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HATMAN
 Post subject: Paper Solvable 12
PostPosted: Wed Jan 16, 2013 3:14 pm 
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 694
Location: Saudi Arabia
PS 12 W X Sevens 8 Cager

Version A just has the 8 cages on the centre horizontal and vertical and the diagonals SS gives it 8+ and JS uses about 25 small fishes. (Andrew asked me to post it for completeness.)

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Version B does not have a cage in r1c678 SS gives it 7+ but JS does it with four small fishes.

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Version C is Paper Solvable with SS giving 0.9

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1C JS Code:
3x3:3:k:4101:13:7940:7940:7940:6156:6156:6156:4615:5131:4101:14:15:7940:16:17:4615:18:5131:19:4101:20:7940:21:4615:22:7937:5131:23:24:7683:7940:7940:25:26:7937:7683:7683:7683:7683:27:7937:7937:7937:7937:7683:28:29:7682:7682:7937:30:31:5385:7683:32:4360:33:7682:34:2822:35:5385:36:4360:37:38:7682:39:40:2822:5385:4360:5642:5642:5642:7682:7682:7682:41:2822:

Version D has an SS score of 3.2 but JS does not have too much trouble. I would rate it at the bottom end of the assassins.

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1D JS Code:
3x3:3:k:4101:13:7940:7940:7940:4364:4364:14:4615:5131:4101:15:16:7940:17:18:4615:19:5131:20:4101:21:7940:22:4615:23:7937:5131:24:25:7683:7940:7940:26:27:7937:7683:7683:7683:7683:28:7937:7937:7937:7937:7683:29:30:7682:7682:7937:31:32:5385:7683:33:4360:34:7682:35:2822:36:5385:37:4360:38:39:7682:40:41:2822:5385:4360:5642:5642:5642:7682:7682:7682:42:2822:


Last edited by HATMAN on Sat Feb 09, 2013 8:14 am, edited 2 times in total.

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Andrew
 Post subject: Re: Paper Solvable 12
PostPosted: Wed Jan 23, 2013 10:03 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
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Thanks HATMAN for your latest puzzles.

I tried to get started on version C in Paper Solvable mode but for some time I couldn’t find a way in so I decided to resort to using elimination solving. Then on my fourth step I found something which also works for Paper Solvable mode, so I started again that way.

Version C is Paper Solvable, but probably the hardest puzzle I've solved this way. It felt easier to start using elimination solving, but that way is probably at least as easy for all killers.

Here is my walkthrough for Paper Solvable 12, version C:
The Windows at R2C2, R2C6, R6C2 and R6C6 are numbered W1, W2, W3 and W4

In this walkthrough I’ve used Windoku properties, the four given windows and five hidden ones, as in my post in the Standard Techniques forum here

In some cases I've referred back to deductions in previous steps, rather than repeating all the deductions. Then in later steps, when cages were clearly limited to one combination, it wasn't necessary to refer back to earlier steps.

1. 24(3) cage at R1C6 contains 9 for R1, 22(3) cage at R9C2 contains 9 for R9 -> hidden window R159C159 must contain 9 in R5C159
45 rule on D/ 3 innies R4C6 + R5C5 + R6C4 = 10 -> no 9 in R5C5
24(3) cage at R1C6 contains 7,8 for R1
Hidden killer pair 7,8 in 22(3) cage at R9C2 and R5C234 for hidden window R159C234, 22(3) cage contains one of 7,8 -> R5C234 must contain one of 7,8
30(7) cage at R4C4 = {1234578} (cannot be {1234569} which doesn’t contain 7 or 8) -> no 9 in R5C1
-> R5C9 = 9

2. 24(3) cage at R1C6 contains 7,8 for hidden window R159C678
21(3) cage at R6C9 = {678} (only combination not containing 9) contains 7,8 for C9
31(7) cage at R3C9 must contain 7 -> R6C6 = 7, placed for D/ and W4

3. R5C9 = 9 -> 31(7) cage at R3C9 cannot contain 8 because a 31(7) cage is missing either 5,9 or 6,8
R4C6 + R5C5 + R6C4 = 10 -> no 8 in R5C5
-> 8 in R5 only in R5C1234, locked for 30(7) cage at R4C4 -> no 8 in R4C4
30(7) cage at R4C4 = {1234578} (deduced in step 1), no 6,9 in R4C4
45 rule on D\ 2 remaining innies R4C4 + R5C5 = 11 = [56] (because no 6,8,9 in R4C4, no 8,9 in R5C5) -> R4C4 = 5, placed for D\ and W1, R5C5 = 6, placed for D\

4. 11(3) cage at R7C3 can only be {128} because 5,6,7 already placed on D\, contains 8 for N9
21(3) cage at R6C9 can only be {678} -> R6C9 = 8

5. 45 rule on N3 4 innies R2C79 + R3C89 = 1 outie R1C6 +3
R1C6 can only contain 8,9 -> R2C79 + R3C89 = 11,12 must contain 1,2 for N3
7 in 24(3) cage at R1C6 must be in R1C78 for N3
45 rule on D/ 2 remaining innies R4C6 + R6C4 = 4 = {13} for D/
18(3) cage at R1C9 can only be {459} for N3
24(3) cage at R1C6 = {789} -> R1C6 = 9

6. 24(3) cage at R1C6 contains 7,8,9 for hidden window R159C678 -> no 7,8,9 in R9C678
21(3) cage at R6C9 contains 6,7,8 for hidden window R678C159 (because R5C9 = 9) -> no 6,7,8 in R678C5, also contains 6 for N9 because R6C9 = 8
30(7) cage at R6C4 must contain both of 6,9 or both of 7,8 but only place for 7,8 is in R9C5 -> 30(7) cage must contain both of 6,9 -> R9C6 = 6 (only place in 30(7) cage)
9 in 30(7) cage must be in C5
45 rule on D/ 2 remaining innies R4C6 + R6C4 = 4 = {13}
-> R6C5 = 9 (hidden single in N5)

7. 18(3) cage at R1C9 can only be {459} (deduced in step 5) for D/
45 rule on D/ 2 remaining innies R4C6 + R6C4 = 4 = {13} for D/
-> 17(3) cage at R7C3 can only be {278} for N7
22(3) cage at R9C2 can only be {589} -> R9C4 = 8 (only remaining place), placed for hidden window R159C234 -> no 8 in R5C234
R5C9 = 9 -> 31(7) cage at R3C9 cannot contain 8 because a 31(7) cage is missing either 5,9 or 6,8
R4C5 = 8 (hidden single in N5), R5C1 = 8 (hidden single in R5)

8. 11(3) cage at R7C3 can only be {128} (because 5,6,7 already placed on D\), locked for D\ -> 16(3) cage at R1C1 can only be {349} for N1
-> 20(3) cage at R2C9 can only be {569} -> R4C9 = 9 (only remaining place), R23C9 = {56} for N1
31(7) cage at R1C3 contains 8 so must also contain 6 -> R1C4 = 6 (only remaining place)
17(3) cage at R7C3 can only be {278} (deduced in step 7) for N7 -> R9C1 = 7 (hidden single in C1)

9. 16(3) cage at R1C1 can only be {349}, 20(3) cage at R2C9 can only be {569} (both deduced in step 8) -> 1,2 in C1 only in R678C1 for hidden window R678C159 -> no 1,2 in R678C5
45 rule on D/ 2 remaining innies R4C6 + R6C4 = 4 = {13}
30(7) cage at R6C4 contains 2, must be in R9C57 for R9
11(3) cage at R7C3 can only be {128} (because 5,6,7 already placed on D\) -> R9C9 = 1
30(7) cage at R6C4 contains 1 -> R6C4 = 1 (only remaining place), placed for W3, R4C6 = 3, placed for W2

10. 31(7) cage at R3C9 contains 1, must be in R5C78 for R5
30(7) cage at R4C4 contains 1 -> R7C1 = 1 (only remaining place), placed for hidden window R678C159, no 1 in R8C5
R8C6 = 1 (hidden single in N8)

11. R5C78 contain 1 and 5 (hidden singles in R5), 31(7) cage at R3C9 contains 3 -> R3C9 = 3 (only remaining place), placed for hidden window R234C159, no 3, in R2C5 -> R2C4 = 3 (hidden single in N2)

12. 3 in R5 only in R5C23 for hidden window R159C234, no 3 in R1C2 -> R1C1 = 3 (hidden single in R1)

13. 20(3) cage at R2C1 = {569} (only possible combination) contains 5 for N1
31(7) cage at R1C3 cannot contain 5 because a 31(7) cage is missing either 5,9 or 6,8
24(3) cage at R1C6 = {789}
-> R1C9 = 5 (hidden single in R1)

14. 20(3) cage at R2C1 = {569} (only possible combination) contains 5,6 for N1, R1C1 = 3 -> 16(3) cage at R1C1 = {349} for N1
24(3) cage at R1C6 = {789} for R1
R2C3 + R3C2 = {78} (only remaining places for 7,8 in N1) -> R1C23 = {12} (only remaining places in N1)
-> R1C5 = 4 (hidden single in R1)
-> R9C5 = 2 (only remaining place in hidden window R159C159)
31(7) cage at R1C3 contains 2 -> R1C3 = 2 (only remaining place) -> R1C2 = 1 -> R4C3 = 1 (hidden single in C3)
17(3) cage at R7C3 = {278} (deduced in step 7) -> R8C2 = 2, R7C3 = 8, R3C2 = 8, R2C3 = 7, R2C6 = 8 (hidden single in N2)

15. 31(7) cage at R1C3 contains 1,7 -> R3C5 = 7, R2C5 = 1
R5C2 = 7, R5C3 = 3 (hidden singles in R5)

16. 22(3) cage at R9C2 = {589} (only possible combination) -> R7C2 = 3 (hidden single in N7), R8C5 = 3, R7C5 = 5 (hidden singles in C5)
20(3) cage at R2C1 = {569} (only possible combination) contains 6 for C1 -> R8C3 = 6, R8C1 = 4 (hidden singles in N7), R6C1 = 2 (hidden single in C1), R5C6 = 2, R5C4 = 4 (hidden singles in N5)

17. 31(7) cage at R3C9 contains 4 -> R4C9 = 4 (only remaining place)
21(3) cage at R6C9 = {678} (only possible combination) -> R7C9 = 6, R8C9 = 7, R2C9 = 2 (hidden single in C9), R7C4 = 7, R7C6 = 4, R8C4 = 9, R8C6 = 1 (hidden singles in N8), R3C4 = 2 (hidden single in C4), R3C6 = 5 (hidden single in N2), R2C1 = 5, R3C1 = 6 (hidden singles in C1)

18. 11(3) cage at R7C7 = {128} (only possible combination) -> R8C8 = 8, R7C7 = 2, R8C7 = 5 (hidden single in R8)
24(3) cage at R1C6 = {789} -> R1C7 = 8, R1C8 = 7

19. 18(3) cage at R1C9 = {459} (only possible combination) contains 4,9 for N3
R3C8 = 1, R2C7 = 6 (hidden singles in N3)
R6C8 = 6, R7C8 = 9, R6C7 = 3 (hidden singles in W4)

and the rest is hidden singles (not naked singles, which don’t exist in Paper Solvable mode), without using diagonals, given or hidden windows.

Rating Comment:
I'll rate my Paper Solvable walkthrough at 1.25. In most steps I had to string together several sub-steps to get a placement. Based on my first few steps using elimination solving, I'll rate it at Hard 1.0 that way.

I'm surprised that the SS score is so low. I can only think that it doesn't give any weighting to using the hidden windows. I'd put using them at least Hard 1.0, which is why I rated my elimination solving steps at that level. The solving path would, I think, be a lot longer and harder without using the hidden windows.
I'm currently busy, but I'll try version D when I've got time. I can see why that small change in R1 has a significant effect on the puzzle's difficulty.
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Andrew
 Post subject: Re: Paper Solvable 12
PostPosted: Sun Feb 03, 2013 3:42 am 
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Joined: Wed Apr 23, 2008 6:04 pm
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As promised, I had a try at version D, after being busy on other things. Of the three puzzles on this forum that I've tried after that break from this site, this was definitely the easiest one. Thanks HATMAN for a nice puzzle! I'll agree, at the easy end of Assassins
but only if:
one uses the hidden windows
Since only version C was intended to be Paper Solvable, I've used elimination solving for version D. The change to the cage at R1C6 took away the way I used to get into the puzzle in version C; however after I'd found a different way in it wasn't much harder.

Here is my walkthrough for Paper Solvable 12, version D:
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) R1C67 = {89}
b) 20(3) cage at R2C1 = {389/479/569/578}, no 1,2
c) 21(3) cage at R6C9 = {489/579/678}, no 1,2,3
d) 11(3) cage at R7C7 = {128/137/146/236/245}, no 9
e) 22(3) cage at R9C2 = {589/679}

Steps resulting from Prelims
1a. Naked pair {89} in R1C67, locked for R1 and hidden window R159C678, no 8,9 in R59C678
1b. 22(3) cage at R9C2 = {589/679}, 9 locked for R9 and hidden window R159C234, no 9 in R5C234

2. 45 rule on D/ 3 innies R4C6 + R5C5 + R6C4 = 10 = {127/136/145/235}, no 8,9

3. 18(3) cage at R1C9 = {279/369/378/459/468} (cannot be {189} which clashes with R1C7, cannot be {567} which clashes with R4C6 + R5C5 + R6C4), no 1
3a. Killer pair 8,9 in R1C7 and 18(3) cage, locked for N3

4. 31(5) cage at R1C3 must contain one of 8,9 in R234C5
4a. 20(3) cage at R2C1 = {479/569/578} (cannot be {389} which clashes with R234C5 in hidden window R234C159), no 3
4b. Killer pair 8,9 in 20(3) cage and R234C5, locked for hidden window R234C159, no 8,9 in R4C9

[Slightly optimised. I didn’t see the next step until I’d done two other steps but moving it here simplifies the step which follows it.]

5. Hidden killer pair 8,9 in 30(7) cage at R4C4 and 31(7) cage at R3C9 for R5, each of the 30(7) and 31(7) cages can only contain one of 8,9 which must be in R5 -> no 8,9 in R4C4 + R67C1 + R6C6
5a. 9 in N5 only in R46C5, locked for C5

6. 45 rule on D\ 3 innies R4C4 + R5C5 + R6C6 = 18 = {567} (only remaining combination), locked for N5 and D\

7. 31(7) cage at R1C3 contains one of 8,9 -> R5C9 = {89}
7a. 21(3) cage at R6C9 = {579/678} (cannot be {489} which clashes with R5C9), no 4, 7 locked for C9 and hidden window R678C159, no 7 in R678C1 + R78C5
7b. Killer pair 8,9 in R5C9 and 21(3) cage, locked for C9

8. 11(3) cage at R7C7 = {128} (only remaining combination), locked for N7 and D\, 8 locked for W4
8a. 8 in C9 only in R56C9, locked for N6

9. Naked triple {349} in 16(3) cage at R1C1, locked for N1, 9 locked for W1

10. 20(3) cage at R2C1 (step 4a) = {569/578} (cannot be {479} because 4,9 only in R4C1), no 4, 5 locked for C1 and hidden window R234C159, no 5 in R23C5 + R234C9
10a. 9 of {569} must be in R4C9 -> no 6 in R4C9

11. 30(7) cage at R4C4 contains 5 in R4C4 + R5C23, CPE no 5 in R5C5

12. 31(7) cage at R3C9 contains 7 in R5C78 + R6C6, CPE no 7 in R5C5

13. R5C5 = 6, placed for D/
13a. R4C6 + R5C5 + R6C4 (step 2) = {136} (only remaining combination), locked for N5 and D/

14. 18(3) cage at R1C9 (step 3) = {279/459}, 9 locked for N3 and D/ -> R1C7 = 8, R1C6 = 9

15. R8C8 = 8 (hidden single in N9)

16. 8 on D/ only in 17(3) cage at R7C3, locked for N7

17. 5 on R5 and D\ only in 31(7) cage at R3C9 and 30(7) cage at R4C4 -> both cages must contain 5 (because no 5 in R5C5) -> 31(7) cage = {1234579} (only remaining combination) -> R5C9 = 9

18. 21(3) cage at R6C9 (step 7a) = {678} (only remaining combination) -> R6C9 = 8, R78C9 = {67}, locked for C9, N9 and hidden window R678C159, no 6 in R678C1

19. R1C9 = 5 (hidden single in C9), placed for D/
19a. 18(3) cage at R1C9 (step 14) = {459} (only remaining combination), locked for D/ and N3
19b. Naked triple {278} in 17(3) cage at R7C3, locked for N7

20. R4C9 = 4 (hidden single in C9) -> R5C6 = 2

21. 30(7) cage at R4C4 contains 2 -> R6C1 = 2

22. R7C7 = 2 (hidden single in W4), R9C9 = 1, placed for hidden window R159C159, no 1 in R1C5
22a. R23C9 = [23], both placed for hidden window R234C159, no 3 in R2C5, no 2 in R3C5

23. 31(7) cage at R3C9 contains 1 in R5C78, locked for R5, N6 and hidden window R159C678, no 1 in R1C8

24. 30(7) cage at R4C4 contains 1 -> R7C1 = 1, locked for hidden window R678C159, no 1 in R8C5

25. R8C2 = 2 (hidden single in N7), R1C3 = 2 (hidden single in C3) , R3C4 = 2 (hidden single in N2), R9C5 = 2 (hidden single in R9)

26. 22(3) cage at R9C2 = {589/679}
26a. 7,8 on in R9C4 -> R9C4 = {78}
26b. 22(3) cage at R9C2 = {589/679}, 9 locked for N7
26c. Naked pair {78} in R9C14, locked for R9

27. R4C1 = 9 (hidden single in C1), R4C5 = 8, R5C4 = 4, R6C5 = 9
27a. R23C1 = {56} (hidden pair in C1), locked for N1

28. Naked pair {34} in R18C1, locked for C1

29. 3 in R5 only in R5C123, locked for N4

30. 4 in R6 only in R6C23, locked for W3

31. R8C1 = 4 (hidden single in N7), R1C1 = 3

32. Naked triple {147} in R123C5, locked for C5, N2 and 31(7) cage at R1C3 -> R1C4 = 6, R4C6 = 3

33. Naked pair {35} in R78C5, locked for N8 and 30(7) cage at R6C4 -> R9C67 = [64], R3C7 = 9, R2C8 = 4, R2C2 = 9, R9C23 = [59], R9C4 = 8 (cage sum)

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

Rating Comment:
I'll rate my walkthrough for PS12, version D at 1.25. That rating is based on the use of hidden windows, rating the use of that feature/technique as 1.25. Step 5 took me some time to spot but isn't a difficult step.

I'm completely confused by the SS scores for version C and version D. If SudokuSolver uses hidden windows then it ought to give relatively low scores for both versions. If it doesn't use hidden windows, then its high score for version D makes sense but I don't know how it could get a very low score for version C without using them.
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Andrew
 Post subject: Re: Paper Solvable 12
PostPosted: Mon Feb 04, 2013 12:02 am 
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After doing version D, I realised that my way in also works for version B so I tried that next. It's longer than version D and a bit harder, but not a lot harder.

I'm still fairly inexperienced at solving Windoku Killer-Xs:
I had to find a new way to use the hidden windows in steps 28 and 29, which cracked this puzzle. I'm guessing this is where JSudoku used small fishes.

Here is my walkthrough for Paper Solvable 12, version B:
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) 20(3) cage at R2C1 = {389/479/569/578}, no 1,2
b) 21(3) cage at R6C9 = {489/579/678}, no 1,2,3
c) 11(3) cage at R7C7 = {128/137/146/236/245}, no 9
d) 22(3) cage at R9C2 = {589/679}

1. 22(3) cage at R9C2 = {589/679}, 9 locked for R9 and hidden window R159C234, no 9 in R15C234

2. 45 rule on D/ 3 innies R4C6 + R5C5 + R6C4 = 10 = {127/136/145/235}, no 8,9

3. Hidden killer pair 8,9 in 30(7) cage at R4C4 and 31(7) cage at R3C9 for R5, each of the 30(7) and 31(7) cages can only contain one of 8,9 which must be in R5 -> no 8,9 in R23C9 + R4C4 + R67C1 + R6C6

4. 45 rule on D\ 3 innies R4C4 + R5C5 + R6C6 = 18 = {567} (only remaining combination), locked for N5 and D\

5. 11(3) cage at R7C7 = {128} (only remaining combination), locked for N7 and D\

6. Naked triple {349} in 16(3) cage at R1C1, locked for N1

7. 20(3) cage at R2C1 = {569/578} (cannot be {389/479} because 3,4,9 only in R4C1), no 3,4, 5 locked for C1 and hidden window R234C159, no 5 in R23C5 + R234C9
7a. 9 of {569} must be in R4C1 -> no 6 in R4C9

8. Hidden killer pair 8,9 in 31(7) cage at R1C3 and 30(7) cage at R6C4 for C5, each of the 30(7) cages can only contain one of 8,9 which must be in C5 -> no 8 in R1C34 + R9C6

9. 30(7) cage at R4C4 contains 5 in R4C4 + R5C23, CPE no 5 in R5C5

10. R4C6 + R5C5 + R6C4 (step 2) = {127/136}, no 4, 1 locked for N5 and D/

11. 5 on R5 and D\ only in 31(7) cage at R3C9 and 30(7) cage at R4C4 -> both cages must contain 5 (because no 5 in R5C5) -> 31(7) cage = {1234579} (only remaining combination), no 6,8, 9 locked for R5

12. 30(7) cage at R4C4 = {1234578} (only remaining combination), no 6

13. R5C5 = 6 (hidden single in R5) -> R4C6 + R5C5 + R6C4 (step 10) = {136} (only remaining combination), locked for N5 and D/

14. 18(3) cage at R1C9 = {279/459}, no 8, 9 locked for N3 and D/

15. 8 on D/ only in 17(3) cage at R7C3, locked for N7

16. 30(7) cage at R4C4 = {1234578}, 5,7 only in R4C4 + R5C123 + R67C1, CPE no 5,7 in R4C1, no 5 in R4C23

17. 20(3) cage at R2C1 = {569/578}
17a. R4C1 = {89} -> R23C1 = {567}, 5 locked for N1

18. 31(7) cage at R3C9 = {1234579}, CPE no 5 in R6C789, no 7 in R6C9

19. 21(3) cage at R6C9 = {489/579/678}
19a. 8 of {489/678} must be in R6C9 -> no 4,6 in R6C9

20. 7 on R5 and D\ only in 31(7) cage at R3C9 and 30(7) cage at R4C4 -> no 7 in R34C9 + R67C1

21. 31(7) cage at R3C9 = {1234579}, CPE no 7 in R6C78

22. 30(7) cage at R4C4 = {1234578}, CPE no 7 in R4C23

23. 1,3 on C5 and D/ only in 31(7) cage at R1C3 and 30(7) cage at R6C4 -> no 1,3 in R1C34 + R9C67

24. 1,3 in 31(7) cage at R1C3 only in R123C5 + R4C6, CPE no 1,3 in R123C6

25. 1,3 in 30(7) cage at R6C4 only in R6C4 + R789C5, CPE no 1,3 in R78C4

26. 5 in W1 only in R234C4, locked for C4

27. 2,4 in 31(7) cage at R1C3 and 30(7) cage at R6C4 only in R1C34 + C5 + R9C67, CPE no 2,4 in R1C678 using hidden window R159C678

28. 4 in 31(7) cage at R3C9 and 30(7) cage at R4C4 only in R34C9 + R5 + R67C1, CPE no 4 in R78C9 using hidden window R678C159
[The puzzle seems to be cracked now, as will become obvious after step 31.]

29. 21(3) cage at R6C9 = {579/678}, 7 locked for C9, N9 and hidden window R678C159, no 7 in R78C5 + R8C1

30. 7 in W4 only in R678C6, locked for C6

31. 30(7) cage at R6C4 = {1234569/1234578} contains one of 6,7 in R9C567
31a. 22(3) cage at R9C2 = {589} (only remaining combination, cannot be {679} which clashes with 30(7) cage) -> R9C4 = 8, placed for hidden window R159C234, no 8 in R1C2 + R5C23, R9C23 = {59}, locked for R9, N7 and hidden window R159C234, no 5 in R5C23

32. R5C1 = 8 (hidden single in R5), placed for hidden window R159C159, no 8 in R1C5, R4C1 = 9, placed for hidden window R234C159, no 9 in R23C5
32a. R4C1 = 9 -> R23C1 = 11 = {56}, locked for C1, N1 and hidden window R234C159, no 6 in R2C9

33. 6 in C9 only in R78C9 -> 21(3) cage at R6C9 (step 29) = {678} (only remaining combination) -> R6C9 = 8, R78C9 = {67}, locked for N9 -> R9C78 = [43], placed for hidden window R159C678, no 3 in R15C7, no 4 in R5C68

34. R9C6 = 6 (hidden single in R9)

35. R9C1 = 7 (hidden single in C1) -> 17(3) cage at R7C3 = {278} (only remaining combination), locked for N7 and D/
35a. Naked pair {28} in R7C3 + R8C2, locked for W3, no 2 in R6C23 + R78C4
35b. Naked triple {459} in 18(3) cage at R1C9, locked for N3

36. 30(7) cage at R6C4 = {1234569}, 5,9 locked for C5

37. R6C1 = 2 (hidden single in C1), placed for hidden window R678C159, no 2 in R78C5, R5C4 = 4, R6C5 = 9, R5C6 = 2, R4C5 = 8

38. 30(7) cage at R4C4 = {1234578} -> R4C4 = 5, R6C6 = 7

39. Naked pair {59} in R7C8 + R8C7, locked for W4

40. 6 in W4 only in R6C78, locked for R6 and N6

41. 31(7) cage at R3C9 contains 4 -> R4C9 = 4, placed for R234C159, no 4 in R23C5

42. R2C8 = 4 (hidden single in N3)

43. R1C5 = 4 (hidden single in C5), R1C1 = 3, R2C2 = 9, placed for W1, R3C3 = 4, R9C23 = [59], R7C1 = 1, placed for R678C159, no 1 in R8C5, R8C1 = 4

44. Naked pair {36} in R7C2 + R8C3, locked for W3 -> R6C4 = 1, R4C6 = 3, R6C23 = [45], R6C78 = [36], R78C6 = [41]

45. R9C9 = 1 (hidden single in N9), R9C5 = 2, R23C9 = [23]

46. Naked pair {17} in R23C5, locked for N2 and 31(7) cage at R1C3 -> R1C34 = [26], R23C4 = [32], R7C3 = 8, R8C2 = 2, R7C7 = 2, R8C8 = 8

47. Naked pair {16} in R4C23, locked for R4 and W1 -> R2C3 = 7

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

Rating Comment:
I'll rate my walkthrough for PS12, version B at Hard 1.25. If I'd done steps 28 and 29 as chains I'd have rated them in the 1.5 range but I think seeing them as CPEs using hidden windows allows them to be in the 1.25 range.

Now to try version A sometime. My early steps allow a way in but, with three more important cages removed, I may find that I don't get much further. It may end up in the Unsolvables but I never place a puzzle there until I've tried it.
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Andrew
 Post subject: Re: Paper Solvable 12
PostPosted: Tue Feb 05, 2013 12:37 am 
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Following on from my earlier comments about the SSscores (in the Rating Comment) I decided to try version C again a different way.


Here is my alternative walkthrough for Paper Solvable 12, version C:
The Windows at R2C2, R2C6, R6C2 and R6C6 are numbered W1, W2, W3 and W4

SudokuSolver gave this version a much lower score than the other variants. I can only assume that it doesn’t use hidden windows so I’m attempting to solve it using the given windows but not the hidden windows. This time I’m using elimination solving, rather than attempting to solve it using Paper Solvable steps.

Prelims

a) 24(3) cage at R1C6 = {789}
b) 20(3) cage at R2C1 = {389/479/569/578}, no 1,2
c) 21(3) cage at R6C9 = {489/579/678}, no 1,2,3
d) 11(3) cage at R7C7 = {128/137/146/236/245}, no 9
e) 22(3) cage at R9C2 = {589/679}

Steps resulting from Prelims
1a.Naked triple {789} in 24(3) cage at R1C6, locked for R1
1b. 22(3) cage at R9C2 = {589/679}, 9 locked for R9

2. 45 rule on D/ 3 innies R4C6 + R5C5 + R6C4 = 10 = {127/136/145/235}, no 8,9

3. Hidden killer pair 8,9 in 31(7) cage at R3C9 and 30(7) cage at R4C4 for R5, 30(7) and 31(7) cages can each only contain one of 8,9 so these must be on R5, no 8,9 in R34C9 + R4C4 + R67C1 + R6C6

4. 45 rule on D\ 3 innies R4C4 + R5C5 + R6C6 = 18 = {567} (only remaining combination), locked for N5 and D\

5. 11(3) cage at R7C3 = {128} (only remaining combination), locked for N9 and D\

6. Naked triple {349} in 16(3) cage at R1C1, locked for N1, 9 also locked for W1

7. 20(3) cage at R2C1 = {569/578} (cannot be {389/479} because 3,4,9 only in R4C1), no 3,4, 5 locked for C1
7a. 9 of {569} must be in R4C1 -> no 6 in R4C1

8. 30(7) cage at R4C4 contains 5, CPE no 5 in R5C5

9. R4C6 + R5C5 + R6C4 (step 2) = {127/136}, no 4, 1 locked for N5 and D/

10. 18(3) cage at R1C9 = {369/459/468} (cannot be {279} which clashes with R1C78, ALS block, cannot be {378/567} which clash with R4C6 + R5C5 + R6C4), no 2,7
10a. Killer triple 7,8,9 in R1C78 and 18(3) cage, locked for N3
10b. 7 in N3 only in R1C78, locked for R1

11. 5 in R5 and D\ only in 31(7) cage at R3C9 and 30(7) cage at R4C4 -> both these cages must contain 5 -> 31(7) cage = {1234579} (only remaining cage), no 6,8, 9 locked for R5
11a. 31(7) cage = {1234579}, CPE no 5,7 in R6C9

12. 21(3) cage at R6C9 = {489/579/678}
12a. 8 of {489/678} must be in R6C9, 9 of {579} must be in R6C9 -> R6C9 = {89}

13. 30(7) cage at R4C4 = {1234578} (only remaining combination), no 6
13a. 30(7) cage at R4C4 = {1234578}, CPE no 5 in R4C123, no 7 in R4C1
13b. 5 in C1 only in R23C1, locked for N1

14. R5C5 = 6 (hidden single in N5) -> R4C6 + R5C5 + R6C4 (step 9) = {136} (only remaining combination), locked for N5 and D/

15. 18(3) cage at R1C9 (step 10) = {459} (only remaining combination), locked for N3 and D/, 9 also locked for W2

16. Naked triple {278} in 17(3) cage at R7C3, locked for N7

17. R1C6 = 9 (hidden single in R1)
17a. 9 in N5 only in R46C5, locked for C5
17b. 9 in R5 only in R5C789, locked for N6 -> R6C9 = 8
17c. 21(3) cage at R6C9 = {489/678}, no 5

18. 8 in W2 only in R23C6, locked for C6 and N2

19. 31(7) cage at R1C3 must contain one of {89} -> R4C5 = {89}
19a. Naked pair {89} in R4C15, locked for R4

20. 8 in W1 only in R2C3 + R3C2, locked for N1

21. 31(7) cage at R1C3 contains 7, locked for C5 and N2

22. 7 in W2 only in R4C78, locked for R4 and N4 -> R4C4 = 5, R6C6 = 7, placed for W4

23. 7 in C9 only in R78C9 -> 21(3) cage at R6C9 (step 12) = {678} (only remaining combination) -> R78C9 = {67}, locked for C9 and N9

24. 31(7) cage at R1C3 contains 4, locked for N2
24a. 31(7) cage contains 3 in R1C45 + R23C5 + R4C6, CPE no 3 in R23C6

25. 5 in R5 only in R5C789, locked for N6

26. 5 in R6 only in R6C23, locked for W3

27. 5 in N7 only in R9C23 -> 22(3) cage at R9C2 = {589} (only remaining combination) -> R9C4 = 8, R9C23 = {59}, locked for R9 and N7

28. Naked pair {24} in R5C46, locked for R5 and N5 -> R6C5 = 9, R4C5 = 8, R4C1 = 9

29. Naked pair {34} in R9C78, locked for R9 and N9

30. R9C1 = 7, R9C6 = 6 (hidden singles in R9)
30a. Naked pair {28} in R7C3 + R8C2, locked for W3

31. R78C4 = {79} (hidden pair in N8)

32. Naked pair {56} in R23C1, locked for C1 and N1
32a. Naked pair {12} in R1C23, locked for R1 and N1

33. Naked triple {134} in R178C1, locked for C1, 1 also locked for N7 -> R5C1 = 8, R6C1 = 2, R5C4 = 4, R5C6 = 2

34. Naked triple {123} in R239C9, locked for C9, 3 also locked for N3 -> R4C9 = 4, R1C9 = 5, R5C9 = 9

35. R1C4 = 6 (hidden single in R1)

36. 30(7) cage at R6C4 = {1234569}, 2,5 locked for C5 and N8

37. 31(7) cage at R1C3 contains 2 -> R1C3 = 2, R1C2 = 1, R7C3 = 8, R8C2 = 2, R2C3 = 7, R3C2 = 8

38. Naked triple {134} in R478C6 locked for C6, 4 also locked for N8 -> R3C6 = 5, R2C6 = 8, R23C1 = [56]

39. 30(7) cage at R6C4 = {1234569} -> R9C7 = 4, R3C7 = 9, R2C8 = 4, R8C7 = 5, R7C8 = 9, R9C8 = 3, R78C4 = [79], R78C9 = [67]

40 Naked triple {134} in R7C126, locked for R7 -> R7C7 = 2

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

Rating Comment:
I'll rate my walkthrough for PS12, version C, using elimination solving but not the hidden windows, at 1.25. This rating is based on steps 3 and 11. The SS score still feels too low, unless it's possible to solve this puzzle without using these steps.

This walkthrough confirms my feeling that SudokuSolver doesn't use hidden windows, which is why it gives a low score for version C but very high scores for versions B and D; at this stage I don't know whether the score for version A is valid or too high.
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Andrew
 Post subject: Re: Paper Solvable 12
PostPosted: Wed Feb 06, 2013 12:11 am 
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Andrew wrote:
Now to try version A sometime. My early steps allow a way in but, with three more important cages removed, I may find that I don't get much further. It may end up in the Unsolvables but I never place a puzzle there until I've tried it.

I've now done that and, for the record, I didn't get very far (the steps I managed are given below). After the position which I reached it probably requires a lot of T&E, which I'm not going to attempt, so I'll add version A to the Unsolvables. It seems to fit Mike's final definition

4.0: Puzzles with unique solutions, but which can only realistically be solved by computer programs using backtracking (brute force).

Three more cages had been removed from version B to give version A. I think that the most important of these is:
the 20(3) cage at R2C1. If that cage had been present, I could have locked 5 for C1 and hidden window R234C159, then removed the 5 from R5C5 and reduced the 31(7) cage at R3C9 to one combination.

I checked this with SudokuSolver which scores version A plus the 20(3) cage at 5.3. It must somehow find a shorter solving path than when the 21(3) cage at R6C9 and the 22(3) cage at R9C2 are also present. It still uses lots of T&E.

This is how far I got for version A:
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.

1. 45 rule on D/ 3 innies R4C6 + R5C5 + R6C4 = 10 = {127/136/145/235}, no 8,9

2. Hidden killer pair 8,9 in 30(7) cage at R4C4 and 31(7) cage at R3C9 for R5, 30(7) and 31(7) cages can each only contain one of 8,9 which must be in R5 -> no 8,9 in R34C9 + R4C4 + R67C1 + R6C6

3. 45 rule on D\ 3 innies R4C4 + R5C5 + R6C6 = 18 = {567} (only remaining combination), locked for N5 and D\

4. 11(3) cage at R7C7 = {128} (only remaining combination), locked for N7 and D\

5. Naked triple {349} in 16(3) cage at R1C1, locked for N1

6. 31(7) cage at R1C3 contains 7, CPE no 7 in R1C6

7. 31(7) cage at R3C9 contains 7, CPE no 7 in R6C9

8. 30(7) cage at R4C4 contains 5, CPE no 5 in R4C1

9. 30(7) cage at R6C4 contains 5, CPE no 5 in R9C4

10. 30(7) cage at R4C4 and 30(7) cage at R6C4 must each contain 1,2,3,4, one set of 1,2,3,4 must be in R5 -> the other set of 1,2,3,4 must be in R34C9 + R67C1 -> no 5,6,7 in R34C9 + R67C1
10a. Naked quad {1234} in R34C9 + R67C1, CPE no 1,2,3,4 in R234C1 using hidden window R234C159
10b. Similarly CPE no 1,2,3,4 in R678C9 using hidden window R678C159

11. 31(7) cage at R1C3 and 31(7) cage at R3C9 must each contain 1,2,3,4, one set of 1,2,3,4 must be in C5 -> the other set of 1,2,3,4 must be in R1C34 + R4C4 + R6C6 + R9C67, CPE no 1,2,3,4 in R1C6 using hidden window R159C678
[Note that whichever of 1,2,3,4 is in R6C4 must be in R123C5]
11a. Similarly CPE no 1,2,3,4 in R9C4 using hidden window R159C234

A typo has been corrected and clarifications added to steps 10 and11.


Last edited by Andrew on Mon Oct 28, 2013 8:38 pm, edited 1 time in total.

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wellbeback
 Post subject: Re: Paper Solvable 12
PostPosted: Sun Feb 17, 2013 8:25 pm 
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Here is a walkthrough for version A. I've been working on and off on this for a while when I've had time.

Five Notes

A) I have named the Windoku groups (visible and hidden) based on the nonet in which their center cell is located.

So the visible groups are W1, W3, W7, and W9.

Hidden group r234c159 is H2
Hidden group r159c234 is H4
Hidden group r159c159 is H5
Hidden group r159c678 is H6
Hidden group r678c159 is H8

B) There is a lot of rotational symmetry in the puzzle. Where I apply a set of logic to a particular number and there is an equivalent set of logic to a different number applied 180 degrees rotated - I simply state that with the resultant end conclusion(s). I mark this as "Rotate(X -> Y)". I.e., the previous argument applied to X can also be applied to Y rotated by 180 degrees.

C) There are a few short contradiction chains below. I'm sure a number of them could be replaced by various aquatic creatures.

D) I've proofread this a few times - but that isn't a guarantee I didn't make a mistake somewhere. Let me know if you spot something.

E) Thanks HATMAN for continuing to generate these fine puzzles!


Hidden Text:
I start the same way as Andrew.


1. Innies D/ r4c6,r5c5,r6c4 = +10 (no 89)

30/7@r4c4 and 31/7@r3c9 each contain one of (89)
Whichever (of 89) is in the 30/7 means the other must be in r5c6789 in r5
which means it (8 or 9) must be in r5c1234 in r5.
-> Neither r4c4 or r6c6 can contain an 8 or a 9.

-> Innies D\ r4c4,r5c5,r6c6 = +18 = {567}
-> 16/3@r1c1 = {349}
-> 11/3@r7c7 = {128}

(Edited to remove conclusions not yet reached. Moved lower.)

2. Whatever goes in r4c4 goes in r6c23 in n4 and therefore in r5c789 in n6.
-> The values in both r4c4 and r6c6 are in 31/7@r3c9.
Now, 31/7 cannot contain both a 5 and a 6.
-> One of r4c4 and r6c6 is a 7.

Similarly, Whatever goes in r6c6 goes in r4c78 in n6 and therefore in r5c123 in n4.
-> The values in both r4c4 and r6c6 are in 30/7@r4c4.
Now, 30/7 cannot contain both a 6 and a 7.
-> One of r4c4 and r6c6 is a 5.

-> r4c4,r6c6 = {57} and r5c5 = 6.
-> r4c6,r6c4 = {13}

Also, since 31/7@r3c9 contains a 5
-> 31/7@r3c9 = {1234579} and 30/7@r4c4 = {1234578}

Also 18/3@r1c9 = {9(27|45)} and 17/3@r7c3 = {8(45|27)}
(This moved here from above).

3. Both 31/7@r3c9 and 30/7@r4c4 contain (1234). Only one instance of each can be in r5.
-> r34c9,r67c1 = {1234}

Now, if 3 is in r4c6 -> 3 in n2 in r23c4 -> r1c1 = 3
-> If 3 was in r67c1 -> 3 in (r2c2 or r3c3) -> 3 in r6c4 -> 3 in r7c1 which leaves no place for 3 in n4.

-> 3 not in r67c1 -> 3 in r34c9.

Rotate(3 -> 1) -> 1 in r67c1.

-> r67c1 = {1(2|4)} and r34c9 = {3(4|2)}


4. If 3 is in r4c6 -> 3 in r23c4 in n2 -> in r789c5 in n8. Also implies 3 in r6c78 in n6 -> only place for it in n9 is r9c8.
Else if 3 in r6c4 -> 3 in r78c6 in n8 -> only place for it in n9 is r9c8.

Either way r9c8 = 3.
Rotate (3 -> 1) -> r1c2 = 1.


5. Both 31/7@r1c3 and 30/7@r6c4 contain a 2 and a 4. Only one instance of each can be in c5.
-> r1c34 and r9c67 contains a 2 and 4 somewhere.
If they both were in r9c67 -> both in r1234c5
-> Both in r5c234 in H4 which contradicts the fact that one of them is in r67c1.
Rotate - Similar argument against both going in r1c34.

-> One of (24) in r1c34 and the other in r9c67.
Whichever goes in r9c67 must go in r1234 in c5
-> in r5c234 in H4
-> in r34c9 in 31/7@r3c9 -> Not elsewhere in H2
-> in r1c5 in c5!

Rotate - Similar argument for whichever of (24) goes in r1c34 must also go in r9c5!
-> r19c5 = {24}
-> r5c46 = {24}
-> r46c5 = {89}


6.Further...
Since whichever of (24) in r9c67 also goes in r1c5 and r34c9 and in r5c4
-> only place for it in H8 is r8c1.
-> r678c1 = {124} with 1 in r67c1.
-> 3 in n7 only in r7c2 or r8c3
-> r6c4 = 1
-> r7c1 = 1
-> Also 1 in n8 in r78c6
-> r9c9 = 1

Rotate(24 -> 42)
-> r234c9 = {234}
and r4c6 = 3
and r3c9 = 3
and 3 in r23c4
and r1c1 = 3.


7.Even more...
HS 9 in c1 r4c1 = 9
-> r46c5 = [89]
HS 8 in c9 r6c9 = 8.
Also HS 6 in c1 -> 6 in r23c1.
and HS 6 in c9 -> 6 in r78c9.

Also 31/7@r1c3 = {1234678).
-> r1c4 = 6
-> One of (24) in r1c34 must go in r1c3 and cannot be a 4.
-> r1c345 = [264]
-> r23c5 = {17}

Rotate
30/7@r6c4 = {1234569}
r9c6 = 6.
r9c567 = [264]
r78c5 = {35}


8. Various other immediate conclusions
r5c46 = [42]
r678c1 = [214]
r234c9 = [234]
18/3@r1c9 = {459}
17/3@r7c3 = {278}
r4c3 = 1
r6c7 = 3
r4c2 = 6 (W1)
r8c3 = 6
r7c2 = 3
r5c3 = 3
r8c2 = 2
r9c23 = {59}

+ ROTATES gives a lot more

etc. etc.

Special Note:

Hidden Text:
If you exchange all 1s and 3s, 2s and 4s, 5s and 7s, and 8s and 9s, work out the new totals, and rotate the puzzle 180 degrees, you get the original puzzle. This is similar to the "Rotational Inverse" puzzle we have seen occasionally, except the exchanges are not simply x:10-x. I don't know if this could have been applied to help solve the puzzle.
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Andrew
 Post subject: Re: Paper Solvable 12
PostPosted: Wed Feb 27, 2013 8:58 pm 
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Congratulations wellbeback for solving version A! :applause:

I probably gave up too quickly, but may well not have managed to solve it. Our solving styles are very different.

I'm planning to go through your walkthrough. It might just give me some ideas for trying to finish the harder versions of Old Lace Ls Triple Large Diagonals 1.
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Andrew
 Post subject: Re: Paper Solvable 12
PostPosted: Mon Oct 28, 2013 9:38 pm 
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I wrote:
I probably gave up too quickly, but may well not have managed to solve it. Our solving styles are very different.
As I found, when I had another try at it over the last few days, I definitely gave up too quickly; see comment after step 11.

wellbeback's solving style:
worked well for this puzzle. He got more out of {13} in R4C4 + R6C6 than I did, helped by the contradiction eliminations is step 3. My solving path is a lot longer

Here is my walkthrough for Paper Solvable 12 version A:
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.

1. 45 rule on D/ 3 innies R4C6 + R5C5 + R6C4 = 10 = {127/136/145/235}, no 8,9

2. Hidden killer pair 8,9 in 30(7) cage at R4C4 and 31(7) cage at R3C9 for R5, 30(7) and 31(7) cages can each only contain one of 8,9 which must be in R5 -> no 8,9 in R34C9 + R4C4 + R67C1 + R6C6

3. 45 rule on D\ 3 innies R4C4 + R5C5 + R6C6 = 18 = {567} (only remaining combination), locked for N5 and D\

4. 11(3) cage at R7C7 = {128} (only remaining combination), locked for N7 and D\

5. Naked triple {349} in 16(3) cage at R1C1, locked for N1

6. 31(7) cage at R1C3 contains 7, CPE no 7 in R1C6

7. 31(7) cage at R3C9 contains 7, CPE no 7 in R6C9

8. 30(7) cage at R4C4 contains 5, CPE no 5 in R4C1

9. 30(7) cage at R6C4 contains 5, CPE no 5 in R9C4

10. 30(7) cage at R4C4 and 30(7) cage at R6C4 must each contain 1,2,3,4, one set of 1,2,3,4 must be in R5 -> the other set of 1,2,3,4 must be in R34C9 + R67C1 -> no 5,6,7 in R34C9 + R67C1
10a. Naked quad {1234} in R34C9 + R67C1, CPE no 1,2,3,4 in R234C1 using hidden window R234C159
10b. Similarly CPE no 1,2,3,4 in R678C9 using hidden window R678C159
[Clarifications been added to steps 10 and 11.]

11. 31(7) cage at R1C3 and 31(7) cage at R3C9 must each contain 1,2,3,4, one set of 1,2,3,4 must be in C5 -> the other set of 1,2,3,4 must be in R1C34 + R4C6 + R6C4 + R9C67, CPE no 1,2,3,4 in R1C6 using hidden window R159C678
[Note that whichever of 1,2,3,4 is in R6C4 must be in R123C5]
11a. Similarly CPE no 1,2,3,4 in R9C4 using hidden window R159C234

[First time I tried this puzzle, I got stuck at this stage.
I ought to have spotted steps 12, 13 and 14 when I first tried this puzzle, and got at least as far as step 20.]

12. Hidden killer pair 8,9 in 31(7) cage at R1C3 and 30(7) cage at R6C4 for C5, 30(7) and 31(7) cages can each only contain one of 8,9 which must be in C5 -> no 8,9 in R1C34 + R9C67

13. 30(7) cage at R4C4 contains 5, CPE no 5 in R5C5

14. 31(7) cage at R3C9 contains 7, CPE no 7 in R5C5

15. R5C5 = 6, placed for D/
15a. R4C6 + R5C5 + R6C4 = 10 (step 1) -> R4C6 + R6C4 = 4 = {13}, locked for N5 and D/
15b. Naked pair {13} in R4C6 + R6C4, CPE no 1,3 in R1C4 + R9C6

16. 18(3) cage at R1C9 = {279/459}, no 8, 9 locked for N3 and D/
16a. 8 on D/ only in 17(3) cage at R7C3 = {278/458}, 8 locked for N7

17. 31(7) cage at R3C9 = {1234579}, no 8, 9 locked for R5
17a. 5,7 only in R5C789 + R6C6, CPE no 5,7 in R6C789

18. 30(7) cage at R4C4 = {1234578}
18a. 5,7 only in R4C4 + R5C123, CPE no 5,7 in R4C123

19. 31(7) cage at R1C3 must contain 3 in R123C5 + R4C6, CPE no 3 in R23C6
19a. 30(7) cage at R6C4 must contain 1 in R6C4 + R789C5, CPE no 1 in R78C4

20. 31(7) cage at R1C3 and 30(7) cage at R6C4 must both contain 2,4, one pair of 2,4 must be in C5 -> the other pair of 2,4 must be in R34C1 + R67C9
20a. 2,4 in R34C1 + R67C9, CPE no 2,4 in R9C23 using hidden window R159C234
20b. Similarly CPE no 2,4 in R1C67 using hidden window R159C678

21. Consider placements for 1,3 in R4C6 + R6C4
21a. R4C6 = 1 or R6C4 = 1 => 1 in R78C6 (hidden single in N8) -> 1 in R4C6 + R78C6, locked for C6
21b. R4C6 = 3 => R23C4 = 3 (hidden single in N2) or R6C6 = 3 -> 3 in R23C4 + R6C4, locked for C4
21c. R4C6 = 1 or R6C4 = 1 => 1 in N2 only in R123C5 -> no 1 in R1C3
21d. R4C6 = 3 => 3 in N8 only in R789C5 or R6C4 = 3 -> no 3 in R9C7
21e. R4C6 = 1 => 3 in C6 only in R78C6, locked for W4 => R9C8 = 3 (hidden single in N9) or R4C6 = 3 -> no 3 in R4C8
21f. R4C6 = 1 or R4C6 = 3 => 1 in C6 only in R78C6, locked for W4 => R9C9 = 1 (hidden single in N9) -> no 1 in R4C9
21g. R4C6 = 1 => 1 in C4 only in R23C4, locked for W1 => R1C2 = 1 (hidden single in N1) or R6C4 = 1 -> no 1 in R6C2
21h. R4C6 = 3 => 3 in C4 only in R23C4, locked for W1 => R1C1 = 3 (hidden single in N1) or R6C4 = 3 -> no 3 in R6C1

22. 1 in hidden window R159C678 only in R1C78, locked for N3 or in R5C78, locked for 31(7) cage at R3C9 -> no 1 in R3C9
22a. 1 in 31(7) cage only in R5C789, locked for R5 and N6
22b. 1 in 30(7) cage at R4C4 only in R67C1, locked for C1 and hidden window R678C159, no 1 in R78C5

23. 3 in hidden window R159C234 only in R5C23, locked for 30(7) cage at R4C4 or in R9C23, locked for N7 -> no 3 in R7C1
23a. 3 in 30(7) cage only in R5C123, locked for R5 and N4
23b. 3 in 31(7) cage at R3C9 only in R34C9, locked for C9 and hidden window R234C159, no 3 in R23C5

24. 1 in hidden window R234C159 only in R23C5 + R2C9, CPE no 1 in R2C4
24a. 3 in hidden window R678C159 only in R78C5 + R8C1, CPE no 3 in R8C6

25. 9 in hidden window R159C234 only in R9C234, locked for R9
25a. 8 in hidden window R159C678 only in R1C678, locked for R1

26. 1 in N1 only in R13C2 + R2C3, CPE no 1 in R4C2 using W1
26a. 3 in N9 only in R79C8 + R8C7, CPE no 3 in R6C8 using W4

27. 1 in R4 only in R4C3, placed for W1 => R1C2 = 1 (hidden single in N1) or in R4C6 -> no 1 in R1C5
27a. 3 in R6 only in R6C4 or in R6C7, placed for W4 => R9C8 = 3 (hidden single in N9) -> no 3 in R9C5

28. 1 in R4 only in R4C3, placed for W1 or in R4C6 => R3C4 = 1 (hidden single in N2), placed for W1 -> no 1 in R2C3 + R3C2
28a. R1C2 = 1 (hidden single in N1), placed for hidden window R159C234, no 1 in R9C3
28b. 1 in hidden window R159C678 only in R5C78, locked for R5

29. 3 in R6 only in R6C4 => R7C6 = 3 (hidden single in N8), placed for W4 or in R6C7, placed for W4 -> no 3 in R7C8 + R8C7
29a. R9C8 = 3 (hidden single in N9), placed for hidden window R159C678, no 3 in R1C7
29b. 3 in hidden window R159C234 only in R5C23, locked for R5

30. 3 in 31(7) cage at R1C3 only in R1C5, placed for N2 or in R4C6 => R3C9 = 3 (hidden single in C9) -> no 3 in R3C4
30a. 1 in 30(7) cage at R6C4 only in R6C4 => R7C1 = 1 (hidden single in C1) or in R9C6, placed for N8 -> no 1 in R7C6

31. 2 in N1 only in R12C3 + R3C2, CPE no 2 in R4C3 using W1
31a. 4 in N9 only in R7C8 + R89C7, CPE no 4 in R6C7 using W4

32. 31(7) cage at R1C3 = {1234579/1234678}, 30(7) cage at R6C4 = {1234569/1234578}
32a. Hidden killer pair 8,9 in 31(7) cage at R1C3 and 30(7) cage at R6C4 for C5, 30(7) and 31(7) cages can each only contain one of 8,9 -> both or neither of these cages must contain 6; if neither contain 6 then both contain 5 and 7
32b. Consider combinations for 31(7) cage
31(7) cage = {1234579} => both 31(7) cage and 30(7) cage contain 5,7, one pair of 5,7 in C5 with the other pair in R23C1 + R9C67, CPE no 5,7 in R1C678 using hidden window R159C678 => R1C78 = {68}, locked for R1 and N3, R1C6 = 9
or 31(7) cage = {1234678} => 6 in R1C23, 8 in R234C5, locked for hidden window R234C159
-> no 9 in R123C5, no 6 in R1C6, no 8 in R2C9
32c. Consider combinations for 30(7) cage
30(7) cage = {1234569} => 6 in R9C67, 9 in R678C5, locked for hidden window R678C159
or 30(7) cage = {1234578} => both 30(7) cage and 31(7) cage contain 5,7, one pair of 5,7 in C5 with the other pair in R23C1 + R9C67, CPE no 5,7 in R9C234 using hidden window R159C234 => R9C23 = {69}, locked for R9 and N7, R9C4 = 8
-> no 8 in R789C5, no 9 in R8C1, no 6 in R9C4

33. 31(7) cage at R1C3 = {1234579/1234678}
33a. Consider placements for 9 in C1
R1C1 = 9 => no 9 in R1C6 => 31(7) cage cannot be {1234579} which requires 9 in R1C6 (step 32b)
or R4C1 = 9 => no 9 in R4C5 => 31(7) cage = {1234678}
-> 31(7) cage = {1234678}, no 5,9, 6 locked for R1, 8 locked for C5 and hidden window R234C159, no 8 in R234C1
33b. 9 in C5 only in 30(7) cage at R6C4 = {1234569}, no 7, 6 locked for R9, 9 locked for hidden window R678C159, no 9 in R678C9
33c. 7 in C5 only in R123C5, locked for N2 and 31(7) cage at R1C3, no 7 in R1C3
33d. 5 in C5 only in R789C5, locked for N8 and 30(7) cage at R6C4, no 5 in R9C7
33e. 8 in N1 only in R2C3 + R3C2, locked for W1, no 8 in R23C4 + R4C23
33f. 9 in N9 only in R7C8 + R8C7, locked for W4, no 9 in R6C78 + R78C6

34. Consider placements for 9 in N1
R1C1 = 9 => 9 in N3 only in R2C8 + R3C7, locked for W2, no 9 in R23C6 => 9 in R23C4 (hidden single in N2), locked for W1, no 9 in R4C23
or 9 in R2C2 + R3C3, locked for W1, no 9 in R4C23
-> no 9 in R4C23
[One of these paths could have been taken further as a contradiction move, but it’s easy to continue using the similar forcing chain for N3.]
34a. Consider placements for 9 in N3
R1C9 = 9 => 9 in N1 only in R2C2 + R3C3, locked for W1, no 9 in R23C4 => 9 in R23C6 (hidden single in N2), locked for W2, no 9 in R4C78
or 9 in R2C8 + R3C7, locked for W2, no 9 in R4C78
-> no 9 in R4C78
34b. R4C1 = 9 (hidden single in R4)
34c. 9 in N1 only in R2C2 + R3C3, locked for W1, no 9 in R23C4
34d. 9 in N2 only in R123C6, locked for C6

35. Consider placements for 8 in N7
8 in R7C3 + R8C2, locked for W3, no 8 in R6C23
or 8 in R9C1 => 8 in N9 only in R7C7 + R8C8, locked for W4, no 8 in R78C6 => 8 in R78C4 (hidden single in N8), locked for W3, no 8 in R6C23
-> no 8 in R6C23
35a. Consider placements for 8 in N9
8 in R7C7 + R8C8, locked for W4, no 8 in R6C78
or R9C9 = 8 => 8 in N7 only in R7C3 + R8C2, locked for W3, no 8 in R78C4 => 8 in R78C6 (hidden single in N8), locked for W4, no 8 in R6C78
-> no 8 in R6C78
35b. R6C9 = 8 (hidden single in R6)
35c. 8 in N9 only in R7C7 + R8C8, locked for W4, no 8 in R78C6
35d. 8 in N8 only in R789C4, locked for C4

36. Naked pair {24} in R5C46, locked for R5 and N5 -> R4C5 = 8, R6C5 = 9

37. 31(7) cage at R1C3 = {1234678}, 4 locked for N2
37a. 30(7) cage at R6C4 = {1234569}, 2 locked for N8

38. 18(3) cage at R1C9 (step 16) = {279/459}
38a. R1C78 = {58/78} (cannot be {57} which clashes with 18(3) cage), 8 locked for R1 and N3
38b. Killer pair 5,7 in R1C78 and 18(3) cage, locked for N3

39. 17(3) cage at R7C3 (step 16a) = {278/458}
39a. R9C23 = {59/79} (cannot be {57} which clashes with 17(3) cage), 9 locked for R9 and N7
39b. Killer pair 5,7 in 17(3) cage and R9C23, locked for N7

40. 5 in C4 only in R234C4, locked for W1, no 5 in R2C3 + R3C2
40a. 5 in N1 only in R23C1, locked for C1

41. 7 in C6 only in R678C6, locked for W4, no 7 in R7C8 + R8C7
41a. 7 in N9 only in R78C9, locked for C9

42. Consider combinations for 18(3) cage at R1C9 (step 16) = {279/459}
18(3) cage = {279}
or 18(3) cage = {459}, locked for N3 => R1C78 = {78} => R1C69 = {59} (hidden pair in R1)
-> no 4 in R1C9

43. Consider combinations for 17(3) cage at R7C3 (step 16a) = {278/458}
17(3) cage = {278}, locked for N7 => R9C23 = {59} => R9C14 = {78} (hidden pair in R9)
or 17(3) cage = {458}
-> no 2 in R9C1

44. 4 in C9 only in R234C9, locked for hidden window R234C159, no 4 in R23C5
44a. 31(7) cage at R1C3 = {1234678}, 4 locked for R1 -> R1C1 = 3, R2C4 = 3 (hidden single in N2), R6C4 = 1, placed for W3, R4C6 = 3, R9C9 = 1 (hidden single in R9), R4C3 = 1 (hidden single in R4), R8C6 = 1 (hidden single in N8), R6C7 = 3 (hidden single in R6), R3C9 = 3 (hidden single in R3), R7C1 = 1 (hidden single in R7)

45. 2 in C1 only in R68C1, locked for hidden window R678C159, no 2 in R78C5

46. Naked pair {49} in R2C2 + R3C3, locked for W1, no 4 in R4C2
46a. Naked pair {28} in R7C7 + R8C8, locked for W4, no 2 in R6C8
46b. 4 in R4 only in R4C789, locked for N6 -> R6C8 = 6, placed for W4, no 6 in R7C6 + R8C7
46c. 2 in R6 only in R6C123, locked for N4 -> R4C2 = 6, placed for W1, no 6 in R2C3 + R3C4
46d. 6 in N3 only in R2C79, locked for R2
46e. 6 in N7 only in R8C13, locked for R8

47. 5 in N1 only in R23C1, consider combinations for R23C1 = {56/57}
R23C1 = [56]
or R23C1 = {57}, locked for N1 => naked pair {28} in R2C3 + R3C2, locked for W1 => R3C4 = 5 => R23C1 = [57]
-> R2C1 = 5

48. 7 in N9 only in R78C9, consider combinations for R78C1 = {57/67}
R78C9 = {57}, locked for N9 => naked pair {49} in R7C8 + R8C7, locked for W4 => R7C6 = 7 => R78C9 = [57]
or R78C9 = [67]
-> R8C9 = 7

49. Consider combinations for 18(3) cage at R1C9 (step 16) = {279/459}
18(3) cage = {279}, locked for N3 => naked pair {58} in R1C78, locked for R1 => R1C6 = 9, R1C9 = 2
or 18(3) cage = {459}
-> no 2 in R2C8 + R3C7

50. Consider combinations for 17(3) cage at R7C3 (step 16a) = {278/458}
17(3) cage = {278}
or 17(3) cage = {458}, locked for N7 => naked pair {79} in R9C23, locked for R9 => R9C4 = 8, R9C1 = 4
-> no 4 in R7C3 + R8C2

51. Consider combinations for 18(3) cage at R1C9 (step 16) = {279/459}
18(3) cage = {279} => R1C9 = 2, R4C9 = 4, R5C6 = 2
or 18(3) cage = {459}, 4 locked for W2, no 4 in R4C78 => R4C9 = 4 (hidden single in R4) => R5C6 = 2
-> R4C9 = 4, R5C6 = 2, R5C4 = 4, R6C1 = 2
[Looks like the puzzle is cracked at last.]

52. Naked pair {26} in R1C34, locked for R1 and 31(7) cage at R1C3, naked pair {59} in R1C69, locked for R1, naked pair {78} in R1C78, locked for R1 and N3 -> R1C5 = 4
52a. Naked pair {46} in R9C67, locked for R9, naked pair {78} in R9C14, locked for R9, naked pair {59} in R9C23, locked for R9 and N7 -> R9C5 = 2
52b. Naked pair {59} in R15C9, locked for C9 -> R7C9 = 6, R2C9 = 2, R9C67 = [64], R2C7 = 6, R3C8 = 1, R2C8 = 4 (hidden singles in N3), R23C5 = [17], R3C1 = 6, R8C1 = 4, R1C34 = [26]
52c. R8C3 = 6, R7C2 = 3, R8C2 = 2 (hidden singles in N7)

and the rest is naked singles, without using any windows.

Rating Comment:
I'll rate my walkthrough for PS12 ver A at least 1.75, because I used a lot of forcing chains.
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HATMAN
 Post subject: Re: Paper Solvable 12
PostPosted: Tue Oct 29, 2013 6:05 pm 
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Grand Master
Grand Master

Joined: Wed Apr 30, 2008 9:45 pm
Posts: 694
Location: Saudi Arabia
Well done Andrew. I'll make time to go through your solution and then do Wellbeback's (which I generally find harder to follow, as being a bit more intuitive).

Note I've solved my loading problems by signing up for a virtual ISP - I actually did it for Queen's on-line purchases, but it works here as well.

Is anyone interested in me posting some more "Greater Than Non-Consecutive" puzzles? I have quite a few but I'm not sure if anyone does them.

By the way I'm 65 today - but no intention of retiring as I'd get too bored - Sudoku would not be enough.
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