Prelims
a) 38(6) cage at R1C1 = {356789}, no 1,2,4
b) 39(6) cage at R1C7 = {456789}, no 1,2,3
c) 23(6) cage at R4C1 = {123458/123467}, no 9
d) 22(6) cage at R5C6 = {123457}, no 6,8,9
e) 39(6) cage at R6C1 = {456789}, no 1,2,3
1. R1C56 = {12} (hidden pair in R1), locked for N2
1a. 3 in R1 only in R1C1234, locked for 38(6) cage at R1C1
1b. 4 in R1 only in R1C789, locked for N3 and 39(6) cage at R1C7
2. R45C1 = {12} (hidden pair in C1), locked for N4 and 23(6) cage at R4C1
2a. 23(6) cage at R4C1 = {123458/123467}, 3,4 locked for R4
2b. R1C1 = 3 (hidden single in C1), placed for D\
2c. 4 in C1 only in R6789C1, locked for 39(6) cage at R6C1
3. 36(6) cage at R7C9 can only contain one of 1,2,3, R9 and C9 each require 1,2,3 -> R56C9 = {123}, R9C45 = {123} and R9C9 = {12}
3a. Naked triple {123} in R569C9, locked for C9, 3 also locked for N6 and 33(6) cage at R5C8
3b. Naked triple {123} in R9C459, locked for R9, 3 also locked for N8
3c. R8C7 = 3 (hidden single in N9)
4. 22(6) cage at R5C6 = {123457}, 3 locked for N5
4a. R5C6 + R6C5 = {13/23} (otherwise 22(6) cage would clash with R9C45) -> 4,5,7 in R7C456 + R8C5, locked for N8
4b. Naked triple {123} in R169C5, locked for C5
4c. Naked triple {123} in R5C169, locked for R5
4d. R8C46 + R9C6 = {689} (hidden triple in N8), CPE no 6,8,9 in R8C9
4e. 3 in R4 only in R4C23, locked for N4
5. R56C9 = {123} -> 33(6) cage at R5C8 = {135789/234789/235689}, 8,9 locked for C8
5a. R56C9 = {123} -> no 1,2 in R678C8
5b. R23C8 = {123}, R4C8 = {12} (hidden triple in C8)
5c. Naked triple R4C8 + R56C9 = {123}, locked for N6
5d. Naked pair {12} in R4C18, locked for N4
6. R7C7 + R9C9 = {12} (hidden pair in N9), locked for D\
6a. R2C3 + R3C2 = {12} (hidden pair in N1)
6b. R8C23 = {12} (hidden pair in R8), locked for N7
6c. 4 in N1 only in R2C2 + R3C3, locked for D\
7. Hidden killer pair 1,2 in R5C6 + R6C5 and R6C4 for N5, R5C6 + R6C5 contain one of 1,2 -> R6C4 = {12}
7a. Naked pair 1,2 in R6C4 + R8C2, locked for D/ -> R2C8 = 3, placed for D/
7b. R3C4 = 3 (hidden single in N2)
7c. R7C2 = 3 (hidden single in N7)
7d. R4C3 = 3 (hidden single in N4)
7e. R9C5 = 3 (hidden single in N8)
7f. R5C6 = 3 (hidden single in N5)
7g. Naked pair {12} in R6C45, locked for R6 -> R6C9 = 3
7h. R2C7 + R3C8 = {12} (hidden pair in N3)
7i. Naked pair {12} in R69C4, locked for C4
7j. R7C6 + R9C4 = {12} (hidden pair in N8)
8. 4 in C6 only in R23C6, locked for N2
8a. 24(6) cage at R1C6 = {123459/123468}, no 7
8b. 7 in C6 only in R46C6, locked for N5
8c. Consider combinations for 24(6) cage
24(6) cage = {123459}, CPE no 5,9 in R4C6
or 24(6) cage = {123468} => R46C6 = {57} (hidden pair in C6)
-> no 9 in R4C6
8d. 9 in R4 only in R4C79, locked for N6, CPE no 9 in R1C7
9. 33(6) cage at R5C8 (step 5) = {135789/234789/235689}, 9 locked for N9
9a. 36(6) cage at R7C9 = {156789/246789} -> R9C6 = 9, 6,7,8 locked for N9
9b. 33(6) cage contains 8, locked for N6
9c. 39(6) cage at R6C1 = {456789}, 9 locked for C1
9d. 38(6) cage at R1C1 = {356789}, 9 locked for R1
9e. Naked pair {68} in R8C46, locked for R8
9f. 39(6) cage at R1C7 = {456789}, 8 locked for N3
9g. 9 in N5 only in R5C45, locked for R5
10. 36(6) cage at R7C9 (step 9a) = {156789/246789}
10a. 33(6) cage at R5C8 (step 5) = {135789/234789} (cannot be {235689} which clashes with 36(6) cage = {156789} -> R56C8 = {78}, locked for C8 and N6
10b. Killer pair 5,6 in 23(6) cage at R4C1 and R4C79, locked for R4
10c. 39(6) cage at R1C7 = {456789}, 7 locked for N3
11. 24(6) cage at R1C6 (step 8a) = {123459/123468}
11a. 9 of {123459} must be in R4C7 –> no 5 in R4C7
11b. 6 of {123468} must be in R4C7 -> no 6 in R23C6
12. R19C7 = {78} (hidden pair in C7)
12a. R1C9 + R9C1 = {456} (cannot contain 7 or 8 because of clashes around the outer ring), no 7,8
13. 9 on D/ only in R3C7 + R5C5 + R7C3, CPE no 9 in R3C3
[Step 12a could have been omitted if I’d remembered this type of step sooner; it’s a long time since I’ve used it. Step 14 could have been done immediately after step 3 if I’d spotted it then.]
14. Total around outer ring = 180 with corner cells counting double. Total of the 6-cell cages plus 1,2 pairs in R1 and C1 and placed 3s in R9 and C9 = 164. This leaves 16 for the four corner cells, R5C9 and R9C4. R1C1 = 3 and min R1C9 + R9C1 = 9 -> max R5C9 + R9C49 = 4 -> R5C9 = 1, R9C49 = [12], R1C9 + R9C1 = 9 = {45}, locked for D/
15. R56C7 = {45} (hidden pair in C7), locked for N6
15a. Naked pair {69} in R4C79, locked for R4
15b. 23(6) cage at R4C1 = {123458} (only remaining combination), 8 locked for R4 -> R4C6 = 7, placed for D/
15c. R2C2 + R3C3 = {47} (hidden pair on D\), locked for N1
15d. 38(6) cage at R1C1 = {356789} -> R1C4 = 7, R19C7 = [87]
16. 33(6) cage at R5C8 (step 10a) = {135789} (only remaining combination) -> R78C8 = {59}, locked for C8 and N9 -> R8C9 = 4, R19C8 = [46], R1C9 = 5, placed for D/
16a. Naked pair {58} in R9C23, locked for N7 and 39(6) cage at R6C1
16b. R5C5 = 8 (hidden single on D/), R4C4 = 5, placed for D\, R6C6 = 6, R8C6 = 8
16c. Naked pair {45} in R23C6, locked for N2, R4C7 = 9 (cage sum)
and the rest is naked singles.