SudokuSolver Forum

The Windows at R2C2, R2C6, R6C2 and R6C6 are numbered W1, W2, W3 and W4

In the following walkthrough I’ve used Windoku properties, the four given windows and five hidden ones, as in my post in the Standard Techniques forum here. This puzzle would be much(?) harder without the hidden windows.

Prelims

a) 28(7) cage at R1C1 = {1234567}, no 8,9
b) 42(7) cage at R1C8 = {3456789}, no 1,2
c) 41(7) cage at R6C1 = {2456789}, no 1,3
d) 29(7) cage at R8C7 = {1234568}, no 7,9

Steps resulting from Prelims
1a. 8,9 in N1 only in R3C123, locked for R3 and 36(7) cage at R3C1, no 8,9 in R4C123
1b. 1,2 in N3 only in R123C7, locked for C7, CPE no 1,2 in R2C6 using W2
1c. 1,3 in N7 only in R789C3, locked for C3 and 21(5) cage at R7C3, no 1,3 in R78C4
1d. 7,9 in N9 only in R7C789, locked for R7 and 25(5) cage at R6C7, no 7,9 in R6C78
[and resulting from those steps …]
1e. 8,9 in N2 only in R1C56 + R2C456, locked for 36(7) cage at R1C5, no 8,9 in R12C7
1f. 8,9 in N3 only in R12C89, locked for 42(3) cage at R1C8, no 8,9 in R4C9

2. 45 rule on complete grid 2 zero cells R6C9 + R9C4 = 6 = [15/24/33/42/51], no 6,7,8,9

3. 45 rule on R12345 3 outies R6C234 = 19 = {289/379/469/478/568}, no 1
3a. 1 in W3 only in R78C3, locked for 21(5) cage at R7C3, no 1 in R9C3

4. 45 rule on R12 3 outies R3C89 + R4C9 = 16 = {367/457}, 7 locked for 42(7) cage, no 7 in R12C89
4a. R3C89 + R4C9 = {367/457}, CPE no 7 in R3C15 using hidden window R234C159
4b. 1,2,7 in R12 only in 28(7) cage at R1C1 and 36(7) cage -> 36(7) cage = {1236789/1245789}

5. 45 rule on N9 3 outies R6C78 + R9C6 = 9
[Note. No repeated numbers in R6C78 + R9C6 because these cells “see” all the cells in N9.]
5a. Min R6C7 = 4 -> max R9C6 = 5
5b. Max R6C78 = 8 -> no 8 in R6C7, no 6,8 in R6C8
5c. 29(7) cage at R8C7 = {1234568}, 6,8 locked for N9
5d. 8,9 in N6 only in R4C78 + R5C789, locked for 38(7) cage at R4C6, no 8,9 in R45C6
5e. 7 in N6 only in R45C789, CPE no 7 in R4C6

6. 8,9 in N4 only in R56C123, CPE no 8,9 in R6C4

7. 45 rule on N1 5 outies R13C4 + R4C123 = 19 = {12367/12457/13456}, CPE no 1 in R4C4
[Note. No repeated numbers in R13C4 + R4C123 because these cells “see” all the cells in N1.]

8. 41(7) cage at R6C1, 39(7) cage at R6C5 and 25(5) cage at R6C7 each contain both of 7,9 -> the other 7,9 in R6789 must be in R6C234 and/or 21(5) cage at R7C3
8a. 21(5) cage cannot contain both of 7,9 -> R6C234 must contain at least one of 7,9 -> R6C234 (step 3) = {289/379/469/478} (cannot be {568} which doesn’t contain 7 or 9), no 5
8b. 2 of {289} must be in R6C4 -> no 2 in R6C23
8c. 40(7) cage at R5C1 = {1456789/2356789}, 5 locked for R5
8d. 4 of {1456789} must be in R6C234 -> no 4 in R5C1234
8e. 4 in R5 only in R5C56789, CPE no 4 in R4C6

[I was stuck at this stage until I found the next step, a well hidden 45.]
9. 45 rule on W3 using R6C234 = 19 (step 3), 2 innies R78C2 = 1 outie R9C3 + 5, IOU no 5 in R78C2
9a. R78C2 cannot total 7 -> no 2 in R9C3

10. 5 in W3 only in R78C34, locked for 21(5) cage at R7C3, no 5 in R9C3
10a. 21(5) cage contains 1,3,5 = {13458} (only remaining combination), no 2,6,7,9

11. 45 rule on N7 3 outies R6C1 + R78C4 = 17 = {458} (only possible combination), CPE no 4 in R6C4
[Note. No repeated numbers in R6C1 + R78C4 because these cells “see” all the cells in N7.]
[Also note that R6C1 + R78C4 and R6C234 form a hidden 6-cell cage because they “see” each other using R6 and W3.]
11a. R6C234 (step 8a) = {379} (only remaining combination, cannot be {289/469/478} which clash with R6C1 + R78C4), locked for R6 and W3, no 3 in R78C3, no 7,9 in R8C2, clean-up: no 3 in R9C4 (step 2)
11b. 40(7) cage at R5C1 = {2356789} (only remaining combination) -> R5C1234 = {2568}, locked for R5
11c. R5C56789 = {13479}, CPE no 1,3 in R4C6
11d. 1 in N4 only in R4C12, locked for R4 and 36(7) cage at R3C1, no 1 in R3C124
11e. 1 in N1 only in R12C12, locked for 28(7) cage at R1C1, no 1 in R1C4
11f. 8 in 38(7) cage at R4C6 only in R4C78, locked for R4 and W2, no 8 in R2C68
[I’d seen the relationships used for steps 11 and 11a when I first worked on this puzzle, but wasn’t able to make use of them until I found step 9 today.]
[The puzzle is cracked after step 11a, the rest is fairly long but routine.]

12. R9C3 = 3 (hidden single in N7), placed for hidden window R159C234, no 3 in R1C24
12a. 28(7) cage at R1C1 = {1234567}, 3 locked for N1
12b. R78C2 = R9C3 + 5 (step 9)
12c. R9C3 = 3 -> R78C2 = 8 = {26}, locked for C2 and N7
12d. 29(7) cage at R8C7 = {1234568}, 3 locked for R8 and N9

13. 7,9 in N9 only in 25(5) cage at R6C7 = {12679} (only remaining combination), no 4,5 -> R6C7 = 6, placed for W4
13a. R6C78 + R9C6 = 9 (step 5), R6C7 = 6 -> R6C8 + R9C6 = 3 = {12}, CPE no 1,2 in R6C6 + R9C8

14. 39(7) cage at R6C5 = {1356789/2346789}, 6 locked for C5

15. 1 in R3 only in R3C567, locked for 24(6) cage at R3C5, no 1 in R5C5

16. 1 in R5 only in R5C689 -> 38(7) cage at R4C6 = {1256789/1346789} -> R4C6 = 6, placed for D/ and W2 -> R8C2 = 2, placed for D/, R7C2 = 6
16a. 2 of {1256789} must be in R4C8 -> no 5 in R4C8

17. 29(7) cage at R8C7 = {1234568}, 2 locked for R9, clean-up: no 4 in R6C9 (step 2)

18. 9 in C4 only in R24C4, locked for W1, no 9 in R3C23
18a. R3C1 = 9 (hidden single in N1)
18b. R9C2 = 9 (hidden single in N7)
18c. Naked quad {4578} in R6789C1, locked for C1
18d. R6C3 = 9 (hidden single in R6)

19. 9 in C4 only in R24C4
19a. 45 rule on C1234 3 innies R249C4 = 14 = {149} (only possible combination, cannot be {239} because no 2,3 in R9C4), locked for C4, clean-up: no 1 in R6C9 (step 2)

20. Naked pair {58} in R78C4, locked for C4, N8 and W3
20a. Naked pair {14} in R78C3, locked for C3 and N7
20b. R6C1 = 4 (hidden single in C1)

21. R5C4 = 2, R5C1 = 6, 2 placed for hidden window R159C234, no 2 in R1C3, 6 placed for hidden window R159C159, no 6 in R9C59
21a. Naked pair {58} in R5C23, locked for N4 and hidden window R159C234, no 5 in R1C23

22. Naked pair {67} in R1C34, locked for R1 and 28(7) cage at R1C1, no 6,7 in R2C23
22a. 28(7) cage at R1C1 = {1234567}, 5 locked for R2 and N1, 2,4 also locked for N1

23. 36(7) cage at R3C1 = {1236789} (only remaining combination), 6 locked for R3

24. R3C89 + R4C9 (step 4) = {457} (only remaining combination), locked for 42(7) cage at R1C8, no 4,5 in R12C89
24a. R3C89 + R4C9 = {457}, CPE no 4,5 in R3C5 using hidden window R234C159
24b. R2C1 = 6 (hidden single in N3), R9C8 = 6 (hidden single in N9)
24c. Naked triple {389} in R1C89 + R2C8, locked for N3, 8 also locked for R1

25. R2C5 = 8 (hidden single in R2), R6C6 = 8 (hidden single in R6), placed for D\
25a. R3C2 = 8 (hidden single in N1), R5C23 = [58]

26. R2C3 = 5 (hidden single in N1)
26a. R4C3 = 2 (hidden single in C3)

27. 38(7) cage at R4C6 (step 16) = {1346789} (only remaining combination), no 5
27a. 5 in N6 only in R46C9, locked for C9

28. 5 in W4 only in R8C78, locked for R8 and N9 -> R78C4 = [58], R7C1 = 8

29. R9C1 = 5, R9C5 = 7, R9C7 = 8 (hidden singles in R9), 5 placed for D/, 7 placed for hidden window R159C159, no 7 in R5C9
29a. R4C8 = 8 (hidden single in N6)
29b. Naked pair {39} in R12C8, locked for C8 and N3 -> R1C9 = 8

30. R3C3 = 6 (hidden single on D\), R1C34 = [76]
30a. R7C7 = 7 (hidden single on D\), R7C9 = 9 (hidden single in N9)
30b. Naked pair {12} in R67C8, locked for C8

31. Naked pair {14} in R3C7 + R7C3, locked for D/
31a. R6C4 = 7 (hidden single on D/), R3C4 = 3, R4C12 = [17], R6C2 = 3

32. R5C8 = 7 (hidden single in N6), R3C9 = 7 (hidden single in N3), R2C6 = 7 (hidden single in N2)
32a. 2 in R3 only in R3C56, locked for N2

33. 24(6) cage at R3C5 = {123459} (only remaining combination), 3 locked for C5 and N5
33a. 24(6) cage at R3C5 = {123459}, CPE no 5 in R1C5
33b. 9 on D\ only in R4C4 + R5C5, locked for N5

34. 39(7) cage at R6C5 = {1356789} (only remaining combination, cannot be {2346789} which clashes with R9C46, ALS block), no 2,4 -> R6C5 = 5, R7C5 = 1, R9C46 = [42], R9C9 = 1, placed for D\, R2C2 = 4, placed for D\

and the rest is naked singles, without using the diagonals.

It's hard to know how to rate some of my steps. I'll go for 1.5; several steps seem to be in the 1.5 range.

I stepped through some early steps using SudokuSolver and noticed that it found the CPE in step 4a, so clearly it knows how to use hidden windows; it's possible that it rates them very low - I'd put them at least in the 1.25 range.