Prelims
a) R12C8 = {49/58/57}, no 1,2,3
b) R12C9 = {49/58/57}, no 1,2,3
c) R23C2 = {17/26/35}, no 4,8,9
d) R3C89 = {14/23}
e) R4C56 = {19/28/37/46}, no 5
f) R4C89 = {14/23}
g) R89C1 = {19/28/37/46}, no 5
h) 19(3) cage at R3C6 = {289/379/469/478/568}, no 1
i) 20(3) cage at R4C2 = {389/479/569/578}, no 1,2
j) 10(3) cage at R5C6 = {127/136/145/235}, no 8,9
1. 45 rule on C1 1 outie R1C2 = 9, clean-up: no 4 in R2C8, no 4 in R2C9
1a. R1C2 = 9 -> R12C1 = 8 = {17/26/35}, no 4,8
1b. 4,8 in N1 only in R12C3 + R3C13, CPE no 4,8 in R3C5
1c. 36(7) cage at R1C3 contains 9, locked for N2
1d. Max R45C2 = 15 -> min R4C3 = 5
2. 45 rule on R12 2 outies R3C25 = 13 = {67} (only possible combination), locked for R3
2a. R3C2 = {67} -> R2C2 = {12}
2b. R3C7 = 9 (hidden single in R3), clean-up: no 4 in R1C8, no 4 in R1C9
2c. R3C6 + R4C7 = 10 = {28}/[37/46], no 5, no 3,4 in R4C7
2d. Naked quad {5678} in R12C89, locked for N3
2e. 45 rule on N3 2 outies R1C56 = 11 = {38/47/56}, no 1,2
2f. Naked pair {67} in R3C25, CPE no 6,7 in R12C3
2g. Hidden killer pair 6,7 in R12C1 and R3C2 for N1, R3C2 = {67} -> R12C1 must contain one of 6,7 -> R12C1 (step 1a) = {17/26}, no 3,5
2h. Naked quad {1267} in R12C1 + R23C2, locked for N1
3. 45 rule on R5 3 innies R5C129 = 22 = {589/679}, 9 locked for R5
3a. 13(3) cage at R5C3 = {148/238/247/346} (cannot be {157/256} which clash with R5C129), no 5
3b. 10(3) cage at R5C6 = {127/136/145/235}
3c. {127/136} cannot be 7{12}/6{13} which clash with R4C89 -> no 6,7 in R5C6
4. 45 rule on C9 3 outies R349C8 = 13, max R34C8 = 7 -> min R9C8 = 6
[I won’t continue this step, because I don’t used Unique Rectangles, but I spotted that R3C89 and R4C89 must have different combinations because otherwise it would be impossible to determine their permutations; I think that’s right even though they’re in different nonets. If I’d used that, then R34C8 cannot total 5 so no 8 in R9C8.]
5. 45 rule on C89 2 outies R67C7 = 1 innie R5C8 + 9
5a. Max R67C7 = 15 -> max R5C8 = 6
5b. Min R5C8 = 1 -> min R67C7 = 10, no 1 in R67C7
6. 18(3) cage at R5C9 = {189/279/369/459} (cannot be {378/468/567} which clash with R12C9), 9 locked for C9
7. 45 rule on R12345 2 outies R67C1 = 1 innie R5C9
7a. Max R67C1 = 9, no 9 in R67C1
8. 13(3) cage at R3C3 = {148/157/238/256} (cannot be {139} = [319] which clashes with R3C89, cannot be {247} = [427] which clashes with R3C89, cannot be {346} = {34}6 which clashes with R3C89), no 9
8a. 5 of {157/256} must be in R3C3 -> no 5 in R34C4
8b. 5 in R3 only in R3C13, locked for N1
9. 5 in R4 only in R4C123, locked for N4
[First time through I’d overlooked this, so I’ve reworked from here using a few of my previous steps.]
9a. R5C129 (step 3) = [985]/{679}, R67C1 (step 7) = R5C9
9b. Consider combinations for 20(3) cage at R4C2 = {389/479/569/578}
20(3) cage = {389/479/569} => R4C3 = 9 => R5C9 = 9 (hidden single in R5) => R5C129 = {67}9
or 20(3) cage = {578}, locked for N4 => R345C1 cannot total 22 (because doesn’t contain 7 and 5,8 only in R3C1) => R67C1 cannot total 5 => no 5 in R5C9
-> R5C129 = {679}, locked for R5
9c. Naked pair {67} in R35C2, locked for C2
9d. 20(3) cage = {479/569/578} (cannot be {389} because R5C2 only contains 6,7), no 3
9e. 20(3) cage = {569/578} (cannot be {479} = [497] which clashes with R4C56 + R4C89, killer ALS block), no 4, 5 locked for N4
9f. R5C2 = {67} -> no 6,7 in R4C3
10. 45 rule on R12345 4(3+1/2+2) innies R34C1 + R5C19 = 27
10a. Max R5C19 = 16 -> min R34C1 = 11, no 9 in R3C1 -> no 1,2 in R4C1
11. 45 rule on R123 4 innies R3C1346 = 18 contains 5,8 for R3 = {1458/2358}
11a. 1 of {1458} must be in R3C4 -> no 4 in R3C4
11b. 13(3) cage at R3C3 (step 8) = {148/157/238/256}
11c. 1 of {148} must be in R3C4 (cannot form R3C1346 = {1458} with 4,8 in R3C34), 7 of {157} must be in R4C4 -> no 1 in R4C4
11d. 1 in R4 only in R4C56 = {19} or in R4C89 = {14} -> no 4 in R4C56 (locking-out cages), clean-up: no 6 in R4C56
[That was another step I overlooked first time through.]
12. R67C1 (step 7) = R5C9
12a. Consider permutations for R5C129 (step 9b) = {679}
6 in R5C12 => no 6 in R4C1
or R5C129 = [976] => R67C1 = 6, R567C1 = 15 => R34C1 = 12 cannot contain 6
-> no 6 in R4C1
12b. 45 rule on R123 2 outies R4C47 = 1 innie R3C1 + 5
12c. R3C1 = {3458} -> max R4C47 = 13 contains 6 for R4 -> no 8 in R4C47, clean-up: no 2 in R3C6 (step 2c)
13. R34C1 + R5C19 (step 10) = 27, R67C1 (step 7) = R5C9
13a. Consider combinations for R5C19 = {69/79}
R5C19 = {69} = 15, locked for R5 => R5C2 = 7, R567C1 = 15 => R34C1 = 12 but cannot be [57] which clashes with R5C2
or R5C19 = {79} = 16, R567C1 = 16 => R34C1 = 11 = {38}/[47]
-> no 5 in R3C1
13b. R3C3 = 5 (hidden single in R3) -> R34C4 = 8 = [17/26]
13c. R4C2 = 5 (hidden single in R4)
13d. R4C47 = R3C1 + 5 (step 12b)
13e. R4C47 = [62]/{67} = 8,13 -> R3C1 = {38}
13f. 4 in N1 only in R12C3, locked for C3 and 36(7) cage at R1C3
13g. 36(7) cage = {1245789/1345689/2345679}, 5 locked for N2, clean-up: no 6 in R1C56 (step 2e)
[After those additional steps, a different angle on my original first forcing chain.]
14. 36(7) cage at R1C3 contains 9 and three of {18}, {27}, {36} and {45}
14a. Consider placements for R3C4 = {12}
R3C4 = 1 => R3C89 = {23}, locked for R3 => R3C16 = [84]
or R3C4 = 2, R3C89 = {14}, locked for N3 => R12C7 = {23}, locked for C7, no 2 in R4C7 => no 8 in R3C6 (step 2c) => R3C6 = 3
-> R3C6 = {34}, clean-up: no 2 in R4C7 (step 2c)
14b. R3C1 = 8 (hidden single in R3), clean-up: no 2 in R89C1
14c. Naked pair {34} in R12C3, 3 locked for C3 and 36(7) cage at R1C3
14d. Killer pair 3,4 in R1C3 and R1C56, locked for R1
14e. Killer pair 1,2 in R1C7 and R3C89, locked for N3
14f. Naked pair {67} in R4C47, locked for R4, clean-up: no 3 in R4C56
14g. Killer pair 8,9 in R4C3 and R4C56, locked for R4
15. R5C129 (step 9b) = {679} cannot be [967] (which clashes with 20(3) cage at R4C2) -> no 7 in R5C9
15a. R67C1 (step 7) = R5C9, R7C1 = 5 (hidden single in C1) -> R5C9 = R6C1 + 5, R5C9 = {69} -> R6C1 = {14}
15b. 2 in C1 only in R12C1 = {26}, locked for N1, 6 also locked for C1 -> R23C1 = [17], R5C2 = 6, R4C3 = 9 (cage sum), R3C5 = 6, clean-up: no 1 in R4C56
15c. R357C1 = [875] = 20 -> R46C1 = 7 = [34], clean-up: no 2 in R4C89
15d. Naked pair {28} in R4C56, locked for N5
15e. 1 in N2 only in R13C4, locked for C4
15f. R5C3 = 8 (hidden single in R5) -> R6C23 = [21]
15g. R5C3 = 8 -> R5C45 = 5 = [41]
15h. R5C9 = 9 -> R67C9 = 9 = {36}/[54/72/81], no 7,8 in R7C9
16. 17(3) cage at R9C5 cannot be {179} (which clashes with R9C1) -> no 1 in R9C67
16a. 1 in N8 only in R78C6, locked for 15(4) cage at R7C6, no 1 in R8C7
16b. R1C7 = 1 (hidden single in C7), clean-up: no 4 in R3C89
[I’m happy that I didn’t have to assume that R3C89 and R4C89 had different combinations.]
16c. Naked pair {23} in R3C89, locked for R3, 3 locked for N3 -> R3C46 = [14], R2C7 = 4, R12C3 = [43], R4C4 = 7 (cage sum), R4C7 = 6, clean-up: no 7 in R1C56 (step 2e), no 3 in R7C9 (step 15h)
16d. Naked pair {38} in R1C56, 8 locked for R1 and N2, clean-up: no 5 in R2C8, no 5 in R2C9
16e. 5 in N3 only in R1C89, locked for R1 -> R1C4 = 2, R12C1 = [62], R2C89 = {68} (hidden pair in R2)
16f. 8 in C4 only in R789C4, locked for N8
[It ought to be getting easier now, but still some work to do.]
17. R67C7 = R5C8 + 9 (step 5)
17a. R67C7 cannot total 14 -> no 5 in R5C8
17b. Naked pair {23} in R35C8, locked for C8
17c. R5C8 = {23} -> R67C7 = 11,12 = {38}/[57], no 7 in R6C7, no 2 in R7C7
18. 45 rule on N69 2 remaining innies R89C7 = 1 outie R5C6 + 6
18a. R5C6 = {35} -> R89C7 = 9,11 = {27/38}, no 5
18b. 5 in C7 only in R56C7, locked for N6, clean-up: no 4 in R7C9 (step 15h)
19. 7 in C7 only in R789C7, locked for N9
19a. 17(3) cage at R8C9 = {269/359/368/458}, no 1
19b. R9C1 = 1 (hidden single in R9) -> R8C1 = 9
20. R349C8 (step 4) = 13 = [319/346], no 8 -> R3C89 = [32], clean-up: no 7 in R6C9 (step 15h)
20a. R5C8 = 2 -> R67C7 (step 5) = 11 = {38}, locked for C7, 8 locked 28(5) cage at R6C7 -> R5C67 = [35], R1C56 = [38], R4C56 = [82]
20b. R6C8 = 7, R1C8 = 5 -> R2C8 = 8, R12C9 = [76], R7C9 = 1 -> R6C9 = 8 (cage sum), R67C7 = [38], R4C89 = [14]
20c. R89C9 = {35} -> R9C8 = 9 (cage sum)
20d. 17(3) cage at R9C5 = [467] (only remaining permutation)
20e. R8C6 = 1 (hidden single in C6), R8C7 = 2 -> R7C6 + R8C5 = 12 = [75]
and the rest is naked singles.