Prelims
a) R23C5 = {49/58/67}, no 1,2,3
b) R2C67 = {59/68}
c) R34C2 = {19/28/37/46}, no 5
d) R34C6 = {18/27/36/45}, no 9
e) R7C12 = {29/38/47/56}, no 1
f) R89C1 = {18/27/36/45}, no 9
g) R89C2 = {29/38/47/56}, no 1
h) R8C34 = {15/24}
i) R9C34 = {19/28/37/46}, no 5
j) 10(4) cage at R7C7 = {1234}
k) 38(6) cage at R5C3 = {356789}, no 1,2,4
1a. Naked quad {1234} in 10(4) cage at R7C7, CPE no 1,2,3,4 in R9C89
1b. 45 rule on N78 2(1+1) innies R7C3 + R9C6 = 11 = [74/83/92]
1c. Naked quad {1234} in 10(4) cage at R7C7, 1 locked for C7 and N9
1d. Naked sext {356789} in 38(6) cage at R5C3, CPE no 3,5,6 in R5C12
2. 45 rule on N9 2(1+1) outies R6C9 + R9C6 = 9 = [54/63/72]
3. 45 rule on C9 2 innies R45C9 = 3 = {12}, locked for C9 and N6
3a. 15(3) cage at R1C9 = {348/357/456}, no 9
3b. 1 in N3 only in 12(3) cage at R1C8 = {129/138/156} (cannot be {147} which clashes with 15(3) cage), no 4,7
3c. 9 in C9 only in 27(4) cage at R6C9, locked for N9
4. 1,2 in R6 only in R6C456, locked for N5, clean-up: no 7,8 in R3C6
4a. 24(5) cage at R4C5 = {12489/12579/12678}, no 3
4b. One of 1,2 in {12579} must be in R6C5 (R456C5 cannot be {579} which clashes with R23C5) -> no 5 in R6C5
5. 45 rule on N6 2 outies R3C7 + R5C6 = 1 innies R6C9 + 8
5a. Min R6C9 = 5 -> min R3C7 + R5C6 = 13, no 2,3 in R3C7 + R5C6
6. 45 rule on N8 3 innies R8C4 + R9C46 = 13 must contain one of 6,7,8,9 -> R9C4 = {6789}, R9C3 = {1234}
7. 45 rule on R12 4 outies R3C1589 = 13 = {1237/1246/1345}, no 8,9, 1 locked for R3, clean-up: no 4,5 in R2C5, no 9 in R4C2, no 8 in R4C6
7a. 7 of {1237} must be in R3C5 -> no 7 in R3C19
8. Hidden killer quad 6,7,8,9 in R7C12, R7C3, R89C1 and R89C2 for N7, R7C12, R7C3 and R89C2 each contain one of 6,7,8,9 -> R89C1 must contain one of 6,7,8 -> R89C1 = {18/27/36}, no 4,5
[Alternatively 6 in N7 only in R7C12/R89C2 = {56} or in R89C1 = {36} -> R89C1 cannot be {45}, locking-out cages.]
[Only just spotted …]
9. 17(3) cage at R7C8 must contain one of 2,3,4
9a. Killer triple 2,3,4 in 10(4) cage at R7C7 and 17(3) cage, locked for N9
10. 27(4) cage at R6C9 = {5679} (only remaining combination), locked for C9
10a. Naked triple {348} in 15(3) cage at R1C9, locked for N3, clean-up: no 6 in R2C6
10b. 7 in N3 only in R13C7, locked for C7
10c. 8 in N9 only in 17(3) cage at R7C8, locked for C8
11. 45 rule on C89 2 innies R4C89 = 1 outie R6C7
11a. Min R4C89 = 4 -> min R6C7 = 4
11b. Max R4C89 = 9, no 9 in R4C8
12. 45 rule on N3 3 innies R123C7 = 18 = {279/567}
12a. 9 of {279} must be in R2C7 -> no 9 in R13C7
12b. Min R3C7 + R5C6 = 13 (step 5a), max R3C7 = 7 -> min R5C6 = 6
13. R4C9 = {12} -> 34(6) cage at R3C7 = {136789/145789/235789/245689}, CPE no 9 in R5C8
14. R3C1589 (step 7) = {1237/1246/1345}
14a. 4,6 of {1246} must be in R3C59 -> no 6 in R3C18
15. Hidden killer pair 1,2 in R5C12 and R5C9, R5C9 = {12} -> R5C12 must contain one of 1,2
15a. 13(3) cage at R4C1 contains one of 1,2 = {139/148/157/238/247} (cannot be {256} because 5,6 only in R4C1), no 6
15b. R5C12 contains one of 1,2 -> no 1,2 in R4C1
15c. 3 of {139} must be in R4C1 -> no 9 in R4C1
16. 45 rule on N1 3 innies R1C3 + R3C23 = 1 outie R2C4 + 17
16a. Max R1C3 + R3C23 = 24 -> max R2C4 = 7
17. 45 rule on R7 4 innies R7C3789 = 19 = {1279/1369/1378/1459/1468/1567/2359/2368/2458/2467/3457}
[Unfortunately the only way I can see to make use of this is to use a contradiction move
, although it does prove to be useful in opening up more steps.]
17a. R7C3789 cannot be {1567} = [71]{56} because 15(3) cage at R7C4 contains 4 (only remaining place for 4 in R7 because R7C12 cannot be {47}) clashes with R7C3 + R9C6 = [74] (step 1b), combo blocker
-> R7C3789 = {1279/1369/1378/1459/1468/2359/2368/2458/2467/3457}
17b. R7C3789 contains two of 1,2,3,4 -> R7C8 = {234}
17c. Naked quad {1234} in R789C7 + R7C8, locked for N9
17d. Naked quad {1234} in 10(4) cage at R7C7 and R789C7 + R7C8 -> R7C8 = R9C6
17e. R7C3 + R9C6 = 11 (step 1b) -> R7C38 = 11 -> R7C79 = 8, no 4 in R7C7, no 9 in R7C9
17f. R7C3789 = {1279/1378/2359/2368/2467/3457} (cannot be {1369/1459/1468/2458} which don’t contain pairs totalling 8 and 11)
18. 15(3) cage at R7C4 = {159/168/249/348/456} (cannot be {258/267/357} which clash with R7C3789), no 7
18a. R8C4 + R9C46 = 13 (step 6) = {139/238/247/256/346} (cannot be {148} which clashes with 15(3) cage at R7C4, cannot be {157} because 1,5 only in R8C4)
18b. 15(3) cage = {159/168/348/456} (cannot be {249} which clashes with R8C4 + R9C46), no 2
19. 45 rule on R789 1 innie R7C3 = 1 outie R6C9 + 2
19a. R7C3789 (step 17f) = {1378/2359/2467} (cannot be {1279/2368/3457} which clash with R7C3 + R6C9 = [97/86/75], combo blockers)
19b. 15(3) cage at R7C4 (step 18b) = {159/168/456} (cannot be {348} which clashes with R7C3789), no 3
19c. R8C4 + R9C46 = 13 (step 18a) = {139/238/247/346} (cannot be {256} which clashes with 15(3) cage), no 5, clean-up: no 1 in R8C3
19d. 17(3) cage at R8C5 = {179/278/359/368/467} (cannot be {269/458} which clashes with 15(3) cage at R7C4)
19e. R7C12 = {29/38/47} (cannot be {56} which clashes with 15(3) cage), no 5,6 in R7C12
20. 45 rule on N7 3 innies R789C3 = 14 = {149/158/239/257/347} (cannot be {248} which clashes with R7C12)
20a. R7C3 + R9C6 (step 1b) = [74/83/92], R8C4 + R9C46 (step 19c) = {139/238/247/346}
20b. R789C3 = {158/239/347} (cannot be {149} because R8C4 + R9C46 cannot be [292], cannot be {257} because R8C4 + R9C46 doesn’t contain {148})
20c. R8C4 + R9C46 = {139/247} (cannot be {238/346} because R789C3 doesn’t contain {248}), no 6,8, clean-up: no 2,4 in R9C3
20d. R7C12 (step 19e) = {29/47} (cannot be {38} which clashes with R789C3), no 3,8 in R7C12
20e. R89C2 = {29/47/56} (cannot be {38} which clashes with R789C3), no 3,8 in R89C2
20f. R89C1 (step 8) = {18/36} (cannot be {27} which clashes with R7C12), no 2,7 in R89C1
20g. 8 in N7 only in R7C3 + R89C1, CPE no 8 in R6C1
[The rest is fairly straightforward.]
21. 3 in R7 only in R7C78, locked for N9
21a. R7C3789 (step 19a) = {1378/2359}, no 4,6
21b. R7C9 = {57} -> no 7 in R7C3, clean-up: no 4 in R9C6 (step 1b), no 5 in R6C9 (step 2)
21c. 4 in N9 only in R89C7, locked for C7
21d. 5 in C9 only in R789C9, locked for N9
22. 6 in R7 only in 15(3) cage at R7C4, locked for N8
22a. 15(3) cage at R7C4 (step 19b) = {168/456}, no 9
22b. 9 in R7 only in R7C123, locked for N7, clean-up: no 2 in R89C2
22c. R34C2 = {28/37}/[91] (cannot be {46} which clashes with R89C2), no 4,6 in R34C2
23. 13(3) cage at R1C1 = {148/157/247/256/346} (cannot be {139/238} which clash with R89C1), no 9
24. R789C3 (step 20b) = {158/239}, no 4, clean-up: no 2 in R8C4
24a. Killer pair 1,4 in 15(3) cage at R7C4 and R8C4, locked for N8
25. Naked sext {356789} in 38(6) cage at R5C3, CPE no 7 in R5C12
25a. 13(3) cage at R4C1 (step 15a) = {139/148/238/247} (cannot be {157} because 5,7 only in R4C1), no 5 in R4C1
26. R3C7 + R5C6 = R6C9 + 8 (step 5)
26a. Min R6C9 = 6 -> min R3C7 + R5C6 = 14, no 6,7 in R5C6
27. 24(5) cage at R4C5 (step 4a) = {12579/12678} (cannot be {12489} which clashes with R5C6), no 4, 7 locked for N5, clean-up: no 2 in R3C6
27a. Killer pair 8,9 in 24(5) cage and R5C6, locked for N5
28. 4 in N5 only in R4C46, locked for R4
28a. 4 in N4 only in 13(3) cage at R4C1 (step 25a) = {148/247}, no 3,9, 8 locked for N4, clean-up: no 2 in R3C2
29. 34(6) cage at R3C7 (step 13) = {136789/235789}, 3 locked for N6
30. 45 rule on C8 3 innies R456C8 = 16 = {349/457} (cannot be {367} which clashes with R6C9), no 6
31. 38(6) cage at R5C3 = {356789} -> R7C3 = 8, R9C6 = 3 (step 1b), R6C9 = 6 (step 2), R9C3 = 1, R9C4 = 9, R89C3 = [36], R8C3 = 5 (step 24), R8C4 = 1
32. Naked pair {24} in R89C7, locked for C7 and N9 -> R7C78 = [13]
32a. Naked triple {456} in 15(3) cage at R7C4, locked for R7 and N8 -> R7C9 = 7, R89C9 = [95], R89C8 = [68]
32b. 2 in N3 only in 12(3) cage at R1C8 = {129} (only remaining combination), locked for C8 and N3, clean-up: no 5 in R2C6
32c. Naked triple {567} in R123C7, locked for C7
32d. Naked pair {89} in R25C6, locked for C6
[Just noticed that I missed clean-up: no 6 in R34C6 from step 31 …]
33. Naked pair {45} in R34C6, locked for C6 -> R7C6 = 6
34. R8C5 = 8 (hidden single in N8), clean-up: no 5 in R3C5
35. Naked pair {47} in R89C2, locked for C2, clean-up: no 3 in R34C2
35a. Naked pair {12} in R4C29, locked for R4
36. 38(6) cage at R5C3 = {356789}, 7 locked for N4 -> R4C1 = 8, R5C12 = 5 = [41], R4C2 = 2, R3C2 = 8, R7C12 = [29]
37. Naked triple {157} in 13(3) cage at R1C1, locked for C1 and N1 -> R6C1 = 9, R6C7 = 8
38. Naked pair {36} in R12C2, locked for C2, N1 and 15(4) cage at R1C2 -> R6C2 = 5
38a. R12C2 = {36} = 9 -> R2C34 = 6 = {24}, locked for R2
39. Naked triple {127} in R6C456, locked for R6 and N5 -> R6C3 = 3, R6C8 = 4
and the rest is naked singles.