Prelims
a) 10(2) cage at R1C1 = {19/28/37/46}, no 5
b) 7(2) cage at R1C9 = {16/25/34}, no 7,8,9
c) 8(2) cage at R2C1 = {17/26/35}, no 4,8,9
d) R2C67 = {39/48/57}, no 1,2,6
e) R45C8 = {18/27/36/45}, no 9
f) R5C12 = {19/28/37/46}, no 5
g) R78C1 = {29/38/47/56}, no 1
h) 11(2) cage at R8C2 = {29/38/47/56}, no 1
i) R89C6 = {15/24}
j) 9(2) cage at R8C8 = {18/27/36/45}, no 9
k) 22(3) at R8C7 = {589/679}
l) 26(4) cage at R1C2 = {2789/3689/4589/4679/5678}, no 1
1a. 45 rule on N9 1 innie R7C7 = 2, placed for D\, clean-up: no 8 in 10(2) cage at R1C1, no 9 in R8C1, no 7 in 9(2) cage at R8C8
1b. 12(3) cage at R7C8 = {138/147/345} (cannot be {156} which clashes with 22(3) cage at R8C7), no 6,9
1c. 45 rule on C89 2 innies R19C8 = 17 = {89}, locked for C8, clean-up: no 1 in R45C8, no 1 in R9C9
1d. Min R1C8 = 8 -> max R1C67 = 6, no 6,7,8,9 in R1C67
1e. 45 rule on N7 2 innies R7C23 = 11 = {38/47/56}, no 1,9
1f. 45 rule on C12 2 innies R19C2 = 10 = {28/37/46}/[91], no 5, no 9 in R9C2
1g. 45 rule on N5 2 outies R37C5 = 9 = {18/36/45}/[27], no 9, no 7 in R3C5
1h. 45 rule on N8 3 innies R7C456 = 18 = {189/369/378/567} (cannot be {459} which clashes with R89C6, cannot be {468} which clashes with R7C23), no 4, clean-up: no 5 in R3C5
1i. 45 rule on C123 3 outies R137C4 = 23 = {689}, locked for C4
1j. 40(7) cage at R3C3 = {1456789/2356789}, 5,7 locked for C3
1k. 1 in N7 only in 12(3) cage at R8C3 = {129/138/147}, no 6, clean-up: no 4 in R1C2
1l. 7 of {147} must be in R9C2 -> no 4 in R9C2, clean-up: no 6 in R1C2
1m. 8 of {138} must be in R89C3 (R89C3 cannot be {13} which clashes with 40(7) cage), no 8 in R9C2, clean-up: no 2 in R1C2
1n. 26(4) cage at R1C2 = {2789/3689/4679}
1o. 2,3 of {2789/3689} must be in R1C23 (R1C234 cannot be {689/789} which clash with R1C8), no 2,3 in R2C3
1p. 26(4) cage = {2789/3689/4679}, CPE no 9 in R1C1, clean-up: no 1 in R2C2
2a. 45 rule on C6789 1 innie R6C6 = 2 outies R4C45 + 2
2b. Min R4C45 = 3 -> min R6C6 = 5
2c. Max R4C45 = 7, no 7,8,9 in R4C45
2d. 45 rule on C6789 3 innies R456C6 = 21 = {489/579/678}, no 1,2,3
3a. 45 rule on R12 2 innies R2C19 = 8 = {17/26/35}, no 4,8,9
3b. 45 rule on R89 2 innies R8C19 = 9 = [27/45/54/63/81], no 3,7 in R8C1, no 8 in R8C9, clean-up: no 4,8 in R7C1
4a. 45 rule on D\ 2 remaining innies R3C3 + R4C4 = 1 outie R3C5 + 7
4b. Min R3C3 + R4C4 = 8, max R4C4 = 5 -> min R3C3 = 3
4c. Max R3C3 + R4C4 = 14 -> no 8 in R3C5, clean-up: no 1 in R7C5 (step 1g)
5a. R7C456 (step 1h) = {189/369/378/567}
5b. Consider combinations for R456C6 (step 2d) = {489/579/678}
R456C6 = {489}, 4 locked for C6 => R89C6 = {15}, 5 locked for N8
or R456C5 = {579/678}, 7 locked for C6 => R7C456 = {189/369}/[675/873], no 5 in R7C5
-> no 5 in R7C5, clean-up: no 4 in R3C5 (step 1g)
5c. Consider placement for 6 in N5
6 in R456C5, locked for C5
or 6 in R456C6 = {678} => R5C5 + R6C6 cannot be 14 = {59/68} (the latter CCC)
-> no 3 in R3C5, clean-up: no 6 in R7C5 (step 1g)
5d. 17(3) cage at R3C5 = 1{79}/2{69}/2{78}/
[647] (cannot be [638] which clashes with R37C5 = [63], step 1g) -> R5C5 = {46789}, R6C6 = {6789}
First time I’d got my mental arithmetic wrong! 5e. 17(3) cage = 1{79}/2{69}/2{78}
(cannot be [647] which gives R7C5 = 3 (step 1g), R4C4 = 3 (hidden single in N5) when R4C4 + R5C5 = [34] clash with 1,3,4 in 10(2) cage at R1C1 + 9(2) cage at R8C8, killer ALS block)-> R3C5 = {12}, R5C5 + R6C6 = {69/78/79}, no
4, clean-up: no 3 in R7C5
5f. R3C5 = {12} -> R3C3 + R4C4 (step 4a) = 8,9, no 9 in R3C3
5g. R7C456 = {189/378/567} (cannot be {369} because R7C5 only contains 7,8)
5h. 1,3,5 only in R7C6 -> R7C6 = {135}
5i. Min R7C5 = 7 -> max R5C4 + R6C45 = 11, no 9 in R6C5
5j. Consider permutations for R37C5 = [18/27]
R37C5 = [18] => R7C456 = [981]
or R37C5 = [27] => 2 in C6 only in R89C6
-> R89C6 = {24}, locked for C6 and N8, clean-up: no 8 in R2C7
[Cracked. The rest is fairly straightforward.]
5k. R456C6 = {579/678}, 7 locked for C6 and N5, clean-up: no 5 in R2C7, no 8 in R6C6 (step 5e)
5l. 4 in N2 only in 13(3) cage at R1C5 = {148/247/346}, no 5,9
5m. 5 in N2 only in R123C6, locked for C6, clean-up: no 9 in R456C6
5n. Naked triple {678} in R456C6, 6,8 locked for C6 and N5 -> R5C5 = 9, placed for both diagonals, clean-up: no 1 in R1C1, no 4 in R2C7, no 1 in R5C12, no 2 in 11(2) cage at R8C2
5o. 9 in C6 only in R23C6, locked for N2
5p. Naked pair {68} in R13C4, locked for C4 and N2
5q. 13(3) cage = {247} (only remaining combination), 2 locked for N2
5r. R3C5 = 1, R7C5 = 8 (step 1g) -> R7C456 = [981], clean-up: no 7 in R2C1, no 1 in R2C9 (step 3a), no 3 in R7C23 (step 1e), no 2 in R8C1
5s. R3C5 = 1, R5C5 = 9 -> R6C6 = 7 (cage sum), placed for D\, clean-up: no 3 in 10(2) cage at R1C1
5t. Naked pair {46} in 10(2) cage, locked for N1 and D\, clean-up: no 2 in 8(2) cage at R2C1, no 2,6 in R2C9 (step 3a), no 3,5 in 9(2) cage at R8C8
5u. R8C8 = 1, R9C9 = 8, both placed for D\ -> R9C8 = 9, R1C8 = 8, R1C67 = 6 = [51], R89C7 = 13 = {67}, locked for C7, 7 locked for N9, clean-up: no 6 in R1C9, no 2,6 in R2C8, no 3 in R8C2, no 2 in R9C2 (step 1f)
6a. R8C3 = 9 (hidden single in N7) -> R9C23 = 3 = [12], R89C6 = [24], R1C2 = 9 (step 1f), R12C3 = [38], R1C4 = 6, R3C3 = 5, placed for D\, R1C1 = 4, R2C2 = 6, R1C9 = 2 -> R2C8 = 5, both placed for D/, R2C1 = 1, R2C9 = 7 (step 3a), R3C2 = 7, clean-up: no 4 in R45C8, no 3 in R5C1, no 7 in R7C1, no 4 in R7C3 (step 1e), no 6 in R9C1
6b. R4C4 = 3 -> R89C4 = {57}, locked for N8, 5 locked for C4, clean-up: no 6 in R5C8
6c. R6C4 = 1 (hidden single on D/), R7C5 = 8 -> R5C4 + R6C5 = 9 = [45], clean-up: no 6 in R5C1
6d. R7C3 = 7 (hidden single in R7), R7C2 = 4 (step 1e), R8C2 = 8, R9C1 = 3, both placed for D/
6e. R7C1 = 6 (hidden single in R7) -> R8C1 = 5
6f. R45C6 = [68], clean-up: no 2 in R5C12, no 3 in R5C8
6g. R3C1 = 2 -> R4C12 = 13 = [85]
6h. R234C7 = [349], R2C9 = 7, R3C89 = [69] -> R4C9 = 1 (cage sum)
and the rest is naked singles, without using the diagonals.