The four windows are numbered W1, W2, W3 and W4; the
hidden windows will give their cells, for example hidden window R159C159.
Green bordered cages are non-consecutive, ordered ascending or descending
Red bordered cages are consecutive, fully ordered (CFO)
Prelims
a) CFO cage at R1C1: no 8,9 in R1C1, no 1,9 in R1C2, no 1,2 in R1C3
b) CFO cage at R1C4: no 8,9 in R1C4, no 1,9 in R1C5, no 1,2 in R2C5
c) CFO cage at R1C6: no 9 in R1C6, no 1 in R2C6
d) CFO cage at R1C9: no 8,9 in R1C9, no 1,9 in R2C9, no 1,2 in R3C9
e) CFO cage at R2C4: no 7,8,9 in R2C4, no 1,8,9 in R3C4, no 1,2,9 in R3C5, no 1,2,3 in R3C6
f) CFO cage at R4C1: no 9 in R4C1, no 1 in R5C1
g) CFO cage at R4C4: no 9 in R4C5, no 1 in R5C6
h) CFO cage at R4C6: no 9 in R4C6, no 1 in R4C7
i) CFO cage at R4C9: no 9 in R4C9, no 1 in R5C9
j) CFO cage at R5C2: no 9 in R5C2, no 1 in R5C3
k) CFO cage at R6C8: no 9 in R6C8, no 1 in R7C8
l) CFO cage at R7C1: no 8,9 in R7C1, no 1,9 in R8C1, no 1,2 in R9C1
m) CFO cage at R8C7: no 9 in R8C7, no 1 in R8C8
n) CFO cage at R9C5: no 9 in R9C5, no 1 in R9C6
o) CFO cage at R9C7: no 8,9 in R9C7, no 1,9 in R9C8, no 1,2 in R9C9
p) Non-consecutive ordered cages, no 1,2,8,9 in the middle cell -> R1C8 + R2C1 + R8C9 + R9C2 = {34567}
1. There are three CFO cages in N2 -> CFO cage at R2C4 = [1234/3456/4567/6789], CFO cage at R1C4 = [123/345/567/789], CFO cage at R1C6 = [12/45/56/89]
R1C4 = {1357}, R1C5 = {2468}, R1C6 = {1458}, R2C4 = {1346}, R2C5 = {3579}, R2C6 = {2569}, R3C4 = {2457}, R3C5 = {3568}, R3C6 = {4679}
1a. 9 in N2 only in R2C56 + R3C6, CPE no 9 in R2C78 using W2
[When I first did this puzzle, I carelessly omitted [56] from CFO cage at R1C6. I’ve therefore done some re-work, including the new step 9 which removes this combination and the new step 7 which removed several combinations from the non-consecutive ordered cage at R9C1.]
2. CFO cage at R1C1 cannot be [567] because R1C45 = [12/34] and there’s no possible non-consecutive cage at R1C7
2a. CFO cage at R1C1 cannot be [789] because R1C45 = [12/34/56] and there’s no possible non-consecutive cage at R1C7
2b. CFO cage at R1C1 = [123/234/345/456/678], no 5,7 in R1C1, no 6,8 in R1C2, no 7,9 in R1C3
3. R1C7 = 9 (hidden single in R1), placed for hidden window R159C678, no 9 in R5C68 + R9C6, clean-up: no 7 in R1C9, no 8 in R2C9, no 8 in R4C5, no 8 in R4C6, no 8 in R9C5
3a. Non-consecutive ordered cage at R1C7 contains 9 in R1C7 -> no 6 in R1C9, clean-up: no 7 in R2C9, no 8 in R3C9
3b. From remaining combinations for CFO cage at R1C1 and in R1C45, remaining possible combinations for non-consecutive ordered cage at R1C7 are {169/179/259/269/359/379/469/479}, no 3,4 in R1C8, no 5 in R1C9, clean-up: no 6 in R2C9, no 7 in R3C9
3c. 8 in N3 only in R23C78, locked for W2, no 8 in R4C78, clean-up: no 7 in R4C6
4. Non-consecutive, ordered cage at R1C1 = [146/158/159/169/259/269/379/479/631/641] (cannot be [13x/24x/35x/46x] which clash with CFO cage at R1C1, cannot be [147/257/258/268/368/369/642] which don’t allow two CFO cages in C1) -> R3C1 = {1689}
4a. CFO cage at R4C1 cannot be [89] (which clashes with non-consecutive, ordered cage at R1C1 or with CFO at R7C1 = [789] when non-consecutive, ordered cage at R1C1 = [146/631/641], no 8 in R4C1, no 9 in R5C1
4b. CFO cage at R7C1 cannot be [567], which clashes with non-consecutive, ordered cage at R1C1, no 5 in R7C1, no 6 in R8C1, no 7 in R9C1
5. Non-consecutive, ordered cage at R9C1 cannot be [531/579] which don’t allow two CFO cages in R9 -> no 5 in R9C1, clean-up: no 4 in R8C1, no 3 in R7C1
5a. CFO cage at R7C1 not [345] -> non-consecutive, ordered cage at R1C1 (step 4) = [146/158/159/169/259/379/479/631/641] (cannot be [269] which clashes with CFO cage at R7C1)
[Since working out interactions between two CFOs and a non-consecutive, ordered cage is rather tedious, I’ll use a short forcing chain which I’ve just spotted.]
6. 9 in R4 only in R4C234, locked for W1 => R3C1 = 9 (hidden single in N1) or R4C8 = 9, locked for W2, no 9 in R3C6
-> no 9 in R3C6, clean-up: no 8 in R3C5, no 7 in R3C4, no 6 in R2C4
6a. 9 in N2 only in R2C56, locked for R2
6b. 8 in N2 only in R1C56, locked for R1, clean-up: no 7 in R1C2, no 6 in R1C1
6c. Non-consecutive, ordered cage at R1C1, max R1C1 = 4 -> no 1 in R3C1
6d. CFO cage at R1C1 = [123/234/345/456] -> CFO cage at R1C4 (step 1) = [123/567/789] (cannot be [345] which clashes with CFO cage at R1C1), no 3 in R1C4, no 4 in R1C5, no 5 in R2C5
6e. CFO cage at R2C4 (step 1) = [1234/3456/4567], 4 locked for N2, clean-up: no 5 in R2C6
6f. 4 in N2 only in R2C4 + R3C46, CPE no 4 in R3C23 using W1
7. 8,9 in R4 only in R4C234 and R4C89
7a. R4C234 cannot contain both of 8,9 because R23C23 must contain at least one of 8,9 for N1
7b. R4C89 cannot contain both of 8,9 because R4C89 = [98] clashes with R45C9 = [89]
7c. Hidden killer pair 8,9 in R4C234 and R4C89 for R4, R4C234 and R4C89 must each contain one of 8,9
7d. R4C234 contains one of 8,9 -> R23C23 must contain one of 8,9 for W1 -> R3C1 = {89}
7e. R4C89 contains one of 8,9 -> R4C8 = 9 or R45C9 = [89], 9 in R4C8 + R5C9, locked for N6
8. CFO cage at R7C1 cannot be [789] which clashes with R3C1, no 7 in R7C1, no 8 in R8C1, no 9 in R9C1
[Removing [56] from CFO cage at R1C6 seems to require either a contradiction move or a forcing chain.]
9. Consider combinations for CFO cage at R1C1 = [123/234/345/456]
CFO cage at R1C1 = [123] => CFO cage at R1C9 = [456] => 6 in R1 only in CFO cage at R1C4 = [567]
or CFO cage at R1C1 = [234] => CFO cage at R1C9 = [123], CFO cage at R1C4 not [123] => 3 in N2 only in R2C4 => CFO cage at R2C4 = [3456]
or CFO cage at R1C1 = [345/456], no 5 in R1C6
-> no 5 in R1C6, no 6 in R2C6
10. Consider combinations for CFO cage at R1C4 = [123/567/789]
CFO cage at R1C4 = [123] => R1C8 = 7 (hidden single in R1)
or CFO cage at R1C4 = [567] => R1C8 = 7 (hidden single in R1)
or CFO cage at R1C4 = [789] => CFO cage at R1C6 = [12] => CFO cage at R1C1 = [345/456] (cannot be [234] which clashes with R1C69, ALS block), 5 locked for R1
-> no 5 in R1C8
10a. 5 in R1 only in R1C234, locked for hidden window R159C234, no 5 in R59C234, clean-up: no 4 in R5C2, no 6 in R5C3
10b. CFO cage at R6C8 cannot be [67], which clashes with R1C8, no 6 in R6C8, no 7 in R7C8
11. 5 in R9 only in CFO cage at R9C5 = [45/56] or in CFO cage at R9C7 = [345/456/567] -> CFO cage at R9C7 cannot be [234] (which doesn’t allow a valid combination for non-consecutive ordered cage at R9C1), no 2 in R9C7, no 3 in R9C8, no 4 in R9C9
11a. CFO cage at R9C5 cannot be [12/34/67] because combined with CFO cage at R9C7 = [345/456/567] they don’t allow valid combinations for non-consecutive ordered cage at R9C1 and CFO at R7C1), no 1,3,6 in R9C5, no 2,4,7 in R9C6
11b. From the remaining combinations for CFO cage at R9C5 = [45/56] and CFO cage at R9C7, combinations for non-consecutive, ordered cage at R9C1 are from three of <1489/4789> (cannot be <1236/1239/1269> because 1,2,9 only in R9C34) = [841/479], R9C4 = {89} clean-up: no 2,5 in R8C1, no 1,4 in R7C1
12. Non-consecutive, ordered cage at R1C1 (step 5a) = [158/159/169/259] (cannot be [146/379/479] which clash with CFO cage at R7C1), no 3,4 in R1C1, clean-up: no 4,5 in R1C2, no 5,6 in R1C3
13. R1C4 = 5 (hidden single in R1) -> CFO cage at R1C4 = [567], 6 placed for hidden window R159C159, no 6 in R5C19 + R9C9
13a. R1C8 = 7, placed for hidden window R159C678, no 7 in R5C67 + R9C7
13b. R3C5 = 3 -> CFO cage at R1C4 = [1234], 1,2 placed for W1, no 1,2 in R234C23, 3 placed for hidden window R234C159, no 3 in R24C19, 4 placed for W2, no 4 in R24C78
13c. CFO cage at R1C6 = [89], 9 placed for W2, no 9 in R4C8
13c. Clean-up: no 2 in R2C9, no 1,2 in R1C9, no 3 in R4C6, no 5 in R4C7, no 8 in R7C8, no 6 in R8C7, no 4,6 in R9C7, no 5,8 in R9C8, no 8,9 in R9C9
[Cracked. The rest is fairly straightforward.
From here on, I’ve only given the most useful hidden window placements.]
14. R1C1 = 1 (hidden single in R1) -> CFO cage at R1C1 = [123], 2,3 placed for hidden window R159C234, no 2,3 in R5C234, CFO cage at R1C9 = [456], 5,6 placed for hidden window R234C159, no 5,6 in R2C1 + R4C15 -> R2C1 = 4, placed for hidden window R234C159, no 4 in R4C5, CFO cage at R7C1 = [678], R3C1 = 9, R4C1 = 2, R4C5 = 1, R5C6 = 2, R9C234 = [419], placed for hidden window R159C234, no 1,4,9 in R5C234, clean-up: no 3 in R4C7, no 8 in R5C2
15. R89C9 = [37] -> non-consecutive, ordered cage at R7C9 = [137], CFO cage at R9C7 = [567], CFO cage at R9C5 = [23], R456C9 = [892], clean-up: no 4,5 in R6C8, no 9 in R7C8
16. R4C234 = {349} (naked triple in W1), locked for R4, 9 and locked for N4 -> R4C8 = 5, R4C67 = [67]
17. Naked triple {678} in R5C234, locked for R5
17a. R6C7 = 6 (hidden single in N6)
17b. R5C2 = 6 (hidden single in N4) -> CFO cage at R5C2 = [67]
17c. R2C23 = [86]
18. R8C8 = 9 (hidden single in N9) -> CFO cage at R8C7 = [89], R3C78 = [18]
19. Naked pair {25} in R8C23 = [52] -> R7C23 = [39], R3C23 = [75], R4C234 = [943], R6C2 = 1, R8C6 = 1
20. R6C8 = 3 -> CFO cage at R6C8 = [34]
and the rest is naked singles, without using the windows.