Prelims
a) R12C5 = {18/27/36/45}, no 9
b) R5C12 = {18/27/36/45}, no 9
c) R5C89 = {18/27/36/45}, no 9
d) R89C5 = {18/27/36/45}, no 9
e) 24(3) cage at R3C8 = {789}
f) 21(3) cage at R8C7 = {489/579/678}, no 1,2,3
g) 9(3) cage at R8C8 = {126/135/234}, no 7,8,9
h) 26(4) disjoint cage R19C19 = {2789/3689/4589/4679/5678}, no 1
1a. 45 rule on NR1C1 + NR2C2 1 innie R1C1 = 8, placed for NR1C1, clean-up: no 1 in R2C5, no 1 in R5C12
1b. 45 rule on NR1C5 + NR2C6 1 innie R1C9 = 5, placed for NR1C5, clean-up: no 4 in R12C5, no 4 in R5C8
1c. 45 rule on NR6C1 + NR6C2 1 innie R9C1 = 7, placed for NR6C1, clean-up: no 2 in R5C2, no 2 in R89C5
1d. 45 rule on NR5C8 + NR6C6 1 innie R9C9 = 6, placed for NR5C8, clean-up: no 3 in R5C89, no 3 in R8C5
[These steps would also have worked with R19C19 as blank cells; however it’s better the way ixsetf presented this puzzle with a disjoint cage as blank cells might have given more of a pointer to these steps.]
2. 9 in R5 and C5 only in NR3C5 -> R5C5 = 9 -> 20(5) cage at R4C5 = {12359}, locked for NR3C5
[That’s probably the only easy step introduced by the changes to the cage pattern.]
3. 21(3) cage at R8C7 = {489/579} (cannot be {678} because 6,7 only in R8C7), no 6 in R8C7
3a. 7 of {579} must be in R8C7 -> no 5 in R8C7
4. 45 rule on R12 2 remaining innies R2C46 = 12 = {39/48/57}, no 1,2,6
4a. Min R2C6 = 3 -> max R3C7 + R4C8 = 9, no 9 in R3C7 + R4C8
5. 45 rule on R89 2 remaining innies R8C46 = 9 = {18/27/36/45}, no 9
6. 45 rule on C12 2 remaining innies R46C2 = 9 = {18/27/36/45}, no 9
7. 45 rule on C89 2 remaining innies R46C8 = 10 = [19]/{28/37/46}, no 5, no 1 in R6C8
8. Naked triple {789} in 24(3) cage at R3C8, CPE no 7,9 in R1C8
9. Law of Leftovers (LoL) for R1234 two outies R5C12 must exactly equal two innies R34C5, no 1,8 in R5C12 -> no 8 in R3C5, no 1 in R4C5
10. LoL for R6789 two outies R5C89 must exactly equal two innies R67C5, no 3,6 in R5C89 -> no 3 in R6C5, no 6 in R7C5
11. LoL for C1234 two outies R89C5 must exactly equal two innies R5C34, no 2,7 in R89C5 -> no 7 in R5C3, no 2 in R5C4
12. LoL for C6789 two outies R12C5 must exactly equal two innies R5C67, no 4,5 in R12C5 -> no 5 in R5C6, no 4 in R5C7
[When I reached this stage for the original TJK 49, I realised that, because there are square jigsaw houses at R2C2, R2C6, R6C2 and R6C6, the Windoku property applies and there are five hidden windows. Some of the hidden window steps can also be made using LoL but hidden window may be quicker, especially for R159C159.]
13. 26(4) disjoint cage R19C19 = [8576], placed for hidden window R159C159, no 5,6,7,8 in R19C5 + R5C19, clean-up: no 2,3 in R2C5, no 3,4 in R5C2, no 1,2 in R5C8, no 1,4 in R8C5
14. 7 in NR5C8 only in R5C8 and R67C9, grouped X-Wing for 7 in 24(4) cage at R3C8, R5C8 and R67C9, no other 7 in C89, clean-up: no 3 in R46C8 (step 7)
[Because of the changes to the cage pattern, I’ve now reached a stage where many of the steps I used for TJK 49 no longer apply.]
15. 45 rule on NR6C6 3 innies R7C8 + R8C78 = 15
15a. Consider combinations for 9(3) cage at R8C8 = {126/135/234}
9(3) cage = {126}, no 3 => 3 in NR5C8 only in 15(3) cage at R6C9 = {348} => R5C8 = 7 (hidden single in NR5C8) or 15(3) cage = {357} => R7C8= 5 => no 5 in R5C8
or 9(3) cage = {135}, 5 locked for C8
or 9(3) cage = {234} with 4 in R8C89 => 21(3) cage at R8C7 (step 3) contains one of 4,5 in R9C67 => R5C89 cannot be [54]
or 9(3) cage = {234} with 4 in R9C8, placed for NR5C8 => R5C89 cannot be [54]
-> no 5 in R5C8, clean-up: no 4 in R5C9
15b. 9(3) cage = {135/234} (cannot be {126} which clashes with R5C9), no 6
15c. 5 in NR5C8 only in R9C678, locked for R9
15d. LoL for R6789 two outies R5C89 must exactly equal two innies R67C5, no 4,5 in R5C89 -> no 5 in R6C5, no 4 in R7C5
16. R7C8 + R8C78 = 15 (step 15), 21(3) cage at R8C7 (step 3) = {489/579}
16a. R67C9 cannot total 14 -> no 1 in R7C8
16b. Consider combinations for 9(3) cage at R8C8 (step 15b) = {135/234}
9(3) cage = {135} => R7C8 + R8C78 cannot be {159/249}
or 9(3) cage = {234} => 5 in NR5C8 only in 21(3) cage => R8C7 = 7
-> no 9 in R8C7
16c. 21(3) cage = {489/579}, 9 locked for R9 and NR5C8
16d. 9 in hidden window R159C234 only in R1C234, locked for R1 and NR1C1
16e. Max R9C34 = 12 -> min R8C3 = 3
17. R7C8 + R8C78 = 15 (step 15) cannot be {249} because 15(3) cage at R6C9 = {249} and R7C8 + R8C78 = [942] clash with 9(3) cage at R8C8 -> no 9 in R7C8
18. 15(3) cage at R6C9 = {168/267/348/357} (cannot be {258} which clashes with R5C89, cannot be {456} because 5,6 only in R7C8)
18a. 6 of {267} must be in R7C8 -> no 2 in R7C8
19. 15(3) cage at R6C9 (step 18) = {168/267/348/357}, 21(3) cage at R8C7 (step 3) = {489/579}, 9(3) cage at R8C8 (step 15b) = {135/234}
19a. R7C8 + R8C78 = 15 (step 15)
19b. Hidden killer triple 7,8,9 in 15(3) cage at R6C6, 15(3) cage at R6C8 and R7C8 + R8C78 for NR6C6, 15(3) cages cannot contain more than one of 7,8,9 -> R7C8 + R8C78 must contain one of 7,8,9
19c. R7C8 + R8C78 = 15 = {168/267/348/357} (cannot be {258} = [582] because 21(3) cage = 8{49} clashes with 9(3) cage = 2{34}, cannot be {456} which doesn’t contain one of 7,8,9)
20. R5C8 = {78}, 15(3) cage at R6C9 (step 18) contains one of 7,8
20a. Grouped X-Wing for 8 in 24(3) cage at R3C8, R5C8 and 15(3) cage, no other 8 in C89, clean-up: no 2 in R46C8 (step 7)
20b. R7C8 + R8C78 (step 19c) = {267/348/357} (cannot be {168} = [681] which clashes with R46C8), no 1
21. 12(3) cage at R1C8 = {129/156/246/345}
21a. 5 of {345} must be in R2C8 -> no 3 in R2C8
22. 45 rule on NR2C6 3 innies R2C78 + R3C8 = 18 = {189/279/459/468/567} (cannot be {369} = 3[69] which clashes with R46C8, cannot be {378} because no 3,7,8 in R2C8), no 3
22a. R2C78 + R3C8 = {189/279/468/567} (cannot be {459} because 12(3) cage at R1C8 cannot contain one of 4,5 and 9, which is only remaining place for 9 in NR1C5)
23. 12(3) cage at R1C8 (step 21) = {129/156/246/345}
23a. Consider permutations for R12C5 = [18/27/36]
R12C5 = [18], 1 placed for NR1C5 => 12(3) cage cannot be {156}
or R12C5 = [27] => R67C5 = [18] => R5C89 = [81] (LoL for R6789) => no 1,5,6 in R2C9 => 12(3) cage cannot be {156}
or R12C5 = [36], 6 placed for NR1C5 => 12(3) cage cannot be {156}
-> 12(3) cage = {129/246/345}
24. 12(3) cage at R1C8 (step 23a) = {129/246/345}
24a. R2C78 + R3C8 (step 22a) = {189/279/567} (cannot be {468} = {46}8, 4,6 locked for R2 => 12(3) cage = {46}2 and R2C9 + R3C8 = [28] clash with R5C89), no 4
25. 12(3) cage at R1C8 (step 23a) = {129/246/345}, R2C78 + R3C8 (step 22a) = {189/279/567}
25a. Consider placements for 6 in NR1C5
R12C5 = [36], 3 placed for NR1C5 => 12(3) cage cannot be {345}
or 6 in R1C67, locked for 13(3) cage at R1C6 => no 6 in R2C7 => R2C78 + R3C78 cannot be [657] => 12(3) cage cannot be {345}
or R1C8 = 6 => 12(3) cage cannot be {345}
-> 12(3) cage = {129/246}, no 3,5
25b. 3 in NR1C6 only in R1C567, locked for R1
25c. 3 in NR1C1 only in R2345C1, locked for C1
25d. 3 in NR6C1 only in R9C2345, locked for R9
25e. 9(3) cage at R8C8 (step 15b) = {135/234}, 3 locked for R8, clean-up: no 6 in R8C46 (step 5)
26. 12(3) cage at R1C8 (step 25a) = {129/246}, R2C78 + R3C8 (step 22a) = {189/279/567}
26a. Consider permutations for R12C5 = [18/27/36]
R12C5 = [18] => R67C5 = [27] => R5C89 = [72] (LoL for R6789) => R2C78 + R3C8 cannot be [567]
or R12C5 = [27], 2 placed for NR1C5 => 2 of 12(3) cage must be in R2C8
or R12C5 = [36]
-> no 6 in R2C8
26b. 6 of 12(3) cage = {246} must be in R1C8 -> no 4 in R1C8
26c. R2C78 + R3C8 = {189/279} (cannot be {567} because no 5,6,7 in R2C8), no 5,6 in R2C7, 9 locked for NR2C6, clean-up: no 3 in R2C4 (step 4)
27. 13(3) cage at R1C6 = {139/148/247} (cannot be {238} which clashes with R12C5 = [18], cannot be {346} because no 3,4,6 in R2C7), no 6
27a. 8,9 of {139/148} must be in R2C7 -> no 1 in R2C7
28. 13(3) cage at R1C6 (step 27) = {139/148/247}, 12(3) cage at R1C8 (step 25a) = {129/246}
28a. Consider combinations for 13(3) cage
13(3) cage = {139} = {13}9
or 13(3) cage = {148/247}, 4 locked for NR1C5 => 12(3) cage = {129}
-> 9 in R2C789, locked for R2, clean-up: no 3 in R2C6 (step 4)
29. 12(3) cage at R1C8 (step 25a) = {129/246}
29a. R2C78 + R3C8 (step 26c) = {189/279}
29b. {189} must be [819] (cannot be [918] which clashes with 12(3) cage = [219]) -> no 8 in R3C8
30. Naked triple {789} in 24(3) cage at R3C8, 8 locked for C9 and NR1C5, clean-up: no 1 in R1C5
30a. LoL for C6789 no 1,8 in R12C5 -> no 1 in R5C6, no 8 in R5C7
31. Max R67C9 = 11 -> min R7C8 = 4
31a. R8C8 = 3 (hidden single in C8) => R7C8 + R8C78 (step 19c) = {348/357}, no 6
31b. R8C8 = 3 -> R8C9 + R9C8 = 6 = [15/24/42], no 1 in R9C8
32. 12(3) cage at R1C8 (step 25a) = {129/246}, R2C46 (step 4) = {48/57}
32a. Consider placements for R2C5 = {67}
R2C5 = 6, placed for NR1C5 => no 6 in R1C8
or R2C5 = 7 => R2C46 = {48}, locked for R2 => no 4 in R2C9
-> 12(3) cage = {129}, 9 locked for R2
32b. R46C8 (step 4) = {46} (cannot be [19] which clashes with 12(3) cage, ALS block), locked for C8, clean-up: no 2 in R8C9 (step 31b)
32c. 3 in NR5C8 only in 15(3) cage at R6C9 = {348/357}, no 1,2
33. R2C5 = 6 (hidden single in NR1C5 => R1C5 = 3
33a. LoL for C6789 R12C5 = [36] -> R5C67 = [36]
33b. 6 in R1 only in R1C234, locked for NR1C1
34. 4 in NR1C5 only in 13(3) cage at R1C6, locked for R1
34a. 4 in C9 only in R678C9, locked for NR5C8 and hidden window R678C159
34b. 21(3) cage at R8C7 (step 3) = {489/579}
34c. 4,7 only in R8C7 -> R8C7 = {47}
[With hindsight I could have got R8C7 = {47} from a clean-up after step 32b.]
35. 12(3) cage at R2C6 must contain one of 1,2,3 -> R3C7 = {123}
35a. Min R3C7 + R4C8 = 5 -> max R2C6 = 7, clean-up: no 4 in R2C4 (step 4)
36. R7C8 + R8C78 (step 31a) = {348/357}
36a. 15(3) cage at R6C8 = {168/249/267} (cannot be {159/258} because R6C8 only contains 4,6, cannot be {456} which clashes with R7C8 + R8C78), no 5
36b. R6C8 = {46} -> no 4 in R7C7 + R8C6, clean-up: no 4,5 in R8C4 (step 5)
36c. 15(3) cage at R6C6 = {159/168/249} (cannot be {258/267/456} which clash with 15(3) cage at R6C8), no 7
36d. Killer triple 1,4,7 in R8C46 and R8C79, locked for R8
37. 1 in NR5C8 only in R58C9, locked for C9
37a. 1 in NR1C5 only in R1C678, locked for R1
37b. 1 in NR1C1 only in R234C1, locked for C1
[Only just spotted, it works after step 34…]
38. 3 in R2 only in 15(3) cage at R1C2 = {357} = 7{35} (cannot be {348} because no 3,4,8 in R1C2) or in 16(3) cage at R1C3 = {367} = {67}3 -> 7 in R1C234 (locking cages), locked for R1 and NR1C1 -> R5C2 = 5, R5C1 = 4, R5C34 = [81], R5C89 = [72], R67C5 = [27], R34C5 = [45], R89C5 = [81], R9C8 = 5 -> R8C9 = 1 (cage sum), clean-up: no 4,7 in R4C2, no 4 in R6C2 (both step 6)
38a. Naked pair {27} in R8C46, locked for R8 -> R8C7 = 4
38b. R6C8 = 6 -> R7C7 + R8C6 = 9 = [27], clean-up: no 3 in R4C2 (step 6)
38c. R2C6 + R4C8 = [54] -> R3C7 = 3 (cage sum), R2C4 = 7 (step 4)
38d. R2C4 = 7 -> R3C3 + R4C2 = 8 = {26}, locked for NR2C2, clean-up: no 1,8 in R6C2 (step 6)
39. 16(3) cage at R6C1 = {169} (only possible combination) -> R67C1 = [96], R7C2 = 1
39a. R8C1 = 5 -> R89C2 = 9 = [63]
and the rest is naked singles, without using the jigsaw nonets.
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