Prelims
a) R1C78 = {17/26/35}, no 4,8,9
b) R3C23 = {29/38/47/56}, no 1
c) R3C67 = {17/26/35}, no 4,8,9
d) R4C23 = {19/28/37/46}, no 5
e) R6C78 = {69/78}
f) R7C34 = {79}
g) R7C78 = {17/26/35}, no 4,8,9
h) R9C23 = {15/24}
i) 10(3) cage at R6C9 = {127/136/145/235}, no 8,9
1. Naked pair {79} in R7C34, locked for R7, clean-up: no 1 in R7C78
1a. 17(3) cage at R7C2 = {269/278/359/368/467} (cannot be {179} which clashes with R7C3, cannot be {458} which clashes with R9C23), no 1
1b. 38(7) cage at R6C1 = {1256789/1346789/2345789} must contain 7,8,9
1c. R6C1234 can only contain one of 7,9 (both would clash with R6C78), R89C1 can only contain one of 7,9 (both would clash with R7C3) -> R6C1234 and R89C1 must each contain one of 7,9
1d. Killer pair 7,9 in R7C3 and R89C1, locked for N7
1d. 17(3) cage = {368} (only remaining combination), locked for N7
1e. 8 in 38(7) cage only in R6C1234, locked for R6, clean-up: no 7 in R6C78
1f. Naked pair {69} in R6C78, locked for R6 and N6
1g. 9 in 38(7) cage only in R89C1, locked for C1 and N7 -> R7C34 = [79], clean-up: no 4 in R3C2, no 3 in R4C2
1h. 38(7) cage at R6C1 = {2345789} (only remaining combination), no 1
1i. 1 in N7 only in R9C23 = {15}, locked for R9 and N7
1j. Naked triple {249} in R789C1, locked for C1 and 38(7) cage
1k. Naked quad {3578} in R6C1234, locked for R6
1l. R4C23 = {19/28/46} (cannot be [73] which clashes with R6C123 which must contain at least one of 3,7), no 7 in R4C2, no 3 in R4C3
2. 45 rule on N9 1 innie R8C7 = 1 outie R6C9 + 3, R6C9 = {124} -> R8C7 = {457}
2a. 10(3) cage at R6C9 = {127/145/235} (cannot be {136} which clashes with R7C78), no 6
2b. 10(3) cage at R6C9 = {127/145} (cannot be {235} = 2{35} which clashes with R6C9 + R8C7 = [25]), no 3, 1 locked for C9
2c. 4 of {145} must be in R6C9 (cannot be 1{45} which clashes with R6C9 + R8C7 = [14]), no 4 in R78C9
2d. 7 of {127} must be in R8C9 -> no 2 in R8C9
2e. 10(3) cage = {12}7
or 10(3) cage = 4{15} => R8C7 = 7
-> 7 in R8C79, locked for R8 and N9
2f. 8,9 in N9 only in 24(4) cage at R8C8 = {1689/3489} (cannot be {2589} which clashes with R7C78), no 2,5
2g. 1 of {1689} must be in R8C8 -> no 6 in R8C8
2h. 2 in N9 only in R7C789, locked for R7 -> R7C1 = 4
3. 45 rule on R6789 1 outie R5C6 = 1 innie R6C5 + 5, R6C5 = {124} -> R5C6 = {679}
3a. 20(4) cage at R5C6 = {1289/1469/1478/1568/2378/2459} (cannot be {3458} because R5C6 only contains 6,7,9, cannot be {1379} because 7,9 only in R5C6, cannot be {2468} because 2,4 only in R6C6, cannot be {2369} = [92]{36} which clashes with R7C78, cannot be {2567} = [72]{56} which clashes with R7C78, cannot be {3467} = [74]{36} which clashes with R7C78)
[Ed pointed out that {2459} should also be eliminated for the same reason as {2468}; having missed that I eliminated it in step 3c for a different reason.]
3b. Hidden killer triple 3,6,8 in R7C2, R7C56 and R7C78 for R7, R7C2 = {368}, R7C78 contains one of 3,6 -> R7C56 must contain ONE of 3,6,8
3c. 20(4) cage = {1289/1469/1478/1568} (cannot be {2378} which contains both of 3,8 in R7C56, cannot be {2459} which doesn’t contain 3,6,8)
3d. 20(4) cage = {1289/1478} (cannot be {1469} = [94]{16} which clashes with R5C6 + R6C5 = [94], cannot be {1568} = [61]{58} which clashes with R5C6 + R6C5 = [61]), no 5,6, clean-up: no 1 in R6C5
3e. 20(4) cage = {1289/1478}, 8 locked for R7 and N8
3f. Killer pair {36} in R7C2 and R7C78, locked for R7
3g. Naked pair {18} in R7C56, locked for R7, N8 and 20(4) cage
3h. Naked pair {24} in R6C56, locked for R6 and N5 -> R6C9 = 1, R8C7 = 4 (step 2), R8C9 = 7 (hidden single in N9), R7C9 = 2 (cage sum), clean-up: no 6 in R7C78
3i. Naked pair {35}, locked for R7 and N9 -> R7C2 = 6, clean-up: no 5 in R3C3, no 4 in R4C3
3j. Naked triple {689} in R9C789, locked for R9 and N9 -> R89C1 = [92], R8C8 = 1, clean-up: no 7 in R1C7, no 5 in R3C3, no 4 in R4C3
3k. Naked pair {37} in R9C56, locked for N8 -> R9C4 = 4
3l. R9C56 = {37} = 10, R8C7 = 4 -> R8C6 = 5 (cage sum), clean-up: no 3 in R3C7
3m. 35(7) cage at R1C9 = {1235789/1245689/1345679/2345678} must contain 5, CPE no 5 in R5C9
3n. 1 of {1235789/1245689/1345679} must be in R4C6, 3 or 8 of {2345678} must be in R4C6 (otherwise 3,4,8 clashes with R5C9, CPE) -> R4C6 = {138}
3o. 6 in C6 only in R123C6, locked for N2
4. 12(3) cage at R1C6 = {129/147/156/237/246/345} (cannot be {138} which clashes with R47C6), no 8
4a. 20(4) cage at R5C6 (step 3e) = {1289/1478} -> R56C6 = [74/92]
4b. 9 in C6 only in 12(3) cage = {129}
or R56C6 = [92]
-> no 2 in R3C6, clean-up: no 6 in R3C7
4c. Hidden killer pair 2,4 in 12(3) cage and R6C6 for C6, R6C6 = {24} -> 12(3) cage must contain one of 2,4 in R12C6 -> 12(3) cage = {129/147/237/246/345} (cannot be {156} which doesn’t contain 2 or 4)
4d. 12(3) cage = {129/147/246/345} (cannot be {237} because R129C6 = {237} clashes with R56C6)
4e. 5 of {345} must be in R2C5 -> no 3 in R2C5
4f. 6 in C6 only in 12(3) cage = {246}, locked for N2
or R3C67 = [62]
-> no 2 in R3C45
5. 23(4) cage at R3C4 = {1589/1679/3569/3578/4568} (cannot be {3479} because 4,9 only in R3C5)
5a. 9 of {1589/1679} must be in R3C5 -> no 1 in R3C5
6. 45 rule on N1 3 innies R2C13 + R3C1 = 10 = {127/136/145/235}, no 8,9
6a. 2,4 of {145/235} must be in R2C3 -> no 5 in R2C3
7. 45 rule on N3 4 innies R123C9 + R3C7 = 21 = {349}5/{469}2/{568}2/{389}1/{569}1 (cannot be {356}7 which clashes with R1C78), no 7 in R3C7, clean-up: no 1 in R3C6
7a. R123C9 + R3C7 = 21 = {469}2/{568}2/{389}1/{569}1 (cannot be {349}5 because 35(7) cage at R1C9 (step 3m) = {1345679} requires 6 in R123C9), no 5 in R3C7, clean-up: no 3 in R3C6
7b. Killer triple 6,8,9 in R123C9 and R9C9, locked for C9
7c. 35(7) cage at R1C9 (step 3m) = {1235789/1245689/2345678} (cannot be {1345679} which clashes with R5C9 using CPE), 2 locked for R4 and N6, clean-up: no 8 in R4C23
7d. 3 of {2345678} must be in R4C6 (otherwise 3,4 clashes with R5C9, CPE) -> no 8 in R4C6
7e. R7C6 = 8 (hidden single in C6) -> R7C5 = 1
8. Max R1C5 + R2C3 = 15 -> min R12C4 = 4 must include at least one of 3,5,7,8
8a. 23(4) cage at R3C4 (step 5) = {1589/1679/3569/4568} (cannot be {3578} which clashes with R12C4, CPE)
8b. 4,9 on 23(4) cage only in R3C5 -> R3C5 = {49}
8c. Killer pair 4,9 in 12(3) cage at R1C6 (step 4d) and R3C5, locked for N2
8d. 19(4) cage at R1C4 = {1378/1567/2368/2458/3457} (cannot be {1468/2467} because 4,6 only in R2C3)
8e. 4,6 of {2368/2458} must be in R2C3 -> no 2 in R2C3
8f. R2C13 + R3C1 (step 6) = {136/145}, no 7, 1 locked for N1
8g. R3C23 = {29/38}/[74] (cannot be [56] which clashes with R2C13 + R3C1), no 5 in R3C2, no 6 in R3C3
8h. Consider placement for 6 in C6
6 in 12(3) cage at R1C6 (4d) = {246}, locked for N2 => R3C5 = 9
or 6 in R3C67 = [62]
-> R3C5 = 9 and/or R3C7 = 2 -> R3C23 = {38}/[74], no 2,9
8i. 2 in R3 only in R3C78, locked for N3, clean-up: no 6 in R1C78
8j. Killer pair 3,4 in R2C13 + R3C1 and R3C23, locked for N1
9. R123C9 (step 7a) = {469}/{568}/{389}/{569}
9a. 16(3) cage at R2C7 = {178/349/358} (cannot be {169//259/268} which clash with R123C9, cannot be {367/457} which clash with R1C78), no 2,6
9b. R3C7 = 2 (hidden single in N3) -> R3C6 = 6
9c. 6 in N3 only in R123C9, locked for C9
9d. R4C8 = 2 (hidden single in R4)
10. 12(3) cage at R1C6 (step 4d) = {129/147/345}, 23(4) cage at R3C4 (step 8a) = {1589/1679/3569/4568}
10a. Consider placement for 1 in C4
1 in R12C4 => 12(3) cage = {345}, locked for N2 => R3C5 = 9
or 1 in 23(4) cage = {1589/1679} => R3C5 = 9
-> R3C5 = 9
[I spent some time trying to find a more routine way to reach the result given by step 10a.
Cracked. The rest is fairly straightforward.]
10b. R5C6 = 9 (hidden single in C6) -> R6C6 = 2 (step 4a), R6C5 = 4
10c. 12(3) cage = {147/345}, no 2
10d. R2C5 = {57} -> no 7 in R12C6
10e. R9C6 = 7 (hidden single in C6) -> R9C5 = 3
10f. 2 in N2 only in 19(4) cage at R1C4 (step 8d) = {2368/2458}, no 1,7, 8 locked for N2
10g. 4,6 only in R2C3 -> R2C3 = {46}
10h. Consider combinations for 19(4) cage
19(4) cage = {2368}, 3 locked for N2 => 12(3) cage = {147}
or 19(4) cage = {2458}, 5 locked for N2 => R2C5 = 7 => 12(3) cage = {147}
-> 12(3) cage = {147}, R2C5 = 7, R12C6 = {14}, locked for C6 and N2
[Ed pointed out that I’d missed the simpler 12(3) cage = {147} (cannot be {345} which clashes with 19(4) cage.]
10i. 23(4) cage = {1589/3569} (cannot be {1679} because R3C4 only contains 3,5), no 7, 5 locked for C4
10j. R6C4 = 7 (hidden single in C4) -> R6C123 = {358}, locked for N4
11. R4C6 = 3 -> 35(7) cage at R1C9 (step 7c) = {2345678} (only remaining combination) -> R4C7 = 7, R1234C9 = {4568}, locked for C9 -> R59C9 = [39]
11a. 8 in N6 only in R5C78, locked for R5
11b. 9 in R4 only in R4C23 = {19}, locked for R4 and N4 -> R45C1 = [67]
11c. Naked pair {58} in R4C45, locked for R4 and N5 -> R4C9 = 4, R5C45 = [16], R8C45 = [62]
11d. R3C5 = 9, R5C4 = 1 -> R34C4 = 13 = [58]
11e. R12C4 = {23}, R1C5 = 8 -> R2C3 = 6 (cage sum)
11f. R2C13 + R3C1 (step 8f) = {136} (only remaining combination), 1,3 locked for C1 and N1 -> R1C1 = 5, clean-up: no 3 in R1C78, no 8 in R3C23
and the rest is naked singles.