Non-consecutive (NC) horizontally and vertically in cage pattern. Old Lace. Black cages are Ordered (increasing top to bottom, then left to right), Non-consecutive, Digitised (last digit of cage total must be same as one of the cell values). Red cages are Reverse Ordered (decreasing top to bottom, then left to right), Non-consecutive, Digitised. Note that the cage at R3C5 decreases in order R3C5 – R4C4 – R4C6.
Digitised cages must contain two values totalling 10 with any other value with isn’t consecutive with these two values.
Valid Permutations for ordered in either direction
1,9: <139>, <149>, <159>, <169> and <179>
2,8: <248>, <258> and <268>
3,7: <137>, <357> and <379>
4,6: <146>, <246>, <468> and <469>
First cells of cages (or last cells of reversed cages) = {1234}, middle cells {34567} and last cells (first of reversed cages) = {6789}
Thus R1C4, R3C4, R3C7, R4C5, R4C6, R5C2 and R6C7 = {1234}
R2C4, R3C3, R3C6, R4C4, R4C7, R5C3, R5C5 and R7C7 = {34567}
R2C3, R2C6, R3C5, R5C4, R5C7, R6C3, R6C5 and R8C8 = {6789}
Old Lace property. R37C5 must exactly equal R5C46 and R46C5 must exactly equal R5C37.
1. Old Lace R46C5 must exactly equal R5C37, only common candidates in R4C5 and R5C3 are 3,4 -> R4C5 = R5C3 = {34}
1a. R4C5 = {34} -> no 3,4 in R4C46 + R5C5 (NC)
1b. R4C6 = {12} -> no 1,2 in R5C6 (NC)
1c. R5C3 = {34} -> no 3,4 in R4C3 + R5C2 (NC)
1d. R5C2 = {12} -> no 1,2 in R46C2 + R5C1 (NC)
1e. Old Lace R37C5 must exactly equal R5C46, no 1,2 in R5C46 -> no 1,2 in R7C5
2. Reverse-ordered cage at R2C6, R4C5 = {34} -> no 3,4 in R3C6 (NC)
2a. R3C6 = {567} -> no 6 in R2C6 + R3C5 (NC)
3. R5C5 = {567} -> no 6 in R5C46 + R6C5 (NC)
3a. Old Lace R5C7 = R6C5, no 6 in R6C5 -> no 6 in R5C7
4. Reverse-ordered cage at R3C5, R3C5 + R4C6 must total 10 = [82/91], R4C56 = [31/41/42] (cannot be [32], NC)
4a. Reverse-ordered cage at R2C6 = [753/864/964] (cannot be [973] which clashes with R3C5 + R4C6 = [91] when R4C5 = 3), no 7 in R3C6
4b. Similarly ordered cage at R4C5 = [357/468/469], no 7 in R5C5
5. R3C5 + R4C6 must total 10 = [82/91], R4C56 = [31/41/42] (step 4)
5a. Reverse-ordered cage at R3C5 = [951/961/971] (cannot be [852] because R4C45 cannot be [54], NC, cannot be [862] which clashes with ordered cage at R4C6 = [469]) -> R3C5 = 9, R4C6 = 1
5b. Old Lace R37C5 must exactly equal R5C46, R3C5 = 9 -> R5C46 must contain 9, locked for R5 and N5
6. R5C5 = {56} -> no 5 in R5C6 (NC)
6a. R5C7 = {78} -> no 7,8 in R4C7 + R5C68 (NC)
6b. R6C5 = {78} -> no 7,8 in R6C46 + R7C5 (NC)
7. Old Lace R37C5 must exactly equal R5C46, no 7,8 in R37C5 -> no 7,8 in R5C4 -> R5C4 = 9, R5C6 = {34} -> R7C5 = {34}
7a. Naked pair {34} in R4C5 + R5C6, locked for N5
7b. Naked pair {34} in R47C5, locked for C5
8. R4C4 + R5C7 = [78] (hidden pair in Old Lace)
9. R6C5 = 8 -> ordered cage at R4C5 = [468], R5C36 = [43], R7C5 = 3
9a. R5C6 = 3 -> R6C6 = 5 (cannot be 2, NC), R6C4 = 2, R3C6 = 6, R2C6 = 8 (cannot be 7, NC)
9b. R2C6 = 8 -> no 7,9 in R1C6 + R2C57 (NC)
9c. R1C5 = 7 (hidden single in N2)
10. R3C5 = 9 -> ordered cage at R1C4 = [139/149/159] -> R1C4 = 1
10a. R3C4 = {34} -> no 3,4 in R2C4 -> R2C4 = 5, R2C5 = 2, R1C6 = 4, R3C4 = 3
11. R3C4 = 3 -> reverse-ordered cage at R2C3 = [753/973] -> R2C3 = {79}, R3C3 = {57}, 7 locked for C3 and N1
12. R5C3 = 4 -> ordered cage at R5C2 = [146/149] -> R5C2 = 1
12a. 2 in R5 only in R5C89, locked for N6
13. R5C7 = 8 -> ordered cage at R3C7 = [258/268/468] -> R3C7 = {24}, R4C7 = {56}
13a. R3C7 = {24} -> no 3 in R2C7 (NC)
13b. R4C7 = {56} -> no 5,6 in R4C8 (NC)
14. Now a few more NCs. No 2,8 in R1C3, no 3,5 in R1C7, no 6 in R4C1, no 5,6,8 in R4C3, no 6 in R6C1, no 4 in R6C7, no 4 in R7C4, no 2 in R7C6
14a. 1 in N1 only in R23C1, locked for C1
14b. 8 in C3 only in R789C3, locked for N7
14c. 7 in C7 only in R789C7, locked for N9
15. Ordered cage at R6C7 = [146/149/159/169/179] (cannot be [379] which clashes with R4C8) -> R6C7 = 1, R8C8 = {69}
16. Ordered cage at R3C7 = [258/268] (cannot be [468] which clashes with R2C7) -> R3C7 = 2
16a. R3C7 = 2 -> no 1 in R3C8 (NC)
16b. 4 in R6 only in R6C89 -> no 3 in R6C89 (NC)
16c. 3 in N6 only in R4C89, locked for R4
16d. 7 in N6 only in R5C9 + R6C89 -> no 6 in R6C9 (NC)
16e. 3 in C7 only in R89C7, locked for N9
16f. 3 in C7 only in R89C7 -> no 4 in R89C7 (NC)
17. 8 in R4 only in R4C12 -> no 9 in R4C12 (NC)
17a. Consider placements for 8 in R4
R4C1 = 8 => R4C3 = 2 (hidden single in R4)
or R4C2 = 8 => R4C3 = 2 (not 9, NC)
-> R4C3 = 2
[Alternatively R4C123 cannot be [289] (NC), no 2 in R4C1 -> R4C3 = 2 (hidden single in R4).]
17b. Naked triple {568} in R4C127, locked for R4
17c. Naked pair {39} in R4C89, locked for N6
18. Consider placements for R8C8 = {69}
R8C8 = 6 => R4C7 = 6 (hidden single in N6) => R1C7 = 9
or R8C8 = 9 => R1C7 = 9 (hidden single in C7)
-> R1C7 = 9
18a. R1C7 = 9 -> no 8 in R1C8 (NC)
19. 2 in R1 only in R1C12 -> no 3 in R1C12 (NC)
19a. Consider placements 2 in R1C12
R1C1 = 2 => no 1,3 in R2C1 (NC) => R3C1 = 1 (hidden single in N1)
or R1C2 = 2 => no 3 in R1C3 + R2C2 (NC) => R23C1 = [31] (hidden pair in N1)
-> R3C1 = 1
20. Ordered cage at R6C7 (step 15) = [149/159/169/179] (cannot be [146] which clashes with R8C7 = {357}, NC) -> R8C8 = 9, R4C89 = [39]
20a. R4C8 = 3 -> R5C8 = 5 (not 2, NC), R1C8 = 6, R2C7 = 4, R4C7 = 6, R5C19 = [72]
20b. R6C8 = 7 (not 4, NC), R6C9 = 4, R23C8 = [18]
20c. R3C8 = 8 -> R3C9 = 5 (not 7, NC), R12C9 = [37], R12C3 = [59], R3C23 = [47], R6C3 = 6
20d. R3C2 = 4 -> R4C2 = 8 (not 5, NC), R4C1 = 5, R1C12 = [82]
20e. R1C2 = 2 -> R2C2 = 6 (not 3, NC), R2C1 = 3, R6C12 = [93]
20f. R8C8 = 9 -> no 8 in R8C9 (NC)
21. R7C18 = {24} (hidden pair in R7)
21a. R8C2 = {57} -> no 6 in R8C1 (NC)
21b. R9C1 = 6 (hidden single in C1)
21c. R9C1 = 6 -> no 5,7 in R9C2 (NC) -> R9C2 = 9
21d. R9C2 = 9 -> no 8 in R9C3 (NC)
22. R9C4 = 8 (cannot be 4, NC, because R9C35 cannot both be 1; alternatively consider placements for R9C3 = {13} …)
and the rest is naked singles, without using NC.