This is a Windoku twin killer. Combinations in killer cages (some with and some without totals must use non-consecutive numbers), while the cages in HATMAN’s diagram with red borders use consecutive numbers (not necessarily in order).
The four windows are numbered W1, W2, W3 and W4; the
hidden windows will give their cells, for example hidden window R159C159.
[I was stuck after 16 steps of my original attempt. When I came back to this puzzle to try it again, I checked the forum thread and found that
wellbeback had told
HATMAN that it had multiple solutions.
HATMAN then added an extra 15(3) killer cage at R7C8.]
Prelims
a) 17(3) cage at R2C4 = {179/269/359/368}, no 4
b) 18(3) cage at R3C4 = {279/369/468}, no 1,5
c) 20(3) cage at R4C9 = {479}
d) 19(3) cage at R6C6 = {379/469}
e) 11(3) cage at R7C2 = {137/146}
f) 13(3) cage at R7C7 = {139/148/157/247}, no 6
g) 15(3) cage at R7C8 = {159/168/249/258/357}
Steps resulting from Prelims
1a. Naked triple {479} in 20(3) cage at R4C9, locked for N6
1b. 19(3) cage at R6C6 = {379/469}, 9 locked for C6 and W4
1c. 11(3) cage at R7C2 = {137/146}, 1 locked for C2 and N7
[I next analysed 17(3) cage at R2C4 and 18(3) cage at R3C4 but this proved unnecessary after step 2a …]
2. 45 rule on N2 3 innies R1C456 = 10 = {127/145/235} (cannot be {136} which clashes with 17(3) cage at R2C4), no 6,8,9
2a. R1C123 is a killer cage so cannot contain consecutive numbers and R1C789 is a red border cage which must contain three consecutive numbers -> R1C456 = {235} (only remaining combination, cannot be {127} because R1C123 cannot contain both of 8,9 since R1C789 = {345/456}, cannot be {145} because R1C123 cannot contain both of 2,3 since R1C789 = {678/789}), locked for R1 and N2
2b. 17(3) cage = {179} (only remaining combination), locked for R2
2c. Naked triple {468} in 18(3) cage at R3C4, locked for R3
2d. R1C56 cannot contain consecutive numbers = {25/35}, R1C4 = {23}
2e. R1C789 must contain three consecutive numbers = {678/789}, 7,8 locked for R1 and N3
2f. 1,4 in R1 only in R1C123, locked for N1
3. 7 in W2 only in R24C6, locked for C6 -> 19(3) cage at R6C6 = {469} (only remaining combination), locked for C6 and W4 -> R3C6 = 8, placed for W2
3a. 4 in W2 only in R2C78, locked for R2
3b. 7 in W4 only in R78C78, locked for N9
4. 7 in N9 only in 13(3) cage at R7C7 = {157/247}
or 15(3) cage at R7C8 = {357} -> 13(3) cage at R7C7 = {148/157/247} (cannot be {139}, locking-out cages), no 3,9
4a. 4 of {148/247} must be in R9C7 -> no 2,8 in R9C7
4b. 15(3) cage = {159/168/357} (cannot be {249} because 4,9 only in R9C8, cannot be {258} which clashes with 13(3) cage), no 2,4
4c. 6,9 of {159/168} must be in R9C8 -> no 1,8 in R9C8
4d. 13(3) cage = {148/247} (cannot be {157} which clashes with 15(3) cage), no 5 -> R9C7 = 4, placed for hidden window R159C678
4e. Killer pair 1,7 in 13(3) cage and 15(3) cage, locked for N9
4f. Combining these cages -> R789C9 = {269/368/359}
[The first new step after the 15(3) cage has been added, replacing a different step for the 13(3) cage.]
5. R5C456 is a red border cage, max R5C6 = 5 -> max in R5C45 = 7, no 8,9 in R5C45
6. 1 in C1 only in red border cage R123C1 or in red border cage R4567C1 -> one of these red border cages must contain all of 1,2,3, locked for C1, no 2,3 in R89C1
7. 9 in W2 only in R3C78, locked for R3 and N3
7a. Naked triple {678} in R1C789, locked for R1 and N3
7b. 6 in N1 only in R2C123 -> no 5 in R2C123 (cannot contain consecutive numbers)
7c. 5 in N1 only in R3C123, locked for R3
8. R3C7 = 9 (hidden single in C7)
8a. R3C7 = 9 -> no 8 in R1C7 (killer cage R123C7 cannot contain consecutive numbers)
9. R789C9 (step 4f) = {269/368} (cannot be {359} because killer cage R123C9 cannot contain both of 1,2), no 5, 6 locked for C9 and N9
9a. Killer pair 7,8,9 in R1C9, R45C9 and R789C9, locked for C9
9b. 6 in N3 only in R1C78, locked for hidden window R159C678
9c. 6 in N6 only in R4C78, locked for R4
10. 15(3) cage at R7C8 (step 4b) = {159/357}, no 8, 5 locked for C8
10a. Killer pair 7,9 in R5C8 and 15(3) cage, locked for C8
11. 8 in hidden window R159C678 only in R1C8 + R5C7, CPE no 8 in R6C8
11a. 8 in N6 only in R56C7, locked for C7
11b. 13(3) cage at R7C7 (step 4d) = {247} (only remaining combination), 2,7 locked for C7 and W4 -> R1C7 = 6, R1C8 = 8, placed for hidden window R159C678, R1C9 = 7, placed for hidden window R159C159
11c. R1C7 = 6 -> R2C7 = 3 (no 5 in R2C7 because R123C7 cannot contain consecutive numbers), placed for W2
11d. 15(3) cage at R7C8 (step 10) = {159} (only remaining combination) -> R9C8 = 9, R78C8 = {15}, locked for C8 and W4 -> R3C8 = 2, placed for W2, R3C9 = 1, placed for R234C159, R2C8 = 4, R2C9 = 5, placed for hidden window R234C159, R6C78 = [83], R45C8 = [67], R6C9 = 2, placed for hidden window R678C159
11d. 5 in W2 only in R4C67, locked for R4
11e. 8 in N5 only in R4C45, locked for R4
11f. 2 in N7 only in R789C3, locked for C3
12. R6C9 = 2 -> killer cage R678C9 = 2{68} (cannot contain 3 because killer cage R678C9 cannot contain consecutive numbers), 6,8 locked for C9 and hidden window R678C159
12a. R9C9 = 3, placed for hidden window R159C159
13. 1 in C1 only in red border cage R123C1 = {123} or in red border cage R4567C1 = {1234} -> R123C1 = {123/789} (cannot be {234}, locking-out cages, cannot be {678} because R1C1 only contains 1,4,9), no 4,6
13a. R123C1 = {123} => R4567C1 = {4567/5678/6789} or R123C1 = {789} -> 7 in R1234567C1 (locking cages), locked for C1
14. 6 in N1 only in R2C23, locked for W1 -> R3C4 = 4, locked for W1, R3C5 = 6
15. 4 in R1 only in R1C23, locked for hidden window R159C234
16. R5C456 is a red border cage = {123/234/345/456}
16a. 4 of {345} must be in R5C5, 5 of {456} must be in R5C6 -> no 5 in R5C5
17. R45C2 is a red border cage = {23/78/89} (cannot be {56} because 5,6 only in R5C2, cannot be {67} which clashes with 11(3) cage at R7C2), no 5,6 in R5C2
17a. 8 of {89} must be in R5C2 -> no 9 in R5C2
18. R89C4 is a red border cage = {12/56/67/78/89} (cannot be {23} which clashes with R1C4), no 3 in R8C4
19. R89C5 is a red border cage = {12/23/45/78/89}
19a. 4 of {45} must be in R8C5 -> no 5 in R8C5
[Time to start using forcing chains?
With hindsight, the ones in steps 22 and 23 are the most important ones.]
20. Consider combinations for 11(3) cage at R7C2= {137/146}
11(3) cage = {137} => R45C2 = {89} => R1C2 = 4
or 11(3) cage = {146} => R1C2 = 9
-> no 4,9 in R6C2
21. Consider combinations for red border cage R123C1 (step 13) = {123/789}
R123C1 = {123} => 2 in C2 only in R45C2 = {23}
or R123C1 = {789} => R1C2 = 4, 11(3) cage at R7C2 = {137} => R45C2 = [98]
-> R45C2 = [23/32/98], no 7
21a. R123C1 = {123}
or R123C1 = {789} => R4567C1 = {1234} => R89C1 = [56] => 11(3) cage at R7C2 = {137}
-> no 3 in R7C1
21b. 3 in N7 only in R78C23, locked for W3
21c. 3 in C1 only in R34C1, locked for hidden window R234C159
22. R45C2 (step 21) = [23/32/98]
22a. Consider combinations for 11(3) cage at R7C2= {137/146}
11(3) cage = {137} => R45C2 = [98] => R2C2 = 2 (hidden single in C2)
or 11(3) cage = {146}, locked for C2
-> no 6 in R2C2
22b. R2C3 = 6 (hidden single in N1)
23. R678C3 is a red border cage = {123/234/345/789}
23a. Consider placement for 4 in C3
R1C3 = 4 => R1C2 = 9 => R123C1 = {123}, R45C2 = {23} => 3 in C3 only in R78C3 => R678C3 = {123}
or 4 in R678C3 = {234/345}
-> R678C3 = {123/234/345}, no 7,8,9
23b. R678C3 = {123/234/345}, 3 locked for C3 and N7
23c. 11(3) cage at R7C2= {146} (only remaining combination), locked for C2 and N7 => R1C2 = 9, R1C1 = 1, placed for hidden window R159C159, R1C3 = 4
23d. R45C2 (step 21) = {23} (only remaining combination), locked for C2 and N4 -> R2C2 = 8, placed for W1, R2C1 = 2, placed for hidden window R234C159, R3C1 = 3 (hidden single in C1)
23e. R678C3 = {123} (only remaining combination) -> R6C3 = 1, placed for W3, R78C3 = {23}, locked for C3 and W3
24. R5C1 = 6 (hidden single in C1)
24a. Red border cage R4567C1 = {4567} (only remaining combination), locked for C1 -> R8C1 = 9, placed for hidden window R678C159, R9C1 = 8
24b. Naked pair {57} in R39C3, locked for C3 -> R4C3 = 9, placed for W1, R4C9 = 4, R4C1 = 7, R6C2 = 5, placed for W3, R3C2 = 7, placed for W1, R2C4 = 1, R6C1 = 4, R6C5 = 7
25. Naked pair {46} in R8C26, locked for R8 -> R8C9 = 8, R8C4 = 7, R8C7 = 2, R8C3 = 3, R8C5 = 1
26. R89C4 (step 18) = {67} (only remaining combination) -> R9C4 = 6
26a. R89C5 (step 19) = {12} (only remaining combination) -> R9C5 = 2
26b. R5C4 = 5 (hidden single in C4), R5C5 = 4 -> R5C456 (step 16) = {345} (only remaining combination) -> R5C6 = 3
and the rest is naked singles, with using the windows.
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