Prelims
a) R12C9 = {59/68}
b) R34C1 = {13}
c) R3C67 = {19/28/37/46}, no 5
d) R3C89 = {29/38/47/56}, no 1
e) R7C23 = {19/28/37/46}, no 5
f) R89C1 = {49/58/67}, no 1,2,3
g) R9C56 = {39/48/57}, no 1,2,6
h) 9(3) cage at R1C8 = {126/135/234}, no 7,8,9
i) 19(3) cage at R8C6 = {289/379/469/478/568}, no 1
j) 8(3) cage at R8C8 = {125/134}
k) 26(4) cage at R5C6 = {2789/3689/4589/4679/5678}, no 1
l) 39(8) cage at R1C1 = {12345789}, no 6
Steps Resulting From Prelims
1a. Naked pair {13} in R34C1, locked for C1
1b. 1,3 in 39(8) cage at R1C1 only in R1C234567, locked for R1
1c. 8(3) cage at R8C8 = {125/134}, 1 locked for N9
2. 6 in R1 only in R1C89, locked for N3, clean-up: no 8 in R1C9, no 4 in R3C6, no 5 in R3C89
2a. 45 rule on N3 2 innies R13C7 = 11 = {29/38/47}, no 1,5, clean-up: no 9 in R3C6
2b. 1 in N3 only in 9(3) cage at R1C8 = {126/135}, no 4, 1 locked for R2
2c. 5 of {135} must be in R1C8 -> no 5 in R2C78
2d. 6 of {126} must be in R1C8 -> no 2 in R1C8
2e. 2,4,7,8 in R1 only in R1C1234567, locked for 39(8) cage at R1C1 -> no 2,4,7,8 in R2C1
3. 45 rule on N9 2 outies R78C6 = 7 = [16]/{25/34}, no 7,8,9, no 6 in R7C6
3a. 19(3) cage at R8C6 = {289/379/469/478/568}
3b. 2,3,4 of {289/379/469/478} must be in R8C6 -> no 2,3,4 in R89C7
4. 45 rule on R789 2 innies R7C14 = 10 = {28/46}/[73/91], no 5 in R7C1, no 5,7,9 in R7C4
5a. 45 rule on N6 2 innies R6C78 = 15 = {69/78}
5b. 45 rule on N6 2 outies R56C6 = 11 = {29/38/47/56}
6. 45 rule on R123 2 innies R3C15 = 10 = [19/37]
6a. 23(4) cage at R2C4 = {2489/2678/3569/3578/4568} (cannot be {1679/2579/3479} which clash with R3C5, cannot be {1589} = {589}1 which clashes with R2C1), no 1
7. 45 rule on R89 1 outie R7C5 = 1 innie R8C9, no 1 in R7C5
8. 5 in R3 only in R3C234
8a. 45 rule on R12 3 outies R3C234 = 14 = {158/257/356}, no 4,9
8b. 4 in R3 only in R3C789, locked for N3, clean-up: no 7 in R3C7 (step 2a), no 3 in R3C6
8c. R3C15 (step 6) = [19/37], R3C234 = {158/257/356} -> combined cage R3C15234 = [19]{257}/[19]{356}/[37]{158}
8d. R3C67 = [28/64/82] (cannot be [19/73] which clash with combined cage R3C15234), no 1,7 in R3C6, no 3,9 in R3C7 clean-up: no 2,8 in R1C7 (step 2a)
8e. 1 in R3 only in R3C123, locked for N1
9. Caged X-Wing for 5 in 39(8) cage at R1C1 (no 5 in R1C7) and R3C234 for N12 -> no 5 in R2C23456
9a. 23(4) cage at R2C4 = {2489/3569/3578/4568} (cannot be {2678} which clashes with R3C6)
9b. Consider placements for R2C1 = {59}
9c. R2C1 = 5 => R3C4 = 5 (hidden single in N2) => 23(4) cage = {3569/3578/4568}
or R2C1 = 9 => 23(4) cage = {3578/4568}
-> 23(4) cage = {3569/3578/4568}, no 2
-> R3C4 = 5
9d. R3C234 = 14 (step 8a) -> R3C23 = 9 -> R2C23 = 10 = {28/37/46}, no 9
9e. 9 in N1 only in R1C123 + R2C1, locked for 39(8) cage at R1C1, clean-up: no 2 in R3C7 (step 2a), no 8 in R3C6
9f. 19(4) cage at R2C2 = {1468/2368/2467} (cannot be {1378} which clashes with R3C1), with R2C23 = 10 and R3C23 = 9 = {28}{36}/{46}{18}/{46}{27} -> R2C23 = {28/46}, no 3,7
9g. 7 in R2 only in R2C456 -> 23(4) cage at R2C4 = {3578} (only remaining combination), locked for N2, 3,8 also locked for R2 -> R3C5 = 9, R3C1 = 1 (step 6), R4C1 = 3, clean-up: no 3 in R9C6
9h. Naked pair {12} in R2C78, 2 locked for R2 and N3, R1C8 = 6 (cage sum), clean-up: no 9 in R6C7 (step 5a)
9i. Naked pair {59} in R12C9, locked for C9, 9 also locked for N3, clean-up: no 5 in R7C5 (step 7)
9j. Naked pair {46} in R2C23, locked for N1 -> R3C23 = {27} (only remaining combination), locked for R3 and N1
9k. R3C6 = 6 -> R3C7 = 4, placed for D/, R1C7 = 7 (step 2a), clean-up: no 5 in R56C6 (step 5b), no 8 in R6C8 (step 5a), no 6 in R7C2, no 1 in R7C6 (step 3), no 9 in R8C1
9l. R89C1 = [49/67/76] (cannot be {58} which clashes with R12C1, ALS block), no 5,8
9m. R7C23 = {19/28/37} (cannot be [46] which clashes with R89C1), no 4,6
9n. R3C5 = 9 -> R4C56 = 9 = {18/27}/[45], no 5,6 in R4C5
[At this stage, after the elimination of 1 in step 9k, I missed Ed’s 1 in R7 only in R7C14 (step 4) = [91] or R7C23 = {19}, locking cages, 9 locked for R7 and even more important locked for N7. This would have avoided my difficult, but interesting, later steps.]
10. 19(3) cage at R8C6 (step 3a) = {289/469/568}, no 3, clean-up: no 4 in R7C6 (step 3)
10a. 5 of {568} must be in R8C6 -> no 5 in R89C7
10b. Naked triple {689} in R689C7, locked for C7, 9 also locked for N9
10c. 19(3) cage must contain 9 = {289/469}, no 5, clean-up: no 2 in R7C6 (step 3)
10d. 25(5) cage at R7C6 = {23578/34567} (cannot be {24568} because 2,4 in N9 clash with 8(3) cage at R8C8)
10e. 12(3) cage at R4C9 = {147/237/246} (cannot be {138} which clashes with R3C9), no 8
10f. Killer pair 6,7 in 12(3) cage and R6C78, locked for N6
10g. 5 in N6 only in 18(4) cage at R4C7 = {1359/1458/2358}
10h. 2 of {2358} must be in R45C7 (cannot be in R45C8 which would clash with R2C78 = [12] as 1 would be hidden single in C7), no 2 in R45C8
11. 25(4) cage at R4C2 and R4C56 ‘see’ each other so must form a 34(6) combined cage = {136789/145789/235789/245689/345679}
11a. 12(3) cage at R4C9 (step 10e) = {147/237/246}
11b. Variable hidden killer pair 2,6 for 34(6) combined cage and R4C79 for R4, R4C79 cannot be [26] which clashes with 12(3) cage = 6{24} -> 34(6) combined cage must contain at least one of 2,6 in R4C23456 -> 34(6) combined cage = {136789/235789/245689/345679} (cannot be {145789}
11c. Variable hidden killer pair 4,6 for 34(6) combined cage and R4C79 for R4, R4C89 cannot be [46] which clashes with 12(3) cage = 6{24} -> 34(6) combined cage must contain at least one of 4,6 in R4C23456 -> 34(6) combined cage = {136789/245689/345679} (cannot be {235789}
11d. R4C56 (step 9n) = {18}/[45] (cannot be {27} because 34(6) combined cage only contains one of 2,7), no 2,7
11e. 34(6) combined cage = {136789/245689/345679}, R4C56 = {18}/[45] -> 25(4) cage = {3679/2689}, no 1,4,5
11f. 3 of {3679} must be in R5C4 -> no 7 in R5C4
12. R56C6 (step 5b) = {29/38/47}, R4C56 (step 11d) = {18}/[45]
12a. Consider permutations for R78C6 (step 3) = [34/52]
12b. R78C6 = [34] => R56C6 = {29}
or R78C6 = [52] => R4C45 = {18}, locked for N5 => R56C6 = {47}
-> R56C6 = {29/47}, no 3,8
12c. Killer pair 2,4 in R56C6 and R8C6, locked for C6 -> R1C6 = 1, clean-up: no 8 in R4C5, no 8 in R9C5
12d. R56C6 contains one of 7,9, R6C8 = {79} -> 26(4) cage at R5C6 = {2789/4679}, CPE no 7,9 in R6C45
12e. 3 in N5 only in R56C45, CPE no 3 in R7C4, clean-up: no 7 in R7C1 (step 4)
12f. 6 in N5 only in R4C4 + R56C45, CPE no 6 in R7C4, clean-up: no 4 in R7C1 (step 4)
12g. Consider permutations for R4C56 = [18/45]
R4C56 = [18]
or R4C56 = [45], R56C6 = {29} => R6C78 = [87]
-> no 8 in R4C8
12h. 8 in R4 only in R4C2346, CPE no 8 in R5C4
13. 12(3) cage at R4C9 (step 10e) = {147/237/246}
13a. 45 rule on R1234 3 innies R4C789 = 1 outie R5C4 + 8
13b. Apart from 3, which is already placed for R4, the value in R5C4 must be in R4C789
13c. R5C4 cannot be 2 because R4C789 = 10 cannot be {127} = [217] clashes with 12(3) cage = 7{14}/7{23} (alternatively 18(4) cage at R4C7 (step 10g) cannot contain both of 1,2)
R5C4 cannot be 6 because 6 in R4C9 and R4C78 cannot total 8
R5C4 cannot be 9 because 9 in R4C8 and R4C79 cannot total 8 = [17] because 18(4) cage at R4C7 = [19]{35} clashes with 12(3) cage = 7{14}/7{23})
-> R5C4 = 3
[It took me a long time to spot the power of that 45, which cracks the puzzle.]
13d. R5C4 = 3 -> R4C789 = 11 = {245} (only remaining combination, cannot be {146} which clashes with R4C5), locked for R4 and N6, R4C5 = 1, R4C6 = 8, placed for D/, R5C7 = 1, clean-up: no 2 in R7C2, no 4 in R9C5
13e. R6C9 = 3 (hidden single in N6) -> R45C9 = 9 = [27], R4C78 = [54], R5C8 = 8 (cage sum), R3C89 = [38]
13f. R6C78 = [69] = 15 -> R56C6 = 11 = [47], 7 placed for D\, R3C3 = 2, placed for D\, R6C4 = 2, placed for D/, R6C5 = 5, R5C5 = 6, placed for both diagonals, R4C4 = 9, placed for D\, R2C8 = 1, placed for D/, R2C2 = 4, placed for D\, R8C8 = 5, placed for D\, R9C8 = 2, R7C678 = [537], R7C3 = 9 placed for D/, R7C2 = 1 (cage sum)
and the rest is naked singles, without using the diagonals.