Prelims
a) R12C7 = {14/23}
b) R1C89 = {49/58/67}, no 1,2,3
c) R23C5 = {18/27/36/45}, no 9
d) R34C1 = {19/28/37/46}, no 5
e) R4C23 = {13}
f) R45C5 = {29/38/47/56}, no 1
g) R5C67 = {18/27/36/45}, no 9
h) R78C5 = {29/38/47/56}, no 1
i) 20(3) cage at R2C6 = {389/479/569/578}, no 1,2
j) 10(3) cage at R2C8 = {127/136/145/235}, no 8,9
k) 19(3) cage at R4C6 = {289/379/469/478/568}, no 1
l) 8(3) cage at R7C8 = {125/134}
m) 44(8) cage at R5C1 = {23456789}, no 1
Steps Resulting From Prelims and Initial Placement
1a. Naked pair {13} in R4C23, locked for R4 and N4, clean-up: no 7,9 in R3C1, no 8 in R5C5
1b. 8(3) cage at R7C8 = {125/134}, 1 locked for N9
1c. 45 rule on C89 1 innie R4C8 = 8 -> R4C67 = 11 = {29/47/56}, clean-up: no 5 in R1C9, no 2 in R3C1, no 3 in R5C5, no 1 in R5C6
2a. 10(3) cage at R2C8 = {145/235} (cannot be {127/136} which clash with R12C7), no 6,7
2b. Naked quint {12345} in R12C7 + 10(3) cage, locked for N3, clean-up: no 8,9 in R1C89
2c. Naked pair {67} in R1C89, locked for R1 and N3
2d. Naked pair {89} in R3C79, locked for R3, clean-up: no 1 in R2C5, no 2 in R4C1
2e. Max R3C23 = 13 -> min R2C3 = 2
2f. Max R3C67 = 16 -> min R2C6 = 4
3a. 45 rule on R1234 2 outies R5C59 = 9 = {27/45}/[63], no 9 in R5C5, no 1,6,9 in R5C9, clean-up: no 2 in R4C5
3b. 18(3) cage at R3C9 = {279/369/378/459/468} (cannot be {567} which doesn’t contain one of 8,9 for R3C9)
3c. 8,9 must be in R3C9 -> no 9 in R4C9
3d. 1 in R5 only in R5C78, locked for N6
4. 45 rule on N9 3 innies R789C7 = 21 = {579/678} (cannot be {489} which clashes with R3C7), no 2,3,4, 7 locked for C7 and N9, clean-up: no 4 in R4C6 (step 1c), no 2 in R5C6
4a. Killer pair 8,9 in R3C7 and R789C7, locked for C7, clean-up: no 2 in R4C6 (step 1c)
4b. Min R89C7 = 11 -> max R9C6 = 7
4c. Max R89C7 = 16 (cannot be {89} which clashes with R3C7) -> min R9C6 = 2
4d. 16(3) cage at R8C8 = {259/268/349} (cannot be {358} which clashes with R789C7)
4e. 8 of {268} must be in R9C9 -> no 6 in R9C9
4f. 9 in N6 only in 17(3) cage at R5C8 = {179/269/359}, no 4
4g. 1 of {179} must be in R5C8 -> no 7 in R5C8
5. 45 rule on N1 2 innies R1C3 + R3C1 = 9 = [36/54/81], R1C3 = {358}, R3C1 = {146}, clean-up: no 7 in R4C1
6. 45 rule on C6789 3(1+2) innies R1C6 + R6C67 = 9
6a. Min R6C7 = 2 -> max R16C6 = 7, no 7,8,9 in R16C6
7. 45 rule on N2 3(1+1+1) outies R1C3 + R3C7 + R4C4 = 18
7a. Min R1C3 + R3C7 = 11 -> max R4C4 = 7
8. 45 rule on R123 2 outies R4C14 = 1 innie R3C9 + 2
8a. R3C9 = {89} -> R4C14 = 10,11 = {46}/[47/65/92]
8b. R4C67 (step 1c) = [56/65/74/92]
8c. Consider combinations for R4C14
R4C14 = {46}, locked for R4 => R4C67 = [92], 9 placed for D/ => R3C79 = [89]
or R4C14 = [47/65/92] = 11 => R3C9 = 9
-> R3C9 = 9, R4C4 = {257}
[Not the cleanest of forcing chains but the best I could see at this stage; hope Ed or wellbeback found something better to make these placements.]
8d. R3C7 = 8, placed for D/ -> R23C6 = 12 = [57/75/93], no 4,6, no 3 in R2C6
8e. R9C9 = 8 (hidden single in C9), placed for D\
8f. R789C7 (step 4) = {579} (only remaining combination), 5 locked for C7 and N9, 9 also locked for N9, clean-up: no 6 in R4C6 (step 1c), no 4 in R5C6
8g. 8(3) cage at R7C8 = {134} (only remaining combination), locked for N9
8h. Naked pair {26} in R89C8, locked for C8 -> R1C89 = [76], 6 placed for D/, clean-up: no 5 in R4C5
8i. R3C9 = 9 -> R45C9 = 9 = {27/45}, no 3
8j. 10(3) cage at R2C8 (step 2a) = {145/235}
8k. 2 of {235} must be in R2C9 -> no 3 in R2C9
8l. 17(3) cage at R5C8 (step 4f) = {179/359}, no 2
8m. R89C7 = {579} = 12,14,16 -> R9C6 = {246}
[Or 18(3) cage must contain at least one even number -> R9C6 = {246}]
9. R23C6 (step 8d) = [57/75/93], R4C6 = {579} -> combined half cage R234C6 = {57}9/[935/937], 9 locked for C6
10. 12(3) cage at R9C3 = {129/147/156/237/345} (cannot be {246} which clashes with R9C6)
10a. Killer triple 2,4,6 in 12(3) cage, R9C6 and R9C8, locked for R9
11. R23C6 (step 8d) = [57/75/93], R234C6 (step 9) = {57}9/[935/937]
11a. 45 rule on N9 3 outies R789C6 = 15 = {168/267/348/456} (cannot be {357} because R9C6 only contains 2,4,6, cannot be {258} = {58}2 which clashes with 18(3) cage at R5C6 = {58}5)
11b. Consider combinations for R789C6
R789C6 = {168}, locked for C6 => R5C6 = {357} => naked quad 3,5,7,9 in R2345C6, locked for C6
or R789C6 = {267/456}, locked for C6 => R23C6 = [93]
or R789C6 = {348}, locked for C6
-> no 3 in R16C6
[Similarly]
11c. Consider combinations for R789C6
R789C6 = {168}, locked for C6 => R5C6 = {357} => naked quad 3,5,7,9 in R2345C6, locked for C6
or R789C6 = {267}, locked for C6 => R234C6 = [935]
or R789C6 = {348}, locked for C6 => R23C6 = {57}, locked for C6
or R789C6 = {456}, locked for C6
-> no 5 in R16C6
11d. Consider combinations for R789C6
R789C6 = {168/267/456}, locked for C6
or R789C6 = {348}, locked for C6 => R23C6 = {57}, locked for C6 => R5C6 = 6
-> no 6 in R6C6
11e. 3 on D\ only in R1C1 + R2C2 + R3C3, locked for N1, clean-up: no 6 in R3C1 (step 5), no 4 in R4C1, no 7 in R4C4 (step 8c)
11f. R1C6 + R6C67 = 9 (step 6), max R16C6 = 6 -> min R6C7 = 3
11g. Max R34C4 = 12 -> min R2C4 = 2
[I ought to have spotted Ed’s hidden killer triple 1,2,4 in R16C6 and R789C6 for C6, R789C6 only contains one of 1,2,4 -> R6C6 = {124}. Much simpler than my forcing chains!]
12. R5C59 = 9 (step 3a), R45C9 (step 8i) = {27/45}
12a. Consider placements for 1 in N6
1 in R5C67 = [81]
or in 17(3) cage at R5C8 = [179] => R45C9 = {45} => R5C59 = {45}, locked for R5
-> no 5 in R5C6, no 4 in R5C7
13. 15(3) cage at R2C3 = {168/249/267/357/456} (cannot be {159/348} which clash with R1C3 + R3C1, cannot be {258} which clashes with R1C3)
13a. Consider placement for 9 in C6
R2C6 = 9
or R4C6 = 9 => R4C1 = 6 => R3C1 = 4 => 15(3) cage cannot be {249}
-> 15(3) cage = {168/267/357/456}, no 9
13b. Consider combinations for R23C6 = {57}/[93]
R23C6 = {57} => caged X-wing for 5 in R23C6 and 10(3) cage at R2C8, no other 5 in R23
or R23C6 = [93]
-> 15(3) cage = {168/267/456} (cannot be {357}), no 3, 6 locked for N1
13c. 3,9 in N1 only in 21(4) cage at R1C1 = {1389/2379/3459}
13d. 45 rule on R1 3 outies R2C127 = 14 = {149/239/347} (cannot be {158} = [851] which clashes with R1C3, cannot be {248/257} because the two digits in 21(4) cage which aren’t 3 or 9 don’t total 9), no 5,8 in R2C1, no 5 in R2C2
13e. 45 rule on C1 3 innies R125C1 = 13
13f. 45 rule on R1 3 innies R1C127 = 12 = {129/138/345}
[Ed spotted that this leaves 20(4) cage at R1C3 with one of 3,9 in N2, which cracks the puzzle immediately!]
13g. R1C127 = {129} must be [291] (cannot be because [192/912] because 8 of {1389} can only be in R1C2, cannot be [921] because 21(4) cage = [9273] would make R125C1 more than 13), no 9 in R1C1, no 2 in R1C7, clean-up: no 3 in R2C7
13h. 8,9 of {129/138} must be in R1C2 -> no 1,2 in R1C2
13i. R1C127 = {138} must be [381] (cannot be [183] because 21(4) cage = [1893] and cannot make R125C1 total 13 because no 3 in R5C1), no 1 in R1C1
13j. R1C127 = {345} must be [354] (because 21(4) cage = {45}[93] would make R125C1 total more than 13
13k. So the remaining permutations for R1C127 are [291/381/354] -> R1C1 = {23}, R1C2 = {589}, R1C7 = {14}, clean-up: no 2 in R2C7
13l. Naked pair {14} in R12C7, locked for C7 and N3, clean-up: no 7 in R4C6 (step 1c), no 8 in R5C6
13m. Naked pair {35} in R23C8, locked for C8 and N3 -> R2C9 = 2, R56C8 = [19], R6C9 = 7 (cage sum), R4C5 = 7 (hidden single in R4) -> R5C5 = 4, placed for both diagonals, R45C9 = [45], clean-up: no 5 in R2C5, no 2,5 in R3C5, no 2 in R5C7
13n. Naked pair {36} in R5C67, locked for R5
13o. R4C7 = 2 (hidden single in N6) -> R4C6 = 9 (cage sum), placed for D/, R4C1 = 6 -> R3C1 = 4, R1C3 = 5 (step 5), R4C4 = 5, placed for D\, clean-up: no 3 in R3C6 (step 8d)
[Getting easier now; routine clean-ups omitted from here.]
13p. Naked pair {57} in R23C6, locked for C6, 7 also locked for N2
13q. R4C4 = 5 -> R23C4 = 9 = {36}/[81]
13r. R1C456 = {249} (hidden triple in N2), locked for R1 -> R1C12 = [38], R12C7 = [14]
13s. R12C3 = [38] = 11 -> R2C12 = 10 = {19}, 1 locked for N1
13t. R125C1 = 13, R1C1 = 3 -> R25C1 = 10 = [19], R2C2 = 9, placed for D\ -> R7C7 = 7, placed for D\
13u. R7C7 = 7 -> R78C6 = 11 = {38}, locked for C6 and N8 -> R5C67 = [63]
13v. R89C7 = {59} -> R9C6 = 4 (cage sum)
13w. R1C456 = [492], R78C5 = {56}, locked for C5 and N8, R23C5 = [81], R9C5 = 2, R9C8 = 6, R8C8 = 2, placed for D\, R6C56 = [31], R6C4 = 2, placed for D/, R5C4 = 8
13x. R23C3 = [76], R23C6 = [57], R2C8 = 3, placed for D/
13y. R7C3 = 1 -> R8C34 = 11 = [47]
and the rest is naked singles, without using the diagonals.