Prelims
a) R1C12 = {17/26/35}, no 4,8,9
b) R12C5 = {14/23}
c) R1C78 = {49/58/67}, no 1,2,3
d) R12C9 = {49/58/67}, no 1,2,3
e) R2C12 = {29/38/47/56}, no 1
f) R45C1 = {39/48/57}, no 1,2,6
g) R45C5 = {39/48/57}, no 1,2,6
h) R4C78 = {17/26/35}, no 4,8,9
i) R45C9 = {16/25/34}, no 7,8,9
j) R78C2 = {39/48/57}, no 1,2,6
k) R7C67 = {18/27/36/45}, no 9
l) R89C9 = {16/25/34}, no 7,8,9
m) R9C78 = {49/58/67}, no 1,2,3
n) 19(3) cage at R1C6 = {289/379/469/478/568}, no 1
o) 26(4) cage at R3C1 = {2789/3689/4589/4679/5678}, no 1
1. 45 rule on N3 2 innies R23C7 = 7 = {25/34}/[61], no 7,8,9, no 6 in R3C7
1a. Killer triple 4,5,6 in R1C78, R12C9 and R23C7, locked for N3
2. 19(3) cage at R1C6 = {289/379/469/478/568}
2a. 2,3 of {289/379} must be in R2C7 -> no 2,3 in R12C6
3. 45 rule on C123 3 outies R127C4 = 19 = {289/379/469/478/568}, no 1
4. 45 rule on N7 2(1+1) outies R6C1 + R7C4 = 11 = {29/38/47/56}, no 1
5. Killer triple 4,5,6 in R12C9, R45C9 and R89C9, locked for C9
6. 45 rule on R12 1 outie R3C3 = 1 innie R2C8 + 5, R2C8 = {123}, R3C3 = {678}
7. 45 rule on N4 2 outies R3C12 = 1 innie R6C1 + 8, IOU no 8 in R3C2
8. 45 rule on N6 2 outies R7C89 = 1 innie R5C7 + 1
8a. Min R7C78 = 3 -> min R5C7 = 2
[And now for a 45 which I only spotted when I came back to this puzzle. While it’s not a breakthrough, I think it’s essential to make significant further progress …]
9. 45 rule on N689 2(1+1) innies R5C7 + R7C4 = 16 = [79/88/97], clean-up: R6C1 = {234} (step 4)
9a. Min R7C4 = 7 -> max R78C3 = 7, no 7,8,9 in R78C3
10. 18(3) cage at R6C1 = {279/369/468} (cannot be {189/567} because R6C1 only contains 2,3,4, cannot be {378/459} which clash with R45C1), no 1,5
10a. R6C2 = {234} -> no 2,3,4 in R78C1
11. 45 rule on N89 3 innies R7C489 = 17 = {179/269/278/359/368} (cannot be {458/467} because no 4,5,6 only in R7C8), no 4
11a. 5,6 of {359/368} must be in R7C8 -> no 3 in R7C8
12. 45 rule on N4 3 innies R4C23 + R6C1 = 18 = {279/369/468} (cannot be {378/459} which clash with R45C1, cannot be {567} because R6C1 only contains 2,3,4), no 5
12a. R6C1 = {234} -> no 2,3,4 in R4C23
13. R4C23 + R6C1 (step 12) = {279/369/468}
13a. R3C12 = R6C1 + 8 (step 7)
13b. R6C1 = {234} -> R3C12 = 10,11,12 = {28/46/38/47/39/57} (cannot be {37} which clashes with R4C23 + R6C1 = {79}2, cannot be {29/56} which clash with R4C23 + R6C1 = {69}3, cannot be {48} which clashes with R4C23 + R6C1 = {68}4)
[I originally analysed the 26(4) cage at R3C1 but I think the above approach is easier to follow.]
14. R3C12 (step 13b) = {28/46/38/47/39/57}
14a. 45 rule on R12 3 outies R3C389 = 17 = {179/269/278/368}
14b. Consider combinations for R3C389
R3C389 = {179/278}, 7 locked for R3
or R3C389 = {269/368} => R3C3 = 6, R1C12 = {17/35} => R3C12 cannot be {57} which clashes with R1C12
-> R3C12 = {28/46/38/47/39}, no 5 in R3C12
15. R3C389 = 17 (step 14a)
15a. Consider combinations for 12(3) cage at R2C8 = {129/138/237}
12(3) cage at R2C8 = {129/237}, 2 locked for N3 => no 5 in R23C7 (step 1)
or 12(3) cage at R2C8 = {138} = 1{38} (cannot be 3{18} because R3C389 cannot be 8{18}), locked for N3 => R23C7 (step 1) = {25}, R3C3 = 6 (step 6) => 6 in N2 only in 19(3) cage at R1C6 = {68}5 => R2C7 = 5
-> no 5 in R3C7, clean-up: no 2 in R2C7 (step 1)
15b. 5 in R3 only in R3C456, locked for N2
16. 19(3) cage at R1C6 = {379/469/478/568}
16a. Consider combinations for R12C5
R12C5 = {14}, locked for N2
or R12C5 = {23} => 1 in N2 only in R3C456, locked for R3, no 6 in R2C7 (step 1) => 19(3) cage cannot be {49}6, 4 of {478} must be in R2C7
-> no 4 in R12C6
16b. 3,4,5 of 19(3) cage only in R2C7 -> R2C7 = {345}, clean-up: no 1 in R3C7 (step 1)
16c. 1 in N3 only in 12(3) cage at R2C8 (step 16a) = {129/138}, no 7
[After these three forcing chains, the rest is fairly straightforward.]
17. R3C389 (step 14a) = {179/269/368} (cannot be {278} because 12(3) cage at R2C8 cannot be 2{28})
17a. 6,7 only in R3C3 -> R3C3 = {67}, clean-up: no 3 in R2C8 (step 6)
17b. R3C12 = {28/46/38/47} (cannot be {39} which clashes with R3C389), no 9 in R3C12
18. R2C12 = {29/38/47} (cannot be {56} which clashes with R1C12 + R3C3, killer ALS block), no 5,6
19. 26(4) cage at R3C1 = {2789/3689/4679}, 9 locked for R4 and N4, clean-up: no 3 in R45C1, no 3 in R5C5
19a. 9 in N4 only in R4C23 + R6C1 (step 12) = {279/369}, no 4,8
19b. 8 in 26(4) cage only in R3C1 -> no 2,3 in R3C1
19c. R6C1 = {23} -> R7C4 = {89} (step 4), R5C7 = {78} (step 9)
19d. Min R7C4 = 8 -> max R78C3 = 6, no 6 in R78C3
19e. 9 in N6 only in R5C8 + R6C789, locked for 31(6) cage at R5C8, no 9 in R7C89
20. 18(3) cage at R6C1 (step 10) = {279/369}, no 8, 9 locked for C1 and N7, clean-up: no 2 in R2C2, no 3 in R78C2
21. 1 in C1 only in R19C1
21a. 45 rule on C1 4 innies R1239C1 = 15 contains 1 = {1248/1257/1356} (cannot be {1347} which clashes with R45C1)
21b. R1239C1 = {1248/1257} (cannot be [1365] => R1C12 = [17] and R1C2 + R3C1 clash with R3C3, cannot be [5361] which clashes with R1C12 = [53] -> combination {1356} eliminated), no 3,6, clean-up: no 2,5 in R1C2, no 8 in R2C2
22. 18(3) cage at R6C1 = {369} (hidden triple in C1) -> R6C1 = 3, R78C1 = {69}, locked for N7, R7C4 = 8 (step 4), R5C7 = 8 (step 9), clean-up: no 5 in R1C8, no 4 in R4C1, no 4 in R4C5, no 1 in R6C78, no 4 in R8C2, no 5 in R9C8
23. R4C23 + R6C1 (step 19a) = {369} (only remaining combination), 6 locked for R4, N4 and 26(4) cage at R3C1, no 6 in R3C2, clean-up: no 2 in R4C78, no 1 in R5C9
23a. R4C23 = {69} = 15 -> R3C12 = 11 = [47/74/83], no 2
24. 45 rule on N1 4(2+2) outies R12C4 + R4C23 = 26, R4C23 = 15 (step 23a) -> R12C4 = 11 = {29/47}, no 3,6
24a. Killer pair 2,4 in R12C4 and R12C5, locked for N2
25. 2 in R3 only in R3C789, locked for N3 -> R2C8 = 1, R3C3 = 6 (step 6), R4C23 = [69], clean-up: no 2 in R1C1, no 4 in R1C5, no 7 in R4C7
26. 6 in N2 only in R12C6, locked for C6, clean-up: no 3 in R7C7
26a. 19(3) cage at R1C6 (step 16) contains 6 = {469/568}, no 3,7, clean-up: no 4 in R3C7 (step 1)
27. 4 in R3 only in R3C12 (step 23a) = {47}, locked for R3 and N1, clean-up: no 1 in R1C12
27a. R1C12 = [53], R2C2 = 9, R2C1 = 2, R12C3 = [18], R1C5 = 2, R2C5 = 3, R2C6 = 6, clean-up: no 8 in R1C8, no 4,7 in R1C9, no 7 in R45C1, no 9 in R5C5
27b. R45C1 = [84], R3C12 = [74], R9C1 = 1, clean-up: no 3 in R4C9, no 8 in R8C2, no 6 in R8C9
28. Naked triple {159} in R3C456, locked for R3 and N2 -> R12C6 = 8, R2C7 = 5 (step 26a), R3C7 = 2 (step 1), clean-up: no 4,7 in R7C6, no 8 in R9C8
28a. Naked pair {47} in R12C4, locked for C4
29. R7C4 = 8 -> R78C3 = 6 = {24}, locked for C3 and N7
29a. Naked pair {57} in R78C2, locked for C2 and N7 -> R9C23 = [83], clean-up: no 4 in R8C9
30. R7C89 = R5C7 + 1 (step 8)
30a. R5C7 = 8 -> R7C89 = 9 = {27}/[63], no 1,5
31. Naked pair {57} in R45C5, locked for C5 and N5
31a. R7C5 = 1 (hidden single in R7), R3C5 = 9
31b. Naked pair {46} in R89C5, locked for C5 and N8 -> R6C5 = 8
31c. Killer pair 4,6 in R9C5 and R9C78, locked for R9, clean-up: no 1,3 in R8C9
32. Naked pair {25} in R89C9, locked for C9 and N9, clean-up: no 7 in R7C89 (step 30a)
32a. R7C89 = [63], R5C9 = 6, R4C9 = 1, R1C9 = 9, R6C9 = 7, R2C9 = 4, R1C78 = [67], R3C89 = [38], R4C78 = [35]
33. R78C1 = [96], R89C5 = [46], R78C3 = [42], R7C7 = 7, R78C2 = [57], R7C6 = 2, R89C9 = [52]
33a. Naked pair {39} in R8C46, locked for R8 and N8 -> R8C78 = [18], R9C46 = [57], R8C4 = 9 (cage sum)
and the rest is naked singles.
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