Prelims
a) R23C7 = {15/24}
b) R5C67 = {69/78}
c) R78C7 = {14/23}
d) 22(3) cage at R1C1 = {589/679}
1. 22(3) cage at R1C1 = {589/679}, 9 locked for N1
2. R23C7 = {15} (cannot be {24} which clashes with R78C7), locked for C7 and N3, clean-up: no 4 in R78C7
2a. Naked pair {23} in R78C7, locked for C7 and N9
2b. Min R6C7 = 4 -> max R6C56 = 9, no 9 in R6C56
3. 45 rule on C89 2 innies R19C8 = 10 = [28/37/46/64/91], no 7,8 in R1C8, no 5,9 in R9C8
4. 45 rule on C9 2 outies R28C8 = 13 = {49/67}/[85], no 2,3 in R2C8, no 1,8 in R8C8
5. 45 rule on N3 1 innie R3C8 = 1 outie R1C6 + 5, R1C6 = {1234}, R3C8 = {6789}
6. 45 rule on N9 1 outie R9C6 = 1 innie R7C8 + 1, no 9 in R7C8, no 1,3,4 in R9C6
7. 18(3) cage at R9C6 = {189/279/459/468/567}
7a. 2,5 of {279/567} only in R9C6 -> no 7 in R9C6, clean-up: no 6 in R7C8 (step 6)
8. 45 rule on N9 3 innies R7C8 + R9C78 = 17 = {179/458/467}
8a. R79C8 cannot total 10 (which would clash with R19C8, CCC) -> no 7 in R9C7
8b. 5 of {458} must be in R7C8 -> no 8 in R7C8, clean-up: no 9 in R9C6 (step 6)
9. 14(3) cage at R1C6 = {149/239/248/347} (cannot be {167} = 1[76] because R1C78 + R3C8 cannot be [76]6), no 6 in R1C78, clean-up: no 4 in R9C8 (step 3)
10. 45 rule on C1234 3 outies R159C5 = 19 = {289/379/469/478/568}, no 1
11. 45 rule on R1 2 innies R1C19 = 1 outie R2C4 + 4, min R1C19 = 7 -> min R2C4 = 3
11a. Max R1C19 = 13, min R1C1 = 5 -> max R1C9 = 8
12. 45 rule on R789 2 innies R7C18 = 1 outie R6C4 + 10
12a. Max R7C18 = 16 -> max R6C4 = 6
12b. Min R7C18 = 11, no 1,2,3 in R7C1, no 1 in R7C8, clean-up: no 2 in R9C6 (step 6)
13. R7C8 + R9C78 (step 8) = {179/458/467}
13a. 8 of {458} must be in R9C8 -> no 8 in R9C7
14. 45 rule on R123 2 innies R3C18 = 1 outie R4C4 + 4
14a. Min R3C18 = 7 -> min R4C4 = 3
14b. Max R3C18 = 13, min R3C8 = 6 -> max R3C1 = 7
15. 45 rule on N3 3 innies R1C78 + R3C8 = 19
15a. R13C8 cannot total 10 (which would clash with R19C8, CCC) -> no 9 in R1C7
15b. R1C78 + R3C8 = {289/379/469} (cannot be {478} because [748] + R9C8 = 6 (step 3) clashes with R28C8 and [847] clashes with R28C8), 9 locked for C8 and N3, clean-up: no 4 in R28C8 (step 3)
15c. R1C7 = {478} -> no 4 in R1C8, no 7,8 in R3C8, clean-up: no 2,3 in R1C6 (step 5), no 6 in R9C8 (step 3)
15d. R3C18 = R4C4 + 4 (step 14)
15e. Max R3C18 = 13 -> no 6 in R3C1
[First try I missed this step, which I’ve inserted here as it’s a logical continuation from the previous step.]
16. R1C78 + R3C8 (step 15b) = {289/379/469}, R19C8 (step 3) = [28/37/91], R28C8 (step 4) = {67}/[85] -> combined cage R1C78 + R3C8 + R28C8 = {289}{67}/{379}[85] (because R19C8 = [37])/{469}[85] -> R1C7 + R2C8 contain 8, locked for N3
17. R7C8 + R9C78 (step 8) = {179/458/467} = [467/548/791], R28C8 (step 4) = [67/76/85] -> combined cage R28C8 + R79C8 = [67][58]/[76][58]/[85][47]/[85][71], 5,7,8 locked for C8, 5 also locked for N9
18. 5 in N6 only in 15(3) cage at R4C9
18a. Hidden killer triple 1,2,3 in R19C8 and R456C8 for C8, R19C8 contains one of 1,2,3 -> R456C8 must contain two of 1,2,3
18b. Hidden killer triple 1,2,3 in R456C8 and 15(3) cage for N6, R456C8 contains two of 1,2,3 -> 15(3) cage must contain one of 1,2,3 = {159/258/357}, no 4,6
19. 14(3) cage at R1C6 (step 9) = {149/248/347} (cannot be {239} because 2,3,9 only in R1C8), 4 locked for R1
19a. 4 in N1 only in R2C3 + R3C123, CPE no 4 in R3C4
20. 45 rule on R12 4 innies R2C3567 = 1 outie R3C9 + 7
20a. Min R2C3567 = 10 -> min R3C9 = 3
20b. Max R3C9 = 7 -> max R2C3567 = 14, no 9 in R2C56
21. 27(5) cage at R1C2 = {12789/13689/15678/23589/23679/24678/34578} (cannot be {14589/14679} which clash with R1C6, cannot be {24579/34569} which clash with 14(3) cage at R1C6 = [149])
21a. 5 in R1 only in R1C1 or 27(5) cage (it may be in both R1C1 and R2C4)
R1C1 = 5 => R2C12 = {89}, locked for R2 => R28C8 = {67}, locked for C8 => R3C8 = 9 => 27(5) cage must contain 9
or 5 in R1 in 27(5) cage
-> 27(5) cage = {12789/13689/15678/23589/23679/34578} (cannot be {24678} which doesn’t contain 5 or 9)
[Now for the bit which I didn’t see before, when I got stuck in this position. I’ve also inserted step 16, which prompted me to have another look at the combinations for the 27(5) cage, since all except one contain 8. Note that it wouldn’t work if R2C4 still contained 2.]
21b. R1C1 = 5 => R2C12 = {89}, locked for R2 => R28C8 = {67}, locked for C8 => R3C8 = 9 => R1C6 = 4 (step 5)
or 5 in R1 in 27(5) cage
-> 27(5) cage = {12789/13689/15678/23589/34578} (cannot be {23679} which clashes with 14(3) cage at R1C6 = [473/482])
22. Grouped X-Wing for 8 in 27(5) cage and R1C7 + R2C8, no other 8 in R12
22a. 22(3) cage at R1C1 = {679} (only remaining combination), locked for N1
22b. 5 in R1 only in 27(5) cage at R1C2, no 5 in R2C4
22c. 27(5) cage (step 21b) contains 5 = {15678/23589/34578}
22d. 1 in R1 only in 27(5) cage or 14(3) cage at R1C6 = [149] -> 27(5) cage = {15678/34578} (cannot be {23589} because {23589} + 14(3) cage at R1C6 = [149] clashes with 22(3) cage at R1C1), no 2,9
22e. 7 or 8 of {15678} must be in R2C4 (R1C2345 cannot be {1578} which clashes with 14(3) cage at R1C6 = [473/482]), 4 of {34578} must be in R2C4 -> R2C4 = {478}
22f. 2 in R1 only in R1C89, locked for N3
22g. 2 in N1 only in R2C3 + R3C123, CPE no 2 in R3C4
23. 27(5) cage at R1C2 (step 22d) = {15678/34578}, R1C6 = {14} -> 4 must be in R1C6 + R2C4, locked for N2
24. R2C3567 = R3C9 + 7 (step 20)
24a. 1,2,5 in R2 only in R2C3567 = {1235/1256}, no 4,7 = 11,14 -> R3C9 = {47}
24b. 4 in N1 only in R3C123, locked for R3 -> R3C9 = 7, R2C3567 = 14 = {1256}, no 3, 6 locked for R2
24c. R2C8 = 8, R1C67 = [14], R1C8 = 9 (cage sum), R3C8 = 6, R12C9 = [23], R9C8 = 1 (step 3), R8C8 = 5 (step 4)
24d. Naked pair {79} in R2C12, locked for R2 and N1 -> R1C1 = 6, R2C4 = 4
25. R9C8 = 1 -> R9C67 = 17 = [89]
25a. R7C8 = 7 (hidden single in C8)
25b. 15(3) cage at R4C9 = {159} (hidden triple in N6)
25c. Clean-up: no 6 in R5C6, no 7 in R5C7
26. 6 in N2 only in 22(4) cage at R2C5 = {2569} (only remaining combination, cannot be {3568} which clashes with R1C45, ALS block), locked for N2
26a. 5 in R1 only in R1C23, locked for N1
27. 8 in R3 only in R3C234, locked for 20(5) cage at R2C3, no 8 in R4C4
27a. 20(5) cage = {12368/12458}
27b. 5,6 only in R4C4 -> R4C4 = {56}
27c. Hidden killer pair 3,4 in R3C1 and 20(5) cage for R3, 20(5) cage contains one of 3,4 -> R3C1 = {34}
[With hindsight I could have used step 14, 45 rule on R123.]
28. 45 rule on N8 2 outies R9C23 = 1 remaining innie R7C4 + 3
28a. Min R9C23 = 5 -> min R7C4 = 2
29. 25(5) cage at R8C4 = {23479/23569/34567} (cannot be {12679/13579/14569} because 1,9 only in R8C4), no 1
29a. 9 of {23479/23569} must be in R8C4, 6 of {34567} must be in R8C4 (R9C2345 cannot contain both of 4,6 which would clash with R9C9) -> R8C4 = {69}
29b. 25(5) cage = {23479/23569/34567}, 3 locked for R9
29c. 1 in C4 only in R56C4, locked for N5
29d. 1 in C4 only in R56C4, CPE no 1 in R6C3
30. 18(3) cage at R4C5 = {279/369/378/468} (cannot be {459} because R4C7 only contains 6,7,8, cannot be {567} which clashes with R4C4), no 5
31. 13(3) cage at R6C5 = {238/247/256/346}
31a. R6C7 = {678} -> R6C56 = {23/24/25/34} -> combined cage R6C56 + R6C8 = {234}/{25}3/{25}4, 2 locked for R6
31b. 8 in N5 only in R4C5 + R5C45, CPE no 8 in R4C3
32. 1 in N8 only in 15(4) cage at R7C5 = {1239/1257/1347/1356}
32a. 4 in N8 only in 15(4) cage = {1347} or in R9C5 -> no 3,7 in R9C5 (locking-out cages)
33. 45 rule on N78 1 remaining innie R7C1 = 1 outie R6C4 + 3, R6C4 = {156}, R7C1 = {489}
34. 18(3) cage at R4C5 (step 30) = {279/369/378/468}
34a. 8 in N5 only in R4C5 + R5C45 -> 8 in 18(3) cage or in R5C45 => R5C67 = [96]
-> 18(3) cage = {378/468} (cannot be {279/369} which clash with R5C67 = [96], killer locking-out cages), no 2,9, 8 locked for R4
34b. 9 in N5 only in R5C456, locked for R5
35. 32(7) cage at R3C1 = {1234589/1234679/1235678}, 1,2 locked for N4
36. 45 rule on N4 4(2+2) outies R37C1 + R5C45 = 21, R7C1 = R6C4 + 3 (step 33) -> R3C1 + R5C45 + R6C4 = 18
36a. R3C1 + R5C45 + R6C4 = 18 with 1 in N5 only in R56C4 -> R3C1 + R5C45 + R6C4 = 3+15/4+14 = 3+{159/168}/4+{158/167} (cannot be 4+{149} because R37C1 cannot be [44] and no 4,9 in R6C4), no 2,3,4 in R5C45
36b. 6 of 3+{168}/4+{167} must be in R6C4 (R5C45 cannot be {67/68} which clash with R5C67), no 6 in R5C45
36c. 9 of 3+{159} must be in R5C5 (cannot be 3+[951] which clashes with R48C4), no 9 in R5C4
36d. Killer pair 5,6 in R4C4 and R5C45 + R6C4, locked for N5
36e. 9 in C4 only in R78C4, locked for N8
36f. 9 in C4 only in R78C4, CPE no 9 in R8C3
[After steps 34 and 36, the puzzle is now cracked.]
37. 2 in N5 only in R6C56, locked for R6
37a. Naked triple {234} in R6C568, locked for R6
37b. 13(3) cage at R6C5 = {238/247}, no 6
38. 6 in N5 only in R46C4, locked for C4 -> R8C4 = 9
38a. 45 rule on R9 2 innies R9C19 = 1 outie R8C4 + 2
38b. R8C4 = 9 -> R9C19 = 11 = [56/74]
39. 2 in C4 only in R79C4, locked for N8
39a. 15(4) cage at R7C5 (step 32) = {1347/1356}, 3 locked for N8
39b. Hidden killer pair 4,6 in 15(4) cage and R9C5 for N8, 15(4) cage contains one of 4,6 -> R9C5 = {46}
39c. Naked pair {46} in R9C59, locked for R9
39d. 3 in R9 only in R9C23, locked for N7
40. 15(3) cage at R8C1 = {258/267/456} (cannot be {168} because R9C1 only contains 5,7), no 1
40a. R9C1 = {57} -> no 7 in R8C12
41. 1 in N7 only in R7C23 + R8C3, locked for 24(5) cage at R6C4, no 1 in R6C4, clean-up: no 4 in R7C1 (step 33)
41a. Naked pair {56} in R46C4, locked for C4 and N5 -> R79C4 = [27], R9C1 = 5
41b. Naked pair {38} in R13C4, locked for C4 and N2 -> R1C5 = 7, R5C4 = 1, R5C9 = 5
42. 15(4) cage at R7C5 (step 39a) = {1356} (only remaining combination), locked for N8, 5 also locked for R7 -> R9C5 = 4, R9C9 = 6
43. R8C3 = 7 (hidden single in R8)
43a. Naked pair {23} in R9C23, locked for N7
44. R9C1 = 5 -> R8C12 = 10 = [46], R3C1 = 3, R13C4 = [38]
45. R2C3 + R3C23 = {124} = 7, R3C4 = 8 -> R4C4 = 5 (cage sum)
46. R6C4 = 6, R7C4 = 2, R8C3 = 7 -> R7C23 = 9 = {18}, locked for R7 -> R7C1 = 9
47. 3 in N4 only in 34(6) cage at R4C3 = {136789} (only remaining combination) -> R5C2 = 7, R5C6 = 9, R5C7 = 6
and the rest is naked singles.
Login
Register
FAQ
Search
