Disjoint 30(5) cage in R4C9 + R67C9 + R9C89
Prelims
a) R2C45 = {18/27/36/45}, no 9
b) 11(2) cage at R2C8 = {29/38/47/56}, no 1
c) R4C78 = {17/26/35}, no 4,8,9
d) R56C6 = {29/38/47/56}, no 1
e) R7C12 = {59/68}
f) R89C6 = {29/38/47/56}, no 1
g) R9C34 = {13}
1a. 45 rule on N8 1 innie R9C4 = 3 -> R9C3 = 1, clean-up: no 6 in R2C5, no 8 in R89C6
1b. 45 rule on N7 2 remaining outies R56C3 = 8 = {26/35}
1c. 45 rule on N47 2 remaining outies R3C12 = 3 = {12}, locked for R3 and N1, clean-up: no 9 in R2C8
1d. 45 rule on N147 2 remaining outies R13C4 = 7 = [16/25]
1e. 45 rule on N3 2 innies R12C7 = 8 = {17/26/35}, no 4,8,9
1f. 45 rule on N3 3 outies R1C56 + R2C6 = 19 = {289/379/469/478/568}, no 1
1g. 45 rule on N5 2 outies R3C56 = 10 = {37/46}
1h. 45 rule on N5 3 innies R4C456 = 9 = {126/135} (cannot be {234} which clashes with R3C56), 1 locked for R4 and N5, clean-up: no 7 in R4C78
1i. R56C6 = {29/38/47} (cannot be {56} which clashes with R4C456), no 5,6
1j. 45 rule on N6 R46C9 = 11 = {29/38/47/56} (cannot be {56} which clashes with R4C78), no 1,5,6
1k. Naked quint {12356} in R4C45678, locked for R4, clean-up: no 8,9 in R6C9
1l. Max R3C1 = 2 -> min R45C1 = 12, no 1,2,3 in R5C1
1m. Naked pair 1,2 in R3C12, CPE no 1,2 in R6C1
1n. R2C45 = {18/27/45} (cannot be [63] which clashes with R3C56), no 3,6
1o. 9 in N2 only in R1C56 + R2C6 = {289/379/469}, no 5
1p. Max R1C4 = 2 -> min R1C23 = 14, no 3,4 in R1C23
2a. R3C1 = 1 (hidden single in C1) -> R45C1 = 13 = {49}/[76/85], no 7,8 in R5C1
2b. R3C2 = 2
2c. 2 in N4 only in R56C3 = {26} (step 1b), locked for C3, 6 locked for N4 and 23(5) cage at R5C3, clean-up: no 7 in R4C1
2d. 2 in N7 only in 15(3) cage at R8C1 = {249/267} (cannot be {258} which clashes with R7C12), no 3,5,8
2e. 45 rule in N7 3 innies R7C3 + R8C23 = 15 must contain 3 for N7 = {348/357}, no 9
3a. R3C4 ‘sees’ all of N1 except for R1C23
3b. R13C4 = [16/25] -> 16(3) cage at R1C2 = [691]/{59}2, 9 locked for R1 and N1
3c. R2C6 = 9 (hidden single in N2), clean-up: no 2 in R56C6, no 2 in R89C6
3d. 45 rule on N3 2 remaining outies R1C56 = 10 = {28/37/46}
3e. 3 in N2 only in R1C56 = {37} or R3C56 (step 1g) = 10 = {37} (locking cages), 7 locked for N2, clean-up: no 2 in R2C45
3f. 2 in N2 only in R1C456, locked for R1, clean-up: no 6 in R2C7 (step 1e)
4a. R12C7 (step 1e) = {17/26/35}, R4C456 (step 1h) = {126/135}
4b. Consider placements for {37} in N2 (step 3e)
R1C56 = {37} => R12C7 = [62] => R4C78 = {35}, locked for R4
or R3C56 = {37}, 3 locked for 19(5) cage at R3C5
-> R4C456 = {126}, 2,6 locked for R4 and N5, 6 locked for 19(5) cage, clean-up: no 4 in R3C56 (step 1g)
4c. Naked pair {37} in R3C56, locked for R3 and N2, clean-up: no 4,8 in R2C8
4d. Naked pair {35} in R4C78, locked for N6, clean-up: no 8 in R4C9 (step 1j)
4e. R12C7 = {17} (cannot be {35} which clashes with R4C7, cannot be [62] which clashes with R1C56), locked for C7 and N3, clean-up: no 4 in R3C7
4f. 8 in R4 only in R4C123, locked for N4
4g. Killer pair 3,7 in R3C6 and R56C6, locked for C6, clean-up: no 4 in R89C6
4h. Naked pair {56} in R89C6, locked for C6, clean-up: no 4 in R1C5 (step 3d)
5a. R4C6 = 1 (hidden single on D/)
5b. R8C8 = 1 (hidden single on D\)
5c. R5C9 = 1 (hidden single in N6)
5d. R6C2 = 1 (hidden single in N4)
5e. Killer pair 2,6 in R13C4 and R4C4, 2 locked for C4
[I ought to have seen step 14a at this point; fortunately the delay wasn’t significant.]
6. 45 rule on N9 3 innies R7C9 + R9C89 = 19 = {289/478/568} (cannot be {379/469} which clash with R46C9), no 3, 8 locked for N9
7a. 2 in R2 only in R2C89, whichever of 5,6 is in R3C4 must be in R1C23 and therefore in R2C89 -> R2C89 = {25/26}, clean-up: no 8 in R3C7
7b. 8 in C7 only in R56C7, locked for N6
7c. 3 in N3 only in R1C89, locked for R1
8a. R46C9 = 11 (step 1j) = {47}/[92]
8b. Consider placement of 9 on D\
9 in R5C5 + R7C7 => no 9 in R3C7 => no 2 in R2C8 => R2C9 = 2 (hidden single in R2)
or R9C9 = 9
-> R46C9 = {47}, locked for C9, N6 and 30(5) disjoint cage at R4C9
8c. 4 in C7 only in R789C7, locked for N9
8d. 9 in R4 only in R4C123, locked for N4, clean-up: no 4 in R4C1 (step 2a)
8e. R7C8 = 7 (hidden single in N9)
9. 2 on D/ only in R2C8 + R9C1, CPE no 2 in R9C8
10. R7C9 + R9C89 (step 6) = 19 = {289/568}
10a. Consider placement for 2 in R2
R2C8 = 2 => R56C8 = {69}, locked for C8 and N6, R56C7 = {28}, locked for C7, R8C1 = 2 (hidden single in C1) => 2 in N9 only in R7C9 + R9C89 = {289} => R9C8 = 8
or R2C9 = 2 => R7C9 + R9C89 = {568} => naked triple {568} in R9C689, 8 locked for N9
-> 8 in R9C89, locked for R9 and N9
11a. 3 in C9 only in R18C9, CPE no 3 in R8C2 using D/
11b. 3 in N7 only in R78C3, locked for C3
11c. Consider placement for 3 in R78C3
R7C3 = 3 => R8C9 = 3 (hidden single in C9)
or R8C3 = 3
->no 3 in R8C7
11d. 3 in R7 only in R7C37, CPE no 3 in R5C5 using both diagonals
11e. 3 in N5 only in R6C5 or in R56C6 = {38} -> no 8 in R6C5 (locking-out cages)
12a. 15(3) cage at R8C1 (step 2d) = {249/267}
12b. Consider permutations for R45C1 = 13 (step 2a) = [85/94]
R45C1 = [85]
or R45C1 = [94] => 15(3) cage = {267}, 6 locked for N7 => R7C12 = [59]
-> 5 in R57C1, locked for C1
Also no 8 in R7C1 -> no 6 in R7C2
12c. 5 in N4 only in R5C12, locked for R5
13. Consider placement for 2 in C4
R1C4 = 2 => R7C6 = 2 (hidden single in C6)
or R4C4 = 2, placed for D\
-> no 2 in R7C7
14a. 6 in C4 only in R34C4, CPE no 6 in R1C1 + R2C2 using D\
14b. Hidden killer pair 2,6 in R4C4 and R7C7 + R9C9 for D\, R4C4 = {26} -> R7C7 + R9C9 must contain one of 2,6
[I hope I’ll be able to use this later.]
15a. 33(6) cage at R1C1 = {345678}, R13C4 (step 1d) = [16/25], 15(3) cage at R8C1 (step 2d) = {249/267}
15b. Consider placement for 6 in C2
R1C2 = 6 => R3C4 = 6
or R9C2 = 6 => R89C1 = {27}, 7 locked for C1 => R1C47 = [17] (hidden pair in R1)
-> R13C4 = [16], R12C7 = [71], clean-up: no 8 in R2C45
[Cracked at last.]
15c. Naked pair {45} in R2C45, locked for R2, 4 locked for N2
15d. R1C1 + R3C3 = [45] (hidden pair in N1), both placed for D\
15e. R5C1 = 5 -> R4C1 = 8 (cage sum)
15f. R789C1 = {269} (hidden triple in C1), 6,9 locked for N7
15g. R3C7 = 9 -> R2C8 = 2, both placed for D/
15h. R2C9 = 6, R3C89 = [48]
15i. R789C1 = [926], R89C6 = [65]
15j. R7C9 + R9C89 (step 10) = [289]
15k. R89C7 = [54], R9C2 = 7, R8C9 = 3, R1C9 = 5, placed for D/
15l. R2C4 = 5 (hidden single in C4)
15m. Naked pair {48} in R8C23, locked for R8 and N7
15n. R19C5 = [82], R5C5 = 7, clean-up: no 4 in R5C6
15o. Naked pair {38} in R56C6, locked for C6 and N5, R6C4 = 4, placed for D/
and the rest is naked singles, without using the diagonals.