Prelims
a) R23C1 = {19/28/37/46}, no 5
b) 11(2) cage at R4C4 = {29/38/47/56}, no 1
c) R45C6 = {19/28/37/46}, no 5
d) R56C9 = {13}
e) R7C89 = {16/25/34}, no 7,8,9
f) R9C67 = {29/38/47/56}, no 1
g) 21(3) cage at R5C2 = {489/579/678}, no 1,2,3
1a. Naked pair {13} in R56C9, locked for C9 and N6, clean-up: no 4,6 in R7C8
1b. 45 rule on N36 3 innies R5C8 + R6C78 = 20 = {479/569/578}, no 2
1c. 2 in N6 only in R4C789 + R5C7, locked for 37(7) cage at R1C7
1d. 45 rule on N5 1 innie R6C6 = 2, placed for D\, clean-up: no 9 in 11(2) cage at R4C4, no 8 in R45C6, no 9 in R9C7
1e. 45 rule on N2 1 innie R1C4 = 3, clean-up: no 8 in R5C5
1f. 45 rule on N1 1 remaining innie R3C2 = 2, clean-up: no 8 in R23C1
1g. 45 rule on N4 1 remaining outie R7C1 = 9, R5C1 = 2 (hidden single in N4) -> R6C1 = 4 (cage sum), clean-up: no 1,6 in R23C1
1h. Naked pair {37} in R23C1, locked for C1 and N1
1i. 21(3) cage at R5C2 = {579/678}, 7 locked for N4
2a. 45 rule on R89 2 innies R8C3 + R9C1 = 7 = [16/25/61]
2b. 45 rule on R789 2 remaining innies R7C67 = 10 = {37/46}
2c. R7C89 = [16/25/52] (cannot be [34] which clashes with R7C67)
2d. 45 rule on N9 2 innies R79C7 = 11 = [38/47/65/74]
2e. 45 rule on N78 2 remaining innies R79C6 = 10 = {37/46}
2e. R5C8 + R6C78 (step 1b) = {569/578} (cannot be {479} which clashes with R7C67), no 4, 5 locked for N6
2f. 4 in N6 only in R4C789 + R5C7, locked for 37(7) cage at R1C7
2g. 45 rule on N3 3 innies R123C7 = 16 = {169/178/358} (cannot be {367} which clashes with R79C7)
2h. 12(3) cage at R2C8 = {129/147/237/246/345} (cannot be {138/156} which clash with R123C7), no 8
3. Hidden killer pair 1,3 in R6C45 and R6C9 for R6, R6C9 = {13} -> R6C45 must contain one of 1,3
3a. 22(4) cage at R4C5 = {1489/1579/3478/3568} (cannot be {1678} which clashes with R45C6, cannot be {3469} which clashes with 11(2) cage at R4C4, cannot be {4567} which doesn’t contain 1 or 3)
3b. 1,3 in R6C45 -> no 1,3 in R4C5 + R5C4
3c. 3 of {3568} must be in R6C5 -> no 6 in R6C5
[I can see interactions between 11(2) cage at R4C4, R45C6 when it’s {37/46}, R79C6 and R79C7 but there don’t seem to be any eliminations from them yet.]
[I’ve been a bit slow to use 37(7) cage at R1C7.]
4a. 2,4 in N4 only in 37(7) cage at R1C7 = {1246789/2345689}, CPE no 6,8,9 in R6C7
4b. Combined half cage 37(7) cage + R6C7 = {1246789}5 (cannot be {2345689}7 which clashes with R79C7 because 3,5,7 all in C7) -> R6C7 = 5, 37(7) cage = {1246789}, no 3, 1 locked for C7 and N3, clean-up: no 6 in R7C7 (step 2d), no 4 in R7C6 (step 2b), no 6 in R9C6
4c. 3 in N3 only in 12(3) cage at R2C8 (step 2h) = {345} (only remaining combination, cannot be {237} = [237] which clashes with R3C1), 3 locked for C8, 4,5 locked for N3
4d. R123C7 (step 2g) = {169} (only remaining combination, cannot be {178} which clashes with R79C7), 6,9 locked for C7, N3 and 37(7) cage at R1C7
4e. R56C8 = {69} (hidden pair in N6), locked for C8, 6 locked for 32(6) cage at R5C8, clean-up: no 4 in R7C7 (step 2b), no 4 in R9C6 (step 2e), no 7 in R9C7
4f. Naked pair {37} in R7C67, locked for R7
4g. Naked pair {37} in R79C6, locked for C6 and N8
5. Hidden killer pair 3,7 in 15(3) cage at R8C1 and R9C3 for N7, R9C3 can only contain one of 3,7 -> 15(3) cage must contain at least one of 3,7 = {348/357} (cannot be {168/456} which don’t contain 3 or 7), no 1,6, 3 locked for C2 and N7
5a. 15(3) cage = 5{37}/8{34}, no 5,8 in R89C2
5b. 18(4) cage at R1C1 = {189/468} (cannot be {459} = 5{49} which clashes with 15(3) cage), no 5, 8 locked for N1
5c. 5 in N1 only in R123C3, locked for C3
5d. R4C3 = 3 (hidden single in R4)
5e. 8 in R3 only in R3C456, locked for N2
6. Consider combinations for 21(3) cage at R5C2 = {579/678}
21(3) cage = {579} = 5{79}, 9 locked for R6 => R6C8 = 6
or 21(3) cage = {678}, locked for N4 => 6 in R4 only in R4C456
-> no 6 in R6C4
7. 2 on D/ only in R1C9 + R7C3, CPE no 2 in R7C9, clean-up: no 5 in R7C8
8. Consider placement for 3 in C7
R7C7 = 3 => R9C7 = 8 (step 2d) => 8 in N6 only in R4C89
or R8C7 = 3, R5C5 = 3 (hidden single on D\) => R4C4 = 8
-> 8 in R4C489, locked for R4, also no 8 in R8C7
8a. 20(5) cage at R3C2 = {12359/12368}
8b. 8 of {12368} must be in R5C3 -> no 6 in R5C3
9. R1C4 = 3 -> R123C3 = 15 and contains 5 for N1 = {159/456}
9a. 20(5) cage at R3C2 (step 8a) = {12359/12368} -> R4C12 + R5C3 = {159}/{16}8
9b. Consider placements for 9 in N1
9 in R12C2, locked for C2 => R4C12 = {15/16}, R5C3 = {89}
or 9 in R123C3 = {159}, locked for C3 => R5C3 = 8
-> R5C3 = {89}, R4C12 = {15/16}, 1 locked for R4, clean-up: no 9 in R5C6
9c. 9 in R4 only in R4C56, locked for N5
[At this stage I originally analysed 21(3) cage at R5C2 but it wasn’t very helpful then, so I’ve omitted it
10. 22(4) cage at R4C5 (step 3a) = {1579/3478/3568} (cannot be {1489} which clashes with R45C6)
10a. Consider placement of 7 in C3
R6C3 = 7 => 22(4) cage = {3478/3568} (cannot be {1579} because 5,7,9 only in R4C5 + R5C4)
or R9C3 = 7, R79C6 = [73], R7C7 = 3 (step 2d), placed for D\ => R6C5 = 3 (hidden single in N5)
-> 22(4) cage = {3478/3568}, R6C5 = 3, R56C9 = [31], R5C6 = 1 (hidden single in N5) -> R4C6 = 9, placed for D/, clean-up: no 8 in R4C4
10b. R7C7 = 3 (hidden single on D\), R67C6 = [73] -> R9C7 = 8
10c. R8C2 = 3 (hidden single in N7), placed for D/ -> R3C8 = 3 (hidden single in N3), R23C1 = [37]
10d. 8 in N5 only in R56C4, locked for C4
10e. 8 on D\ only in R1C1 + R2C2, locked for N1
10f. 7 in N7 only in R9C23, locked for R9
11. Consider combinations for R8C3 + R9C1 (step 2a) = [16/25/61]
R8C3 + R9C1 = {16}, locked for N7
or R8C3 + R9C1 = [25], R8C1 = 8 => R9C2 = 4 (cage sum), R9C3 = 7 (hidden single in N7)
-> R9C3 = {247}
11a. 1,6 in N7 only in R7C12 + R8C3 + R9C1, locked for 26(6) cage at R7C2
12. 17(3) cage at R2C4 = {179/269} (cannot be {278} because 2,7 only in R2C4, cannot be {458/467} which clash with R123C6), no 4,5,8, 9 locked for N2
12a. 2,7 only in R2C4 -> R2C4 = {27}
12b. Naked triple {169} in R3C457, locked for R3
12c. R3C6 = 8 (hidden single in R3)
12d. Killer pair 4,5 in R3C3 and 11(2) cage at R4C4, locked for D\
13. 18(3) cage at R1C1 = {189/468}, 26(6) cage at R7C2 = {124568}, R8C3 + R9C1 (step 2a) = [16/25/61]
13a. Consider permutations for R7C89 = [16/25]
R7C89 = [16] => 1 on D\ only in R1C1 + R2C2 => 18(3) cage = {189}
or R7C89 = [25] => 2 in 26(6) cage only in R8C3 => R9C1 = 5, R8C12 = [83], R9C2 = 4 (cage sum) => 18(3) cage = {189}
-> 18(3) cage = {189}, locked for N1, 9 locked for C2
13b. Naked triple {456} in R123C3, 4,6 locked for C3
13c. 1 in C3 only in R78C3, locked for N7
14. Consider placements for R8C1 = {58}
R8C1 = 5 => R9C1 = 6 => R9C9 = 9, placed for D\ => R1C1 + R2C2 = {18}
or R8C1 = 8 => R1C1 = 1, R12C2 = [98]
-> R1C1 + R2C2 = {18}, 1 locked for N1 and D\, R1C2 = 9, R8C8 = 7, placed for D\, clean-up: no 4 in 11(2) cage at R4C4
[Cracked at last; the rest is straightforward.]
14a. Naked pair {56} in 11(2) cage at R4C4, locked for N5 and D\ -> R9C9 = 9, R3C3 = 4, R3C9 = 5, R2C8 = 4, R7C9 = 6 -> R7C8 = 1
14b. R8C3 = 1, R9C1 = 6 (hidden pair in N7), 6 placed for D/, R123C7 = [691], R12C3 = [56], R12C6 = [45], 11(2) cage at R4C4 = [65], R3C45 = [96], R2C4 = 2 (cage sum)
14c. Naked pair {45} in R78C4, locked for C4, 4 locked for N8, R9C345 = [712], R7C5 = 8, R8C56 = [96] -> R8C4 = 5 (cage sum), R7C3 = 2, placed for D/
14d. R18C1 = [18], R1C5 = 7, R1C9 = 8, placed for D/
and the rest is naked singles, without using the diagonals.