean-up: no 6 in R23C5 (step 1d)
4a. R4C1 = 8 -> R23C1 = 13 = [94]
4b. R1C1 = 3, placed for D\ -> R1C23 = 9 = [81], clean-up: no 5 in R23C2
4c. R23C2 = [26]
4d. R23C3 = [75] = 12 -> R23C4 = 10 = [19]
and the rest is naked singles, without using the diagonals.
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