Prelims
a) R1C78 = {19/28/37/46}, no 5
b) R12C9 = {19/28/37/46}, no 5
c) R45C1 = {19/28/37/46}, no 5
d) R45C4 = {12}
e) R5C23 = {19/28/37/46}, no 5
f) R56C6 = {29/38/47/56}, no 1
g) R5C78 = {16/25/34}, no 7,8,9
h) R56C9 = {39/48/57}, no 1,2,6
i) R89C1 = {19/28/37/46}, no 5
j) R9C23 = {89}
k) 20(3) cage at R4C5 = {389/479/569/578}, no 1,2
Steps resulting from Prelims
1a. Naked pair {12} in R45C4, locked for C4 and N5, clean-up: no 9 in R56C6
1b. Naked pair {89} in R9C23, locked for R9 and N7, clean-up: no 1,2 in R89C1
1c. 5 in N3 only in R2C78 + R3C789, CPE no 5 in R3C56
1d. 1,2,5 in N7 only in R7C123 + R8C23, CPE no 1,2,5 in R7C45
2. 45 rule on N5 2 innies R4C6 + R6C4 = 11 = {38/47/56}, no 9
2a. 9 in N5 only in 20(3) cage at R4C5, locked for C5
3. 45 rule on R1234 3 innies R4C145 = 13
3a. R4C4 = {12} -> no 1,2 in R4C1 (because 13(3) cage can only contain one of 1,2), clean-up: no 8,9 in R5C1
4. 45 rule on R6789 3 innies R6C569 = 15 = {348/357/456}, no 9, clean-up: no 3 in R5C9
4a. R6C56 cannot total 11 (which clashes with R56C6, CCC) -> no 4 in R6C9, clean-up: no 8 in R5C9
4b. 6 of {456} must be in R6C6 (cannot be [645] because R56C6 = [74] clashes with R56C9 = [75], combo blocker), no 6 in R6C5
4c. 9 in N5 only in 20(3) cage at R4C5 = {389/479/569}
4d. 5 of {569} must be in R6C5 -> no 5 in R45C5
5. 12(3) cage at R6C2 = {138/156/237/246} (cannot be {129} because no 1,2,9 in R6C4, cannot be {147/345} which clash with R6C569), no 9
5a. Hidden killer pair 1,2 in R6C178 and 12(3) cage for R6, 12(3) cage contains one of 1,2 -> R6C178 must contain one of 1,2
5b. 45 rule on R789 3 outies R6C178 = 18 = {189/279} (other combinations don’t contain 1 or 2), no 3,4,5,6
6. 15(3) cage at R6C1 = {159/168/249/258/357} (cannot be {267/348} which clash with R89C1, cannot be {456} because no 4,5,6 in R6C1)
6a. 8,9 only in R6C1, 7 of {357} must be in R6C1 -> R6C1= {789}, no 7 in R7C12
7. 45 rule on N6 3 innies R4C9 + R6C78 = 1 outie R4C6 + 11, IOU R6C78 cannot total 11 (because R4C6 must be different from R4C9)
7a. R6C178 = 18 (step 5b), R6C78 not 11 -> no 7 in R6C1
7b. 15(3) cage at R6C1 (step 6) = {159/168/249/258}, no 3
8. Consider combinations for R6C178 (step 5b) = {189/279}
8a. R6C178 = {189}, 1 in R6C78
or R6C178 = {279}, 2,7 locked for N6 => R5C78 = {16} (cannot be {34} which clashes with R56C9 = [48/93] when 7 in R6C78)
-> 1 must be in R56C78, locked for N6
9. 45 rule on N6789 2(1+1) outies R4C6 + R6C1 = 1 innie R4C9 + 7
9a. Min R4C6 + R6C1 = 12 (cannot be 11 because R4C6 + R6C4 (step 2) = 11, CCC) -> min R4C9 = 5
10. Hidden killer pair 1,2 in 15(3) cage at R6C1 and 28(6) cage at R7C3 for N7, 15(3) cage contains one of 1,2 in N7 -> 28(6) cage must contain one of 1,2 = {134569/134578/234568} (other combinations contain both of 1,2)
10a. {134569/234568} must have one of 4,5,6 in R8C4, {134578} must have one of 4,5 in R8C4 (otherwise clash with R7C12 because R8C4 is the only cell of the 28(6) cage which doesn’t “see” R7C12) -> R8C4 = {456}
10b. Hidden killer pair 3,7 in 28(6) cage and R89C1 for N7, R89C1 contains both or neither of 3,7 -> 28(6) cage must contain both or neither of 3,7 in N7
10c. 3,7 of {134578} must be in N7 (R7C3 + R8C23 cannot be {145} which clashes with 15(3) cage at R6C1) -> no 7 in R7C45
10d. 3,8,9 of {134569/234568} must be in R7C45 (3 cannot be in N7 because these combinations don’t contain 7) -> no 6 in R7C45
11. 45 rule on N78 2(1+1) outies R6C1 + R9C7 = 1 innie R7C6 + 7
11a. Min R6C1 + R9C7 = 9 -> min R7C6 = 2
11b. 1 in N8 only in 27(6) cage at R8C5, no 1 in R9C7
11c. Min R6C7 + R9C7 = 10 -> min R7C6 = 3
11d. 2 in N8 only in 27(6) cage at R8C5, no 2 in R9C7
11e. Min R6C7 + R9C7 = 11 -> min R7C6 = 4
[I enjoy iterative steps like this; it’s a long time since I’ve used them.]
12. 1,2 in N8 only in 27(6) cage at R8C5 = {123579/123678/124569/124578} (cannot be {123489} which clashes with 28(6) cage at R7C3)
12a. Killer pair 8,9 in 28(6) cage at R7C3 and 27(6) cage, locked for N8
13. R6C1 + R9C7 = R7C6 + 7 (step 11)
13a. Max R7C6 = 7 -> max R6C1 + R9C7 = 14, min R6C1 = 8 -> max R9C7 = 6
14. R6C78 contains one of 1,2 (step 5b), 1,2 in N9 only in 28(5) cage at R6C7 and 28(6) cage at R6C8, 28(5) cage cannot contain both of 1,2 -> 28(5) contains one of 1,2 and 28(6) cage must contain both of 1,2
14a. 28(5) cage at R6C7 = {13789/14689/15679/23689/24589/24679/25678} (cannot be {34579/34678} which don’t contain 1 or 2)
14b. 28(6) cage at R6C8 = {123589/123679/124579/124678} (cannot be {134569/134578/234568} which only contain one of 1,2), CPE no 1,2 in R8C8
15. 45 rule on N23 2(1+1) outies R1C3 + R4C9 = 1 innie R3C4 + 8
15a. Min R3C4 = 3 -> min R1C3 + R4C9 = 11, no 1 in R1C3
16. R6C569 (step 4) = {348/357/456}, R6C178 (step 5b) = {189/279}
16a. 5 in R5 only in R5C6789, consider placements for 5 in R5
R5C6 = 5 => R6C6 = 6 => R6C59 = [45]
or 5 in R5C78 = {25} (locked for N6) => R6C178 = {189}, locked for R6 => R6C569 = {357/456}
or 5 in R5C9 => R6C9 = 7 => R6C56 = {35}
-> R6C569 = {357/456}, no 8, 5 locked for R6, clean-up: no 6 in R4C6 (step 2), no 3 in R5C6, no 4 in R5C9
16b. 6 of {456} must be in R6C6 -> no 4 in R6C6, clean-up: no 7 in R5C6
17. 5 in N4 only in R4C23, locked for R4, CPE no 5 in R2C2, clean-up: no 6 in R6C4 (step 2)
17a. 45 rule on N4 3(2+1) innies R4C23 + R6C1 = 1 outie R6C4 + 13, IOU R4C23 cannot total 13 (because R6C1 must be different from R6C4), R4C23 contains 5 -> no 8 in R4C23
17b. R6C4 = {3478} -> R4C23 + R6C1 = 16,17,20,21 with 5 in R4C23 = {25}9/{35}8/{35}9/{45}8/{56}9/{57}8/{57}9, no 1,9 in R4C23
[Now my first forcing chain in step 8 proves useful. The rest is fairly straightforward.]
18. R4C4 = 1 (hidden single in R4), R5C4 = 2, clean-up: no 8 in R4C1, no 8 in R5C23, no 5 in R5C78
18a. 8 in N4 only in R6C123, locked for R6, clean-up: no 3 in R4C6 (step 2)
19. 5 in N6 only in R56C9 = {57}, locked for C9 and N6, clean-up: no 3 in R12C9
20. R6C178 (step 5b) = {189} (only remaining combination) -> R6C1 = 8, R6C78 = {19}, locked for R4 and N6, clean-up: no 6 in R5C78
20a. Naked pair {19} in R6C78, CPE no 9 in R8C8
20b. Naked pair {34} in R5C78, locked for R5 and N6, clean-up: no 6,7 in R4C1, no 6,7 in R5C23, no 7 in R6C6
20c. Naked pair {19} in R5C12, locked for R5 and N4
20d. R4C5 = 9 (hidden single in R4)
20e. 20(3) cage at R4C5 = [965/974/983], no 7 in R6C5
21. 15(3) cage at R4C6 must contain an odd number -> R4C6 = 7, R6C4 = 4 (step 2)
21a. R4C6 = 7 -> R4C78 = 8 = {26}, locked for R4 -> R4C9 = 8, clean-up: no 2 in R12C9
21b. R6C4 = 4 -> R6C23 = 8 = {26}, locked for R6 and N4 -> R5C1 = 7, R4C1 = 3, R56C9 = [57]
21c. Naked pair {45} in R4C23, CPE no 4 in R2C2
22. Naked pair {46} in R89C1, locked for C1 and N7
22a. R6C1 = 8 -> R7C12 = 7 = {25}, locked for R7 and N7
22b. 1,9 in C1 only in R123C1, locked for N1
23. Naked triple {137} in R7C3 + R8C23 -> 28(6) cage (step 10) = {134578} (only remaining combination) -> R7C5 = 4, R7C4 = 8, R8C4 = 5, R7C6 = 6, R5C6 = 8, R6C6 = 3, R56C5 = [65]
24. R8C6 = 9 (hidden single in N8), R8C5 + R9C45 = {1237} = 13 -> R9C7 = 5 (cage sum)
25. 28(6) cage at R6C8 (step 14b) = {123589/123679/124579/124678}, 2 locked for N9
25a. R7C6 = 6, 8 in N9 only in R8C78 -> 28(5) cage at R6C7 (step 14a) = {14689} (only remaining combination) -> R67C7 = {19}, locked for C7, R8C78 = {48}, locked for R8 and N9 -> R89C1 = [64], clean-up: no 1,9 in R1C8
26. R4C9 = 8 -> R3C89 = 6 = [24/42/51]
26a. Killer pair 1,4 in R12C9 and R3C89, locked for N3, clean-up: no 6 in R1C78
27. 45 rule on N3 3 remaining outies R2C6 + R3C56 = 11 = {245} (only remaining combination, cannot be {128} which clashes with R9C6, cannot be {137} because 3,7 only in R3C5) -> R2C6 = 5, R3C56 = [24], R3C89 = [51], R3C1 = 9, clean-up: no 9 in R12C9
28. Naked pair {46} in R12C9, locked for C9 and N3
28a. Naked pair {23} in R89C9, locked for N9 -> R7C9 = 9, R6C8 = 1, R7C8 = 7, R9C8 = 6, R4C78 = [62]
29. R1C7 = 2 (hidden single in N3), R1C8 = 8, R8C78 = [84], R5C78 = [43], R2C8 = 9
30. R1C6 = 1, R1C1 = 5, R7C12 = [25], R2C1 = 1, R4C2 = 4
30a. R123C1 + R4C2 = [5194] = 19 -> R13C2 = 10 = {37}, locked for C2 and N1 -> R8C2 = 1
and the rest is naked singles.
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