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Andrew
 Post subject: nd's Killers
PostPosted: Tue Apr 17, 2012 9:31 pm 
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Posts: 1895
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I first came across Killer sudokus in April 2006 on a "newspaper puzzles" site and was immediately hooked on them. Shortly after I used a link which took me to the techniques and puzzles pages of nd's blog (links which unfortunately no longer work) and I was introduced to harder Killers a few months before coming across another link to Ruud's Assassins.

At that time I solved the first six of nd's puzzles using insertion solving. For some time I'd been meaning to have another go at all of them and this time write walkthroughs. I eventually found time to do them last month.

Since the link to nd's blog no longer works, I've taken the liberty of posting all the puzzles here, since I feel they deserve a larger audience. I hope that other forum members and guests will enjoy these puzzles. I've tried several ways in the last month to contact nd, but unfortunately with no success.

Another reason I say unfortunately is that the blogs contained walkthroughs for each of the puzzles; from memory I think they were written by J-C, udosuk and nd. If any site member has access to these walkthroughs, please send them to me by PM and I'll include them in the appropriate puzzle posts.

Each of nd's puzzles had a theme for a particular technique. In the posts below I've included my March 2012 walkthroughs and also a note (warning, it might be a spoiler) of what I thought the theme might be. Please feel free to post your walkthroughs in this thread.

Members of this site may be familar with two of nd's puzzles, nd#9 (aka Night of the Living Sudoku) and A New One (nd#10), which were posted on Ruud's site. I've included them in this thread for completeness and have given links to the archive entries, which include walkthroughs posted at the time.
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Andrew
 Post subject: Re: nd's Killers
PostPosted: Tue Apr 17, 2012 9:36 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
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nd #1

Image

Solution: 
+-------+-------+-------+
| 7 6 9 | 2 3 4 | 5 8 1 |
| 3 1 4 | 8 6 5 | 7 9 2 |
| 5 2 8 | 1 7 9 | 3 6 4 |
+-------+-------+-------+
| 1 9 7 | 6 2 8 | 4 5 3 |
| 8 3 5 | 9 4 1 | 2 7 6 |
| 2 4 6 | 3 5 7 | 8 1 9 |
+-------+-------+-------+
| 4 7 3 | 5 1 6 | 9 2 8 |
| 9 5 1 | 4 8 2 | 6 3 7 |
| 6 8 2 | 7 9 3 | 1 4 5 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:5633:5633:5633:7683:7683:7683:8450:8450:8450:2820:5125:5125:7683:7683:5638:8450:8450:2311:2820:2820:5125:2314:7683:5638:8450:2824:2311:2820:3081:5125:2314:2314:5638:1803:2824:2311:7436:3081:6927:6927:6927:1803:1803:5646:5646:7436:2829:6927:6927:8976:8976:8976:1553:5646:7436:2829:4370:4370:4370:8976:8976:1553:5139:7436:3349:1300:1300:4370:3606:2840:1553:5139:7436:3349:2327:2327:3606:3606:2840:2840:5139:
nd’s puzzles each featured a particular technique, often the key to getting into the puzzle. For this puzzle I used:
a large innie-outie difference. This step, available from the starting position, cracks this puzzle.

This puzzle is an ideal one for Newbies, who know how to solve easier Killers and want to try harder ones.

Here is my 2012 walkthrough for nd#1:
This puzzle, which I originally solved using insertion solving, may not require Prelims but I’ll do them anyway.

Prelims

a) R34C8 = {29/38/47/56}, no 1
b) R45C2 = {39/48/57}, no 1,2,6
c) R67C2 = {29/38/47/56}, no 1
d) R89C2 = {49/58/67}, no 1,2,3
e) R8C34 = {14/23}
g) R9C34 = {18/27/36/45}, no 9
h) 22(3) cage at R1C1 = {589/679}
i) 9(3) cage at R2C9 = {126/135/234}, no 7,8,9
j) 22(3) cage at R2C6 = {589/679}
k) 9(3) cage at R3C4 = {126/135/234}, no 7,8,9
l) 7(3) cage at R4C7 = {124}
m) 22(3) cage at R5C8 = {589/679}
n) 6(3) cage at R6C8 = {123}
o) 20(3) cage at R7C9 = {389/479/569/578}, no 1,2
p) 11(3) cage at R8C7 = {128/137/146/236/245}, no 9
q) 11(4) cage at R2C1 = {1235}
r) 35(5) cage at R6C5 = {56789}

Steps resulting from Prelims

1. 22(3) cage at R1C1 = {589/679}, 9 locked for R1 and N1
1a. 22(3) cage at R2C6 = {589/679}, 9 locked for C6
1b. 22(3) cage at R5C8 = {589/679}, 9 locked for N6, clean-up: no 2 in R3C8
1c. 6(3) cage at R6C8 = {123}, 1,2,3 locked for C8, no 8,9 in R3C8, no 8 in R4C8

[And now the step which I would have started with if I’d just been doing insertion solving.]
2. 45 rule on N9 1 innie R7C7 = 1 outie R6C8 -> R6C8 = 1, R7C7 = 9, clean-up: no 2 in R6C2
2a. Naked pair {23} in R78C8, locked for N9
2b. Naked pair {24} in R45C7, locked for C7, N6 and 7(3) cage at R4C7 -> R5C6 = 1, clean-up: no 7 in R3C8

3. 20(3) cage at R7C9 = {578} (only remaining combination), locked for C9 and N9

4. Naked pair {16} in R89C7, locked for C7 and N9 -> R9C8 = 4, clean-up: no 7 in R4C8, no 9 in R8C2, no 5 in R9C34

5. Naked pair {69} in R56C9, locked for C9 and N6 -> R5C8 = 7 (cage sum), R4C9 = 3, R34C8 = [65], R12C8 = [89], R6C7 = 8, clean-up: no 9 in R5C2, no 3 in R7C2

6. 11(4) cage at R2C1 = {1235}, 3,5 locked for N1

7. Naked triple {679} in 22(3) cage at R1C1, locked for R1 and N1

8. R4C9 = 3 -> R23C9 = 6 = {24}, R1C9 = 1

9. 9(3) cage at R3C4 = {126} (only remaining combination, cannot be {135} because 1,3,5 only in R3C4, cannot be {234} because 3{24} clashes with R4C7) -> R3C4 = 1, R4C45 = {26}, locked for R4 and N5 -> R4C1 = 1, R45C7 = [42], clean-up: no 8 in R5C2, no 4 in R8C3, no 8 in R9C3

10. Naked pair {57} in R6C56, locked for R6, N5 and 35(5) cage at R6C5 -> R7C6 = 6, clean-up: no 4 in R7C2, no 3 in R9C3

11. R2C23 + R3C3 = {148} (hidden triple in N1), R4C3 = 7 (cage sum), clean-up: no 5 in R5C2, no 2 in R9C4

[Only now the second 45 …]
12. 45 rule on C1 1 innie R1C1 = 1 outie R3C2 + 5 -> R1C1 = 7, R3C2 = 2, R23C9 = [24], R3C3 = 8, clean-up: no 9 in R6C2
12a. Naked pair {35} in R23C1, locked for C1
12b. Naked pair {14} in R2C23, locked for R2

13. 45 rule on N2 1 remaining outie R4C6 = 8, R23C6 = 14 = [59], R23C1 = [35], R4C2 = 9, R5C2 = 3, R1C23 = [69], R6C2 = 4, R7C2 = 7, R2C23 = [14], R23C7 = [73], R1C7 = 5, R3C5 = 7, R6C56 = [57]

14. Naked pair {49} in R5C45, locked for R5 and N5 -> R56C9 = [69], R5C3 = 5, R6C4 = 3, R6C3 = 6 (cage sum), R567C1 = [824]

15. Naked pair {24} in R18C4, locked for C4 -> R4C45 = [62], R5C45 = [94], R1C5 = 3, R2C45 = [86], R7C4 = 5, R9C4 = 7, R9C3 = 2

16. R9C6 = 3 -> R8C6 + R9C5 = 11 = [29]

and the rest is naked singles.
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Andrew
 Post subject: Re: nd's Killers
PostPosted: Tue Apr 17, 2012 9:38 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
nd #2

Image

Solution: 
+-------+-------+-------+
| 9 8 2 | 5 1 4 | 6 3 7 |
| 7 1 4 | 3 6 9 | 8 2 5 |
| 5 3 6 | 8 2 7 | 4 9 1 |
+-------+-------+-------+
| 8 4 7 | 1 9 3 | 2 5 6 |
| 2 6 9 | 4 8 5 | 7 1 3 |
| 1 5 3 | 2 7 6 | 9 8 4 |
+-------+-------+-------+
| 3 7 8 | 9 4 1 | 5 6 2 |
| 6 2 5 | 7 3 8 | 1 4 9 |
| 4 9 1 | 6 5 2 | 3 7 8 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:7937:7937:7937:2562:2562:2562:4611:4611:4611:7937:7937:7937:4357:5892:5892:5892:4611:1542:6152:6152:4357:4357:5639:5639:5639:5639:1542:6152:3081:3081:3081:6154:3852:3339:3339:3339:6152:6152:3853:3853:6154:3852:3852:4110:4110:2319:2319:2319:3853:6154:5904:5904:5904:4110:2323:7186:7186:7186:7186:3601:3601:4110:4110:2323:4116:3861:3861:3861:3601:8215:8215:8215:4116:4116:4116:3350:3350:3350:8215:8215:8215:
nd’s puzzles each featured a particular technique, often the key to getting into the puzzle. For this puzzle I used:
a split-cage using 45 rule on R789.
Here is my 2012 walkthrough for nd#2:
Prelims

a) R23C9 = {15/24}
b) R78C1 = {18/27/36/45}, no 9
c) 10(3) cage at R1C4 = {127/136/145/235}, no 8,9
d) 23(3) cage at R2C5 = {689}
e) 24(4) cage at R4C5 = {789}
g) 9(3) cage at R6C1 = {126/135/234}, no 7,8,9
h) 23(3) cage at R6C6 = {689}

1. Naked pair {789} in 24(4) cage at R4C5, locked for C5 and N5 -> R2C5 = 6, R6C6 = 6

2. Naked pair {89} in R2C67, locked for R2

3. Naked pair {89} in R6C78, locked for R6 and N6 -> R6C5 = 7
3a. Naked pair {89} in R26C7, locked for C7

4. 45 rule on R12 2 innies R2C49 = 8 = [35/71], R2C4 = {37}, R2C9 = {15}, clean-up: no 2,4 in R3C9
4a. Naked pair {15} in R23C9, locked for C9 and N3

5. 45 rule on R89 2 innies R8C16 = 14 = [59/68], R8C1 = {56}, R8C6 = {89}, R7C1 = {34}

6. 45 rule on R123 2 innies R3C12 = 8 = {17/26/35}, no 4,8,9
6a. 45 rule on N1 using R3C12 = 8, 1 remaining innie R3C3 = 6, R23C4 = 11 = [38/74], R3C4 = {48}, clean-up: no 2 in R3C12

7. 45 rule on R789 3 outies R5C89 + R6C9 = 8 -> R5C8 = 1, R56C9 = 7 = {34}, locked for C9, N6 and 16(5) cage at R5C8, no 3,4 in R7C89
7a. 45 rule on R789 2 innies R7C89 = 8 = {26} (only remaining combination), locked for R7 and N9

8. 13(3) cage at R4C7 = {256} (only remaining combination), locked for R4 and N6
8a. Naked pair {26} in R47C9, locked for C9

9. R6C7 = 7 -> R45C6 = 8 = [35]

10. 45 rule on N9 using R7C89 = 8 (step 7a), 1 remaining innie R7C7 = 5, R78C6 = 9 = [18] , R2C67 = [98], R6C78 = [98], R8C1 = 6 (step 5), R7C1 = 3, R7C5 = 4
10a. R9C9 = 8 (hidden single in N9)

11. 18(4) cage at R1C7 cannot contain more than one of 7,9, R1C9 = {79} -> no 7,9 in R12C8
11a. 18(4) cage = {2349/2367}, 2,3 locked for N3 -> R3C7 = 4
11b. 18(4) cage = {2367} (only remaining combination) -> R1C9 = 7, R3C8 = 9, R8C9 = 9
11c. Naked pair {13} in R89C7, locked for C7 and N9

12. R3C4 = 8, R2C4 = 3 (step 6a), R2C8 = 2, R1C78 = [63], R7C89 = [62], R4C789 = [256]

13. 10(3) cage at R1C4 = {145} (only remaining combination) -> R1C6 = 4, R1C45 = {15}, locked for R1 and N2 -> R3C56 = [27], R9C6 = 2

14. R3C12 = 8 (step 6) = [53], R23C9 = [51]

15. Naked pair {35} in R89C5, locked for C5 and N8 -> R1C45 = [51], R8C4 = 7, R7C4 = 9, R9C4 = 6, R9C5 = 5 (cage sum), R8C5 = 3, R8C3 = 5 (cage sum), R8C78 = [14], R8C2 = 2, R9C78 = [37]

16. 9(3) cage at R6C1 = {135/234} -> R6C3 = 3, R56C9 = [34], R6C12 = [15], R56C4 = [42], R5C3 = 9 (cage sum), R45C5 = [98], R5C12 = [26], R4C1 = 8 (cage sum)

and the rest is naked singles.
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Andrew
 Post subject: Re: nd's Killers
PostPosted: Tue Apr 17, 2012 9:39 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
nd #3

Image

Solution: 
+-------+-------+-------+
| 2 9 3 | 6 1 4 | 7 5 8 |
| 7 4 6 | 2 5 8 | 9 3 1 |
| 5 8 1 | 7 9 3 | 6 4 2 |
+-------+-------+-------+
| 1 7 5 | 3 4 2 | 8 9 6 |
| 3 2 4 | 9 8 6 | 1 7 5 |
| 9 6 8 | 1 7 5 | 3 2 4 |
+-------+-------+-------+
| 8 1 7 | 5 2 9 | 4 6 3 |
| 6 5 9 | 4 3 1 | 2 8 7 |
| 4 3 2 | 8 6 7 | 5 1 9 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:5889:5889:3074:3074:3074:9222:7683:9221:9221:5889:2567:2567:3074:9222:9222:7683:9221:2308:5889:4873:2568:9222:9222:9994:7683:9221:2308:4873:4873:2568:9222:9994:9994:7683:9221:2308:4873:5900:2568:9994:9994:9486:4623:9221:4363:5901:5900:5900:9994:9994:9486:4623:4363:4363:5901:6160:5900:9994:9486:9486:4623:4363:5137:5901:6160:6160:9486:9486:4882:4623:4623:5137:6160:6160:6160:9486:4882:4882:4882:5137:5137:
nd’s puzzles each featured a particular technique, often the key to getting into the puzzle. For this puzzle I used:
limited combinations for large cages and hidden singles.
Here is my 2012 walkthrough for nd#3:
Prelims

a) R2C23 = {19/28/37/46}, no 5
b) 9(3) cage at R2C9 = {126/135/234}, no 7,8,9
c) 10(3) cage at R3C3 = {127/136/145/235}, no 8,9
d) 23(3) cage at R6C1 = {689}
e) 12(4) cage at R1C3 = {1236/1245}, no 7,8,9
f) 30(4) cage at R1C7 = {6789}
g) 39(8) cage at R3C6 = {12345789}, no 6

Steps resulting from Prelims
1. Naked quad {6789} in 30(4) cage at R1C7, locked for C7
1a. Naked triple {689} in 23(3) cage at R6C1, locked for C1

2. 45 rule on C123 1 innie R1C3 = 3, clean-up: no 7 in R2C23
2a. 12(4) cage at R1C3 = {1236} (only remaining combination), 1,2,6 locked for N2

3. 45 rule on C789 1 innie R9C7 = 5

4. 45 rule on C89 1 innie R8C8 = 8

5. 36(6) cage at R1C8 contains 8 -> R1C9 = 8
5a. Naked triple {679} in R123C7, locked for C7 and N3 -> R4C7 = 8
5b. 36(6) cage contains 7,9 -> R45C8 = {79}, locked for C8 and N6
5c. 36(6) cage = {345789} (only remaining combination), 3,4,5 locked for C8 and N3

6. Naked pair {12} in R23C9, locked for C9, R4C9 = 6 (cage sum)

7. 7,9 in N9 only in 20(4) cage at R7C9 = {1379} (only remaining combination) -> R9C8 = 1, R789C9 = {379}, locked for C9 and N9, R67C8 = [26]
7a. R56C7 = {13} (hidden pair in N6)

8. 45 rule on N7 3 innies R7C13 + R8C1 = 21 = {489/678} (cannot be {579} because 5,7 only in R7C3) -> R7C3 = {47}, R7C1 = 8

9. 45 rule on N1247 1 innie R3C6 = 1 outie R4C4, no 8 in R3C6, no 1,2,6 in R4C4

10. 39(8) cage at R3C6 = {12345789}, 8 locked for N5

11. 37(7) cage at R5C6 contains 8 -> R9C4 = 8

12. R2C6 = 8 (hidden single in C6), clean-up: no 2 in R2C23

13. 6 in N5 only in R56C6, locked for C6 and 37(8) cage at R5C6, no 6 in R7C5 + R8C45

14. R9C5 = 6 (hidden single in N8)
14a. R9C57 = [65] = 11 -> R89C6 = 8 = [17], clean-up: no 7 in R4C4 (step 9)

15. R7C2 = 1 (hidden single in N7), clean-up: no 9 in R2C3

16. 2,4 in R9 only in R9C123, locked for N7 -> R7C3 = 7, R8C1 = 6 (step 8), R6C1 = 9

17. 10(3) cage at R3C3 = {145} (only remaining combination), locked for C3 -> R2C3 = 6, R2C2 = 4, R89C3 = [92], R9C12 = [43], R8C2 = 5, R6C3 = 8, R789C9 = [379]

18. R67C3 = [87] = 15 -> R45C2 = 8 = [26]

19. R3C2 = 8 (hidden single in 1) -> R4C12 + R5C1 = 11 = {137} (only remaining combination) -> R4C2 = 7, R45C1 = {13}, locked for C1 and N4, R1C2 = 9

20. R1C4 = 6 (hidden single in N2), R123C7 = [796]

21. Naked pair {45} in R1C68, locked for R1 -> R1C1 = 2, R23C1 = [75], R1C5 = 1, R2C4 = 2, R23C9 = [12], R3C3 = 1, clean-up: no 5 in R4C3 (step 9)

22. R4C1 = 1 (hidden single in R4), R5C1 = 3, R56C7 = [13]
22a. R6C4 = 1, R6C5 = 7 (hidden singles in R6)

23. R3C4 = 7 (hidden single in R3)
23a. 9 in R3 only in R3C56, CPE no 9 in R45C5

24. Naked pair {45} in R5C39, locked for R5 -> R5C45 = [98]

25. R7C4 = 5 (hidden single in C4), R6C6 = 5 (hidden single in N5)

and the rest is naked singles.
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Andrew
 Post subject: Re: nd's Killers
PostPosted: Tue Apr 17, 2012 9:42 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
nd #4

Image

Solution: 
+-------+-------+-------+
| 9 3 5 | 2 8 7 | 4 1 6 |
| 1 4 7 | 9 6 5 | 8 3 2 |
| 8 2 6 | 4 3 1 | 5 9 7 |
+-------+-------+-------+
| 6 8 9 | 7 2 3 | 1 4 5 |
| 5 7 2 | 1 4 8 | 9 6 3 |
| 4 1 3 | 5 9 6 | 7 2 8 |
+-------+-------+-------+
| 2 6 1 | 8 7 4 | 3 5 9 |
| 7 9 4 | 3 5 2 | 6 8 1 |
| 3 5 8 | 6 1 9 | 2 7 4 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:4609:8706:8706:5379:5379:5892:5892:5892:5892:4609:8706:8706:5894:5379:5379:5893:5892:5892:4609:4106:8706:5894:5894:4616:5893:5893:4103:4106:4106:8706:5894:4616:4616:5893:4103:4103:4875:4875:2825:2825:4616:4616:6925:4364:4364:4875:4875:2825:2825:5390:5390:6925:6925:4364:4875:5647:5649:5649:6162:5390:6925:3859:3859:5647:5647:5649:5649:6162:5390:6925:4884:3859:4112:4112:4112:5649:6162:6162:6162:4884:4884:
nd’s puzzles each featured a particular technique, often the key to getting into the puzzle. For this puzzle I used:
early single combination innies, then I used a caged X-Wing followed by a nice CPE to crack the puzzle.
Here is my 2012 walkthrough for nd#4:
Prelims

a) 22(3) cage at R7C2 = {589/679}
b) 19(3) cage at R8C8 = {289/379/469/478/568}, no 1
c) 11(4) cage at R5C3 = {1235}

1. 22(3) cage at R7C2 = {589/679}, 9 locked for N7

2. 45 rule on N7 3 innies R7C13 + R8C3 = 7 = {124}, locked for N7
2a. 3 in N7 only in 16(3) cage at R9C1, locked for R9

3. 45 rule on R1234 3 innies R3C6 + R4C56 = 6 = {123}
3a. Naked quad {1235} in R4C56 + R56C4, locked for N5

4. 45 rule on R1234 2 outies R5C56 = 12 = {48} (only remaining combination), locked for R5 and N5

5. 5 in N5 only in R56C4, locked for C4 and 11(4) cage at R5C3, no 5 in R56C3

6. Naked quad {1234} in R5678C3, locked for C3
[Alternatively this comes from 45 rule on C123 4 innies R5678C3 = 10 = {1234} …]

7. 9 in C3 only in R1234C3, locked for 34(6) cage at R1C2, no 9 in R12C2

8. 3 in C3 only in R56C3, locked for N4 and 11(4) cage at R5C3, no 3 in R56C4

9. 4 in C3 only in R78C3, locked for N7 and 22(5) cage at R7C3, no 4 in R789C4
9a. 4 in C4 only in R123C4, locked for N2

10. 3 in N5 only in R4C56, locked for R4 and 18(5) cage at R3C6, no 3 in R3C6

11. 45 rule on N1 3 outies R4C123 = 23 = {689}, locked for R4 and N4 -> R4C4 = 7
11a. 45 rule on N1 1 outie R4C3 = 1 innie R3C2 + 7, R3C2 = {12}, R4C3 = {89}
11b. Naked pair {12} in R3C26, locked for R3

12. Naked pair {69} in R6C56, locked for R6
12a. R6C56 = {69} = 15 ->R78C6 = 6 = {15/24}
12b. Killer pair 1,2 in R3C6 and R78C6, locked for C6 -> R4C6 = 3

13. 45 rule on N2 2 innies R13C6 = 8 = [62/71], R1C6 = {67}

14. 45 rule on C12 1 outie R9C3 = 2 innies R12C2 + 1
14a. Max R9C3 = 8 -> max R12C2 = 7, no 7,8 in R12C2

15. 45 rule on C789 1 outie R1C6 = 1 innie R9C7 + 5, R1C6 = {67} -> R9C7 = {12}

16. 16(3) cage at R3C9 must contain at least one of 7,8,9 -> R3C9 = {789}
16a. 16(3) cage = {259/457} (cannot be {178} because 7,8 only in R3C9), no 1, 5 locked for N6
16b. 7,9 only in R3C9 -> R3C9 = {79}

17. 22(5) cage at R7C3 contains 4 = {12469/13468}, 6 locked for C4 and N8
[udosuk or Simon might have written 22(5) cage at R7C3 must contain 1 because 2+3+4+6+8 > 22.]
17a. Caged X-Wing for 1 in 11(4) cage at R5C3 and 22(5) cage at R7C3, no other 1 in C34

18. 1 in N2 only in R12C5 + R3C6, CPE no 1 in R4C5 -> R4C5 = 2, R3C6 = 1, R3C2 = 2, R4C3 = 9 (step 11a), clean-up: no 5 in R78C6 (step 12a)
18a. Naked pair {15} in R56C4, locked for C4 and 11(4) cage at R5C3, no 1 in R56C3
18b. Naked pair {23} in R56C3, locked for C3 and N4
18c. Naked pair {14} in R78C3, locked for N7 -> R7C1 = 2, R78C6 = [42] -> R5C56 = [48], R78C3 = [14]

19. Naked pair {45} in R4C89, locked for N6 -> R4C7 = 1, R9C7 = 2, R1C6 = 7 (step 15)
19a. R4C89 = {45} = 9 -> R3C9 = 7
19b. R9C5 = 1 (hidden single in R9)

20. R4C7 = 1 -> R2C7 + R3C78 = 22 = {589} (only remaining combination), locked for N3

21. R8C9 = 1 (hidden single in N9), R7C89 = 14 = {59/68}, no 3,7

22. 22(5) cage at R7C3 (step 17) = {13468} (only remaining combination), no 9, 3,8 locked for C4 and N8
22a. Naked triple {249} in R123C4, locked for N2

23. R4C4 = 7 -> R2C4 + R3C45 = 16 = {259/349} (cannot be {268/358} because 3,6,8 only in R3C5), no 6,8
23a. 6 in R3 only in R3C13, locked for N1

24. Naked triple {157} in R5C12, 7 locked for R5 and N4

25. 9 in N1 only in 18(3) cage at R1C1, locked for C1
25a. 18(3) cage = {189/369/459}, no 7

26. R2C3 = 7 (hidden single in N1)

27. 4 in R9 only in 19(4) cage at R8C8 = {478} (only remaining combination, cannot be {469} which clashes with R7C89), locked for N9, clean-up: no 6 in R7C78 (step 21)
27a. Naked pair {59} in R7C89, locked for R7 and N9 -> R7C5 = 7

28. Naked pair {36} in R78C7, locked for C7 and 27(5) cage at R5C7, no 3 in R6C8 -> R1C7 = 4, R5C7 = 9

29. Naked pair {58} in R23C7, locked for C7 and N3 -> R3C8 = 9, R6C7 = 7, R6C8 = 2 (cage sum), R56C3 = [23], R6C9 = 8, R7C89 = [59], R9C9 = 4, R4C89 = [45]

30. R3C4 = 4 -> R2C4 + R3C45 (step 23) = {349} (only remaining combination) -> R2C4 = 9, R3C5 = 3, R1C4 = 2
30a. R2C9 = 2 (hidden single in N3)

31. R9C6 = 9 (hidden single in R9), R8C5 = 5, R6C56 = [96], R2C6 = 5, R23C7 = [85], R12C5 = [86], R1C3 = 5

32. R1C1 = 9 (hidden single in N1), R23C1 = 9 = [18/36], no 4
32a. R2C2 = 4 (hidden single in N1), R6C1 = 4 (hidden single in N4)

33. 22(3) cage at R7C2 = {679} (only remaining combination) -> R7C2 = 6, R8C12 = [79], R9C3 = 8

34. R3C3 = 6, R3C1 = 8, R2C1 = 1 (step 32)

and the rest is naked singles.
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Andrew
 Post subject: Re: nd's Killers
PostPosted: Tue Apr 17, 2012 9:43 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
nd #5

Image

Solution: 
+-------+-------+-------+
| 8 4 9 | 2 3 5 | 1 6 7 |
| 5 7 1 | 8 4 6 | 2 3 9 |
| 2 6 3 | 7 1 9 | 5 4 8 |
+-------+-------+-------+
| 7 2 4 | 9 5 8 | 3 1 6 |
| 3 1 8 | 6 2 4 | 9 7 5 |
| 9 5 6 | 1 7 3 | 4 8 2 |
+-------+-------+-------+
| 4 3 7 | 5 8 2 | 6 9 1 |
| 1 9 2 | 3 6 7 | 8 5 4 |
| 6 8 5 | 4 9 1 | 7 2 3 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:4353:4353:6658:6658:6149:3075:3075:3075:4868:4353:2822:2822:6658:6149:6149:6149:4868:4868:4872:4872:2822:6658:5897:6149:4359:4359:4359:4872:7437:7437:7437:5897:5897:2570:2570:2570:4872:4872:4875:7437:5897:5897:6156:6156:6156:5902:4875:4875:7437:7437:5897:8719:8719:6156:5902:5902:5902:6160:6160:8719:8719:8719:6156:7441:7441:6160:6160:6160:7442:7442:8719:4115:7441:7441:7441:7442:7442:7442:4115:4115:4115:
nd’s puzzles each featured a particular technique, often the key to getting into the puzzle. For this puzzle I used:
min-max and “outies see all cells”, which could be done in either order; the order in which I happened to spot them is possibly simpler.
Here is my 2012 walkthrough for nd#5:
Prelims

a) 19(3) cage at R1C9 = {289/379/469/478/568}, no 1
b) 11(3) cage at R2C2 = {128/137/146/236/245}, no 9
c) 10(3) cage at R4C7 = {127/136/145/235}, no 8,9
d) 19(3) cage at R5C3 = {289/379/469/478/568}, no 1
e) 26(4) cage at R1C3 = {2789/3689/4589/4679/5678}, no 1
f) 23(6) cage at R3C5 = {123458/123467}, no 9

1. 45 rule on N7 1 outie R6C1 = 1 innie R8C3, R6C1 = {89}, R8C3 = {12}

2. 19(3) cage at R5C3 = {379/469/478/568} (cannot be {289} which clashes with R6C1), no 2
2a. Killer pair 8,9 in 19(3) cage and R6C1, locked for N4
2b. R4C4 = 9 (hidden single in R4)

3. 45 rule on N5 3(1+2) outies R3C5 + R4C23 = 7
3a. Max R4C23 = 6, no 6,7 in R4C23
3b. Min R4C23 = 3 -> max R3C5 = 4

4. Max R4C23 = 6 -> max R4C23789 = 16 = {12345/12346}, no 7, 1,2,3,4 locked for R4
4a. 10(3) cage at R4C7 must contain one of 5,6 -> 5,6 of R4C23789 must be in 10(3) cage, no 5 in R4C23
4b. Min R4C23789 = 15 -> min R4C23 = 5 -> max R3C5 = 2 (step 3)
4c. 8 in R4 only in R4C56, locked for N5

5. R3C5 = {12}, R4C23 = 5,6 (step 4) must contain one of 1,2
5a. R3C5 + R4C23 = 7 must contain both of 1,2 (because R3C5 + R4C23 “see” all cells of N5) = 1{24}/2{14} -> R4C23 = {14/24}, no 3, 4 locked for R4, N4 and 29(6) cage at R4C2, no 4 in R5C4 + R6C45
5b. 3 in R4 only in 10(3) cage at R4C7, locked for N6

6. 23(6) cage at R3C5 contains 8 = {123458}, no 6,7, 3,5 locked for N5
6a. Naked pair {58} in R4C56, locked for R4 and N5

7. 10(3) cage at R4C7 = {136} (only remaining combination), locked for R4 and N6 -> R4C1 = 7
7a. Naked pair {24} in R4C23, locked for N4 and 29(4) cage at R4C2, no 2 in R5C4 + R6C45
7b. R4C23 = {24} -> R3C5 = 1 (step 5a)
7c. 1 in N5 only in R56C4, locked for C4

8. 19(3) cage at R5C3 (step 2) = {568} (only remaining combination), locked for N4 -> R6C1 = 9, R8C3 = 2 (step 1), R4C23 = [24]

9. Naked pair {13} in R5C12, locked for R5 and 19(5) cage at R3C1, no 3 in R3C12
9a. R4C1 = 7, R5C12 = {13} = 4 -> R3C12 = 8 = [26]
9b. Naked pair {24} in R5C56, locked for R5 and N5 -> R6C6 = 3
9c. R6C4 = 1 (hidden single in N5)
9d. 6 in N4 only in R56C3, locked for C3

10. 11(3) cage at R2C2 = {137} (only remaining combination), locked for N1, 1 also locked for R2
10a. 17(3) cage at R1C1 = {458} (only remaining combination), locked for N1 -> R1C3 = 9

11. 1 in R1 only in 12(3) cage at R1C6 = {138/147/156}, no 2
11a. 8 of {138} must be in R1C6 -> no 8 in R1C78

12. 45 rule on N3 3 innies R1C78 + R2C7 = 9 contains 1 = {126/135}, no 4,7,8,9
12a. 2 of {126} must be in R2C7 -> no 6 in R2C7
12b. 12(3) cage at R1C6 (step 11) = {138/156} (cannot be {147} because 4,7 in R1C6), no 4,7

13. 17(3) cage at R1C1 (step 10a) = {458}
13a. 4 of {458} must be in R1C12 (R1C12 cannot be {58} which clashes with 12(3) cage at R1C6), locked for R1 and N1
13b. Killer pair 5,8 in R1C12 and 12(3) cage at R1C6, locked for R1

14. 9 in N6 only in 24(5) cage at R5C7 which must contain one of 1,3,6 -> R7C9 = {136}
14a. Four of {245789} including 9 cannot total 18 or 21 -> no 3,6 in R7C9 -> R7C9 = 1
14b. 24(5) cage = {12579} (cannot be {12489} because 2,4 only in R6C9) -> R6C9 = 2, R5C789 = {579}, locked for R5 and N6 -> R5C4 = 6, R6C5 = 7, R5C3 = 8, R6C23 = [56]
14c. 5 in N1 only in R12C1, locked for C1
14d. Naked pair {48} in R6C78, locked for 24(6) cage at R6C7, no 4,8 in R7C678 + R8C8

15. 17(3) cage at R3C7 = {359/458}, no 7, 5 locked for R3 and N3

16. R1C78 + R2C7 (step 12) = {126} (only remaining combination) -> R2C7 = 2, R1C78 = {16}, locked for R1 and N3 -> R1C5 = 3, R1C9 = 7, R1C4 = 2
16a. R1C78 = {16} = 7 -> R1C6 = 5, R4C56 = [58]
16b. R2C1 = 5 (hidden single in N1)

17. R1C34 = [92] = 11 -> R23C4 = 15 = {78}, locked for C4 and N2
17a. Naked triple {345} in R789C4, locked for N8

18. 23(4) cage at R6C1 contains 9 = {3479/3569}, no 8, 3 locked for R7 and N7

19. 45 rule on N69 1 innie R8C7 = 1 outie R7C6 + 6 -> R7C6 = 2, R8C7 = 8, R5C56 = [24], R6C78 = [48], R23C6 = [69], R2C5 = 4

20. R89C6 = {17} = 8, R8C7 = 8 -> R9C45 = 13 = [49], R78C4 = [53], R78C5 = [86]

21. 17(3) cage at R3C7 (step 15) = {458} (only remaining combination) -> [548]

22. R6C78 = [48] = 12, R7C6 = 2 -> R7C78 + R8C8 = 20 = {569} (only remaining combination) -> R8C8 = 5, R7C78 = {69}, locked for R7 and N9 -> R9C9 = 3

and the rest is naked singles.
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Andrew
 Post subject: Re: nd's Killers
PostPosted: Tue Apr 17, 2012 9:43 pm 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
nd #6

Image

Solution: 
+-------+-------+-------+
| 6 9 3 | 8 5 7 | 2 1 4 |
| 8 7 1 | 4 3 2 | 6 9 5 |
| 2 5 4 | 9 1 6 | 7 8 3 |
+-------+-------+-------+
| 4 3 8 | 7 9 1 | 5 2 6 |
| 9 6 7 | 3 2 5 | 8 4 1 |
| 1 2 5 | 6 4 8 | 9 3 7 |
+-------+-------+-------+
| 3 8 9 | 5 6 4 | 1 7 2 |
| 5 4 2 | 1 7 9 | 3 6 8 |
| 7 1 6 | 2 8 3 | 4 5 9 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:5377:5122:5122:5122:3331:4868:4868:4868:4868:5377:5377:3331:3331:3331:2309:9222:9222:4868:5640:5640:7433:7433:2309:2309:9222:9222:3079:5640:5640:5640:7433:6154:9222:9222:3079:3079:5901:6924:6924:7433:6154:6154:6154:4107:3079:5901:5901:6924:7433:5390:5390:5390:4107:4107:5901:5901:6924:4879:4879:4879:4879:5392:4107:4881:4881:4881:4881:8467:8467:4879:5392:5392:4881:8467:8467:8467:8467:5394:5394:5394:5394:
nd’s puzzles each featured a particular technique, often the key to getting into the puzzle. For this one I used:
large innies-outies, 19(5) cages and an ALS block which I only spotted at step 13, but ought to have seen earlier since it’s there after the Prelims.
Here is my 2012 walkthrough for nd#6:
Prelims

a) 21(3) cage at R1C1 = {489/579/678}, no 1,2,3
b) 20(3) cage at R1C2 = {389/479/569/578}, no 1,2
c) 9(3) cage at R2C6 = {126/135/234}, no 7,8,9
d) 21(3) cage at R6C5 = {489/579/678}, no 1,2,3
e) 21(3) cage at R7C8 = {489/579/678}, no 1,2,3
f) 13(4) cage at R1C5 = {1237/1246/1345}, no 8,9
g) 12(4) cage at R3C9 = {1236/1245}, no 7,8,9
h) 27(4) cage at R5C2 = {3789/4689/5679}, no 1,2

Steps resulting from Prelims
1. 12(4) cage at R3C9 = {1236/1245}, CPE no 1,2 in R6C9
1a. 27(4) cage at R5C2 = {3789/4689/5679}, CPE no 9 in R4C3
1b. 1,2 in N1 only in R2C3 + R3C123, CPE no 1,2 in R4C3

2. 45 rule on R89 1 outie R7C8 = 1 innie R8C7 + 4, no 4 in R7C8, no 6,7,8,9 in R8C7

3. 45 rule on R9 2 outies R8C56 = 1 innie R9C1 + 9
3a. Max R8C56 = 17 -> max R9C1 = 8

4. 45 rule on N1 4 innies R2C3 + R3C123 = 1 outie R1C4 + 4
4a. Min R2C3 + R3C123 = 10 -> min R1C4 = 6
4b. Max R2C3 + R3C123 = 13, no 8,9 in R3C123
[1,2 in N1 only in R2C3 + R3C123 = 10,11,12,13 = {1234/1235/1236/1245/1237/1246} contains at least one of 3,4 may be useful later.]

5. 13(4) cage at R1C5 and 19(5) cage at R1C6 both contain 1, Caged X-Wing for 1 in R12, no other 1 in R12

6. 19(5) cage at R7C4 contains 1, CPE no 1 in R7C9

7. 45 rule on N5 4(2+2) outies R3C34 + R56C7 = 1 innie R4C6 + 29
7a. Max R3C34 + R56C7 = 16 + 17 = 33 -> max R4C6 = 4
7b. Min R3C34 + R56C7 = 30, max R3C34 = 16 -> min R56C7 = 14, no 1,2,3,4
7c. Min R3C34 + R56C7 = 30, max R56C7 = 17 -> min R3C34 = 13, no 1,2,3, no 4,5 in R3C4
7d. 36(6) cage at R2C7 contains 7,8,9 in R23C78 + R4C7, CPE no 7,8,9 in R1C7

8. Max R2C3 + R3C123 = 13 (step 4b), min R3C3 = 4 -> max R2C3 + R4C12 = 9, no 7 in R2C3 + R3C12

9. 45 rule on C1234 3 outies R8C56 + R9C5 = 3(2+1) innies R2C34 + R7C4 + 14
9a. Max R8C56 + R9C5 = {789} = 24 -> max R2C34 + R7C4 = 10 -> max R7C4 = 6 (R2C34 + R7C4 cannot be {12}7 which clashes with {789})
9b. Min R2C34 + R7C4 = {12}1 = 4 -> min R8C56 + R9C5 = 18 cannot be {189} which clashes with R2C34 + R7C4 = {12}1), no 1 in R8C56 + R9C5

10. 19(5) cage at R7C4 and 19(5) cage at R8C1 both contain 1
10a. Consider placement of 1 in 19(5) cage at R8C1
1 in R8C123 + R9C1 => no 1 in R7C12
or 1 in R8C4 => 1 in 19(5) cage at R7C4 must be in R7C7 => no 1 in R7C12
-> no 1 in R7C12
10b. 1 in R7 only in R7C4567, locked for 19(5) cage at R7C4, no 1 in R8C7, clean-up: no 5 in R7C8 (step 2)
[This is simpler with the contradiction move R8C7 cannot be 1 because 1 in 19(5) cage at R8C1 would have to be in R9C1 and then cannot place 1 in R7. However I prefer to use forcing chains, rather than contradiction moves, these days.]
10c. 1 in R8 only in R8C1234, locked for 19(5) cage at R8C1, no 1 in R9C1

11. 45 rule on N9 3 innies R7C79 + R8C7 = 1 outie R9C6 + 3
11a. Min R7C79 + R8C7 = 6 -> min R9C6 = 3
11b. Max R7C79 + R8C7 = 12, min R7C9 + R8C7 = 5 -> max R7C7 = 7

12. 45 rule on N6 3 innies R456C7 = 2 outies R37C9 + 17
12a. Min R37C9 = 3 -> min R456C7 = 20, no 1,2 in R4C7
12b. Max R456C7 = 24 -> max R37C9 = 7, no 7,8,9 in R7C9, no 6 in R3C9

[I should have spotted the next step earlier …
However I would probably still have needed the earlier harder steps.]
13. 9(3) cage at R2C6 = {126/135} (cannot be {234} which clashes with 13(4) cage at R1C5, ALS block), no 4, 1 locked for R3 and N2

14. R2C3 = 1 (hidden single in N1)
14a. 2 in N1 only in R3C12, locked for R3 and 22(5) cage at R3C1, no 2 in R4C12
14b. 9(3) cage at R2C6 (step 13) = {126/135}
14c. 2 of {126} must be in R2C6 -> no 6 in R2C6
14d. 2 in C3 only in R89C3, locked for N7, CPE no 2 in R8C56

15. 13(4) cage at R1C5 = {1246/1345} (cannot be {1237} which clashes with 9(3) cage at R2C6), no 7, 4 locked for N2
15a. R1C46 + R3C4 = {789} (hidden triple for N2)

16. 19(5) cage at R1C6 = {12349/12358/12367/12457} (cannot be {13456} because R1C6 only contains 7,8,9), 2 locked for N3
16a. R1C6 = {789} -> no 7,8,9 in R1C89 + R2C9

17. 7,8,9 in N3 only in R23C78, locked for 36(6) cage at R2C7, no 7,8,9 in R4C7

18. 12(4) cage at R3C9 = {1236/1245}, 1,2 locked for N6

19. 16(4) cage at R5C8 = {2347/2356}, no 8,9 -> R7C9 = 2, clean-up: no 6 in R7C8 (step 2)
19a. R4C8 = 2 (hidden single in N6)
19b. 1 in N6 only in R45C9, locked for C9
19c. R1C78 = [21] (hidden pair in N3)

20. 16(4) cage at R5C8 = {2347/2356}, 3 locked for N6
20a. R56C7 = {89} (hidden pair in N6), locked for C7

21. R89C9 = {89} (hidden pair in C9), locked for N9 -> R7C8 = 7, R8C7 = 3 (step 2)
21a. 21(3) cage at R7C8 contains 7 = {579/678}, no 4
21b. 3 in R7 only in R7C123, locked for N7

22. R6C9 = 7 (hidden single in C9)
22a. 16(4) cage at R5C8 (step 20) = {2347} (only remaining combination) -> R56C8 = {34}, locked for C8 and N6
22b. Naked pair {56} in R89C8, locked for C8 and N9
22c. Naked pair {14} in R79C7, locked for C7

23. R234C7 = {567} = 18, R23C8 = {89} = 17 -> R4C6 = 1
23a. Naked pair {56} in R4C79, locked for R4 and N6 -> R5C9 = 1
23b. R3C5 = 1 (hidden single in N2)

24. 21(3) cage at R6C5 = {489} (only remaining combination), locked for R6, 4 also locked for N5 -> R56C8 = [43]

25. 19(5) cage at R7C4 = {13456} (only remaining combination), no 8,9, 4,5,6 locked for R7, 5,6 also locked for N8

26. Naked triple {389} in R7C123, locked for N7

27. R3C48 = {89} (hidden pair in R3)

28. R4C6 = 1, R56C7 = {89} = 17 -> R3C34 = 13 (step 7) = [49/58], no 6,7 in R3C3
28a. R3C7 = 7 (hidden single in R3)

29. 27(4) cage at R5C2 = {5679} (cannot be {3789} because R6C3 only contains 5,6) -> R7C3 = 9, R5C23 + R6C3 = {567}, locked for N4, 7 also locked for R5

30. R6C12 = {12} = 3, R7C12 = {38} = 11 -> R5C1 = 9 (cage sum), R56C7 = [89]

31. R4C45 = {79} (hidden pair in R4), R6C34 = {56} (hidden pair in R6)

32. R3C34 = 13 = [49/58] (step 28) -> R456C4 = 16 = {367} (only remaining combination, cannot be {259} which clashes with R3C4) -> [736], R4C5 = 9, R6C3 = 5, R3C3 = 4, R3C4 = 9, R1C4 = 8, R1C6 = 7, R23C8 = [98]

33. 21(3) cage at R1C1 = {678} (only remaining combination) -> R1C1 = 6, R2C12 = {78}, R1C3 = 3, R1C2 = 9 (hidden single in N1), R4C3 = 8

34. R1C678 = [721] = 10 -> R12C9 = 9 = {45} (only remaining combination, cannot be {36} because 3,6 only in R2C9), locked for C9 and N3 -> R24C7 = [65], R34C9 = [36]

35. R3C6 = 6 (hidden single in R3), R2C6 = 2 (cage sum), R5C56 = [25], R7C67 = [41], R7C45 = [56], R6C56 = [48], R8C9 = 8

36. 21(3) cage at R7C8 (step 21a) = {678} (only remaining combination) -> R8C8 = 6

and the rest is naked singles.
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Andrew
 Post subject: Re: nd's Killers
PostPosted: Tue Apr 17, 2012 9:44 pm 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
nd #6 alt

This puzzle is a harder variant of nd#6, with a modified cage pattern and different solution.

Image

Solution: 
+-------+-------+-------+
| 5 3 9 | 8 6 7 | 1 2 4 |
| 6 8 1 | 4 2 3 | 7 9 5 |
| 2 7 4 | 9 5 1 | 6 8 3 |
+-------+-------+-------+
| 9 5 3 | 7 8 2 | 4 1 6 |
| 1 4 7 | 3 9 6 | 8 5 2 |
| 8 6 2 | 1 4 5 | 9 3 7 |
+-------+-------+-------+
| 3 9 6 | 2 7 8 | 5 4 1 |
| 4 2 5 | 6 1 9 | 3 7 8 |
| 7 1 8 | 5 3 4 | 2 6 9 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:6657:5122:5122:5122:3331:4868:4868:4868:4868:6657:6657:3331:3331:3331:2309:9222:9222:4868:4872:6657:8713:8713:2309:2309:9222:9222:3079:4872:4872:4872:8713:8713:9222:9222:3079:3079:5133:5133:5133:8713:7946:7946:6667:4108:3079:5646:5647:5133:8713:7946:7946:6667:4108:4108:5646:5647:5133:8713:7946:7186:6667:6667:4108:5646:5647:5647:5904:7186:7186:7186:7186:7441:5646:5904:5904:5904:5904:7441:7441:7441:7441:
nd’s puzzles each featured a particular technique, often the key to getting into the puzzle. With this cage pattern being a variant of #6 I again used:
an ALS block and large innies-outies, but not the same 45s that I used for #6; I also used a “must be even” step near the end.
Here is my 2012 walkthrough for nd#6 alt:
Prelims

a) 20(3) cage at R1C2 = {389/479/569/578}, no 1,2
b) 9(3) cage at R2C6 = {126/135/234}, no 7,8,9
c) 26(4) cage at R1C1 = {2789/3689/4589/4679/5678}, no 1
d) 13(4) cage at R1C5 = {1237/1246/1345}, no 8,9
e) 12(4) cage at R3C9 = {1236/1245}, no 7,8,9
f) 26(4) cage at R5C7 = {2789/3689/4589/4679/5678}, no 1

[The step I ought to have spotted earlier when working on #6 …]
1. 9(3) cage at R2C6 = {126/135} (cannot be {234} which clashes with 13(4) cage at R1C5, ALS block), no 4, 1 locked for N2

2. 1 in 13(4) cage at R1C5 only in R2C3 -> R2C3 = 1
2a. 1 in N2 only in R3C56, locked for R3
2b. 13(4) cage at R1C5 = {1246/1345} (cannot be {1237} which clashes with 9(3) cage at R2C6), no 7

3. R1C46 + R3C4 = {789} (hidden triple in N2)
3a. 19(5) cage at R1C6 = {12349/12358/12367/12457} (cannot be {13456} because R1C6 only contains 7,8,9), 2 locked for N3
3b. R1C6 = {789} -> no 7,8,9 in R1C789 + R2C9

4. 7,8,9 in N3 only in R23C78, locked for 36(6) cage at R2C76, no 7,8,9 in R4C67

5. 12(4) cage at R3C9 = {1236/1245}, 1,2 locked for N6

6. 45 rule on N1 1 outie R1C4 = 2 remaining innies R3C13 + 2
6a. R1C4 = {789} -> R3C13 = 5,6,7 = {23/24/25/34}, no 6,7,8,9

7. 16(4) cage at R5C8 must contain at least one of 1,2 -> R7C9 = {12}
7a. R689C9 = {789} (hidden triple in C9)

8. Killer pair 1,2 in 12(4) cage at R3C9 and R7C9, locked for C9
8a. Hidden killer pair 1,2 in 12(4) cage at R3C9 and R7C9 for C9, R7C9 = {12} -> 12(4) cage at R3C9 only contains one of 1,2 in C9 -> R4C8 = {12}

9. R1C78 = {12} (hidden pair in N3), locked for R1
9a. Naked pair {12} in R14C8, locked for C8

10. 16(4) cage at R5C8 = {1348/1357/2347} (cannot be {1249/1258/1267} because 1,2 only in R7C9, cannot be {1456/2356} because no 1,2,3,4,5,6 in R6C9), no 6,9, 3 locked for C8 and N6
10a. R6C9 = {78} -> no 7,8 in R56C8
10b. 3 in C9 only in R123C9, locked for N3
10c. 9 in C9 only in R89C9, locked for N9 and 29(5) cage at R8C9, no 9 in R9C6
10d. 9 in C8 only in R23C8, locked for N3

11. R56C7 + R6C9 = {789} (hidden triple in N4)

12. 7,8,9 in N3 only in R23C78 -> R23C7 must contain at least one of 7,8
12a. Killer pair 7,8 in R23C7 and R56C7, locked for C7

13. 45 rule on N6 3 innies R456C7 = 2 outies R37C9 + 17
13a. Max R456C7 = 23 -> max R37C9 = 6, no 6 in R3C9

14. 45 rule on N3 3(2+1) outies R1C6 + R4C67 = 1 innie R3C9 + 10
14a. Max R3C9 = 5 -> max R1C6 + R4C67 = 15, min R1C6 = 7 -> max R4C67 = 8 -> R4C6 = {123} (because R4C67 cannot be [44])

15. 19(5) cage at R1C6 (step 3a) = {12367/12457} (cannot be {12349/12358} which clash with 12(4) cage at R3C9) -> R1C6 = 7

16. Naked pair {89} in R13C4, locked for C4

17. 20(3) cage at R1C2 = {389/569}, no 4, 9 locked for R1

18. 7 in N1 only in 26(4) cage at R1C1 = {2789/4679/5678}, no 3

19. 31(5) cage at R5C5 must contain 9, CPE no 9 in R4C5
19a. 9 in N5 only in R56C56, locked for 31(5) cage at R5C5, no 9 in R7C5

20. 9 in R4 only in R4C123, locked for N4
20a. 19(4) cage at R3C1 = {1279/1369/1459/2359}, no 8

21. R4C5 = 8 (hidden single in R4), R3C4 = 9, R1C4 = 8

22. R2C8 = 9 (hidden single in N2)
22a. Hidden killer pair 7,8 in R23C7 and R56C7 for C7, R56C7 contain one of 7,8 -> R23C7 contains one of 7,8 -> R3C8 = {78}

23. R1C4 = 8 -> 20(3) cage at R1C2 (step 17) = {389} (only remaining combination), 3,9 locked for R1 and N1

24. 26(4) cage at R1C1 = {5678} (only remaining combination), locked for N1

25. Naked pair {24} in R3C13, locked for R3, CPE no 2,4 in R4C3

26. 9(3) cage at R2C6 (step 1) = {126/135}
26a. 2 of {126} must be in R2C6 -> no 6 in R2C6

27. 34(7) cage at R3C3 contains both of 8,9 = {1234789/1235689}, 1,3 locked for C4

28. 19(4) cage at R3C1 = {1279/1459/2359} (cannot be {1369} because R3C1 only contains 2,4), no 6
28a. R3C1 = {24} -> no 2,4 in R4C12
28b. Killer triple 1,2,3 in 19(4) cage, R4C6 and R4C8, locked for R4

30. 12(4) cage at R3C9 = {1236/1245}
30a. 1,2 only in R4C8 + R5C9 -> R5C9 = {12}
30b. R34C9 = [36/54], no 5 in R4C9

31. 31(5) cage at R5C5 = {45679} (only remaining combination), no 1,2,3, 7 locked for C5

32. 45 rule on N4 2 innies R6C12 = 2(1+1) outies R3C1 + R7C3 + 6
32a. Max R6C12 = 14 (cannot be {78} = 15 which clashes with R6C9) -> max R3C1 + R7C3 = 8, min R3C1 = 2 -> max R7C3 = 6
32b. Min R3C1 + R7C3 = [22] = 4 -> min R6C12 = 10, no 1 in R6C1

33. R4C6 + R56C4 = {123} (hidden triple in N5)

[45 rule on N5 3(1+2) remaining outies R3C3 + R7C45 = 1 innie R4C6 + 11 looks interesting but I can’t quite get anything out of this; it would be different if R4C6 only contained 1,3.]

34. 6 in C8 only in R789C8, locked for N7

35. 6 in C7 only in 36(6) cage at R2C7 = {156789/246789}, no 3
35a. 3 in N5 only in R56C4, locked for C4

36. Naked pair {12} in R4C68, locked for R4

37. 19(4) cage at R3C1 (step 28) = {2359} (only remaining combination) -> R3C1 = 2, R3C3 = 4, R4C123 = {359}, locked for R4 and N4

38. R4C4 = 7 (hidden single in R4), R7C5 = 7 (hidden single in C5)

39. 34(7) cage at R3C3 (step 27) contains both of 4,7 = {1234789} (only remaining combination), no 5,6, 2 locked for C4

40. 13(4) cage at R1C5 (step 2b) = {1246/1345}
40a. 2,3 only in R2C5 -> R2C5 = {23}

41. Naked pair {46} in R4C79, locked for N6

42. Naked pair {35} in R56C8, locked for C8
42a. R56C8 = {35} = 8 -> R67C9 = 8 = [71], R5C9 = 2, R4C8 = 1, R1C78 = [12], R4C6 = 2, R7C4 = 2
42b. R2C5 = 2 (hidden single in N2)

43. 9(3) cage at R2C6 (step 1) = {135} (only remaining combination), locked for N2
43a. Naked triple {135} in R3C56 + R3C9, locked for R3
43b. 5 in C4 only in R89C4, locked for N8 and 23(5) cage at R8C4, no 5 in R9C23

44. Naked pair {89} in R89C9, locked for N9 and 29(5) cage at R8C9, no 8 in R9C6
44a. Naked triple {467} in R789C8, locked for C8 and N8 -> R3C8 = 8
44b. Naked triple {235} in R789C7, locked for C7

45. R6C12 = R3C1 + R7C3 + 6 (step 32)
45a. R3C1 and R6C12 are all even -> R7C3 must be even -> R7C3 = 6
45b. R3C1 + R7C3 = [26] = 8 -> R6C12 = 14 = {68}, locked for R6 and N4 -> R56C3 = [72], R56C7 = [89]
45c. Naked pair {14} in R5C12, locked for R5 -> R56C4 = [31], R56C8 = [53]

46. R7C8 = 4, R56C7 = [89] = 17 -> R7C7 = 5 (cage sum)

47. 8 in C3 only in R89C3, locked for N7
47a. Naked pair {39} in R7C12, locked for R7 and N7 -> R7C6 = 8, R89C3 = [58], R89C9 = [89]

48. 29(5) cage at R8C9 contains 8,9 = {23789/24689} (cannot be {14789} because R9C7 only contains 2,3) -> R9C7 = 2, R9C6 = {34}, R8C7 = 3
48a. R8C2 = 2 (hidden single in R8)

49. R8C23 = [25] = 7 -> R67C2 = 15 = [69]

50. R67C1 = [83] = 11 -> R89C1 = 11 = {47}, locked for C1 and N7 -> R9C2 = 1

51. R9C4 = 5 (hidden single in C4)
51a. R9C234 = [185] = 14 -> R8C4 + R9C5 = 9 = [63]

and the rest is naked singles.
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Andrew
 Post subject: Re: nd's Killers
PostPosted: Tue Apr 17, 2012 9:45 pm 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
nd #7

Image

Solution: 
+-------+-------+-------+
| 7 9 5 | 1 6 8 | 2 3 4 |
| 3 8 4 | 9 5 2 | 7 6 1 |
| 1 2 6 | 7 4 3 | 5 9 8 |
+-------+-------+-------+
| 4 3 9 | 2 8 7 | 6 1 5 |
| 5 7 8 | 3 1 6 | 9 4 2 |
| 2 6 1 | 4 9 5 | 3 8 7 |
+-------+-------+-------+
| 9 1 2 | 6 7 4 | 8 5 3 |
| 8 4 7 | 5 3 9 | 1 2 6 |
| 6 5 3 | 8 2 1 | 4 7 9 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:7425:7425:7425:3586:3586:4355:4355:3588:3588:7425:9221:9221:9221:3586:3586:4355:3588:3588:7425:4362:9221:9221:4105:5382:5382:3591:3848:7425:4362:6667:9221:4105:5382:5900:3591:3848:4362:4362:6667:4105:4105:5382:5900:3591:3848:6667:6667:6667:5902:5902:5900:5900:6413:6413:6928:3599:3599:3599:5902:3347:3347:6413:6413:6928:6928:5905:3599:5902:4114:3347:6413:5652:6928:5905:5905:5905:4114:4114:4114:5652:5652:
nd’s puzzles each featured a particular technique, often the key to getting into the puzzle. For this puzzle I used:
large outies and interactions between the split halves of the 36(6) cage.
Here is my 2012 walkthrough for nd#7:
Prelims

a) 22(3) cage at R8C9 = {589/679}
b) 14(4) cage at R1C4 = {1238/1247/1256/1346/2345}, no 9
c) 14(4) cage at R1C8 = {1238/1247/1256/1346/2345}, no 9
d) 27(4) cage at R7C1 = {3789/4689/5679}, no 1,2
e) 14(4) cage at R7C2 = {1238/1247/1256/1346/2345}, no 9

Steps resulting from Prelims
1. 27(4) cage at R7C1 = {3789/4689/5679}, 9 locked for N7
1a. 22(3) cage at R8C9 = {589/679}, 9 locked for N9

2. 45 rule on N7 1 outie R9C4 = 2 innies R7C23 + 5
2a. Min R7C23 = 3 -> min R9C4 = 8
2b. R9C4 = {89} -> R7C23 = 3,4 = {12/13}, 1 locked for R7, N7 and 14(4) cage at R7C2, no 1 in R8C4

3. 45 rule on N7 3 outies R789C4 = 19 = {289/469/478/568} (cannot be {379} because 14(4) cage at R7C2 cannot contain both of 3,7), no 3

4. 45 rule on N4 1 innie R4C1 = 1 outie R3C2 + 2, no 8,9 in R3C2, no 1,2 in R4C1

5. 45 rule on C1234 3 innies R156C4 = 8 = {125/134}, 1 locked for C4

6. 45 rule on N14 3 outies R234C4 = 18 = {279/369/378/567} (cannot be {459/468} which clash with R789C4), no 4

7. 45 rule on R789 4 outies R6C4589 = 28 = {4789/5689}, no 1,2,3, 8,9 locked for R6
7a. R6C4 = {45} -> no 4,5 in R6C589

8. R156C4 (step 5) = {125/134}
8a. R6C4 = {45} -> no 4,5 in R15C4

9. 45 rule on N14 3 innies R2C23 + R3C3 = 18 = {189/459/468} (cannot be {279/369/378/567} which clash with R234C4), no 2,3,7
9a. R234C4 (step 6) = {279/378/567} (cannot be {369} which clashes with R2C23 + R3C3)

10. 4 in C4 only in R678C4, CPE no 4 in R78C5

11. 45 rule on C7 2 outies R17C6 = 2 innies R39C7 + 8
11a. Min R39C7 = 3 -> min R17C6 = 11, no 1
11b. Max R17C6 = 17 -> max R39C7 = 9, no 9

12. 45 rule on C67 1 outie R9C5 = 1 innie R2C6, no 9 in R9C5

13. 45 rule on R89 4 innies R8C4578 = 1 outie R7C1 + 2
13a. Min R8C4578 = 10 -> min R7C1 = 8
13b. R7C1 = {89} -> R8C4578 = 10,11 = {1234/1235}, 1,2,3 locked for R8

14. Max R6C4 + R8C5 = 9 -> min R67C5 = 14 (but cannot repeat 5 within cage) -> no 2,3,5 in R7C5

15. 45 rule on R9 3 outies R8C369 = 1 innie R9C1 + 16
15a. Max R8C369 = 24 -> max R9C1 = 8

16. 45 rule on R6789 4(2+2) outies R45C37 = 32
16a. Max R45C3 = 17 -> min R45C7 = 15, no 1,2,3,4,5
16b. Naked quad {6789} in R45C7 + R6C89, locked for N6
[I could have got more out of this, see steps 19 and 20.]
16c. Max R45C7 = 16 (cannot be {89} = 17 because R6C89 cannot contain both of 6,7, step 7) -> min R45C3 = 16 -> R45C3 = {79/89}, 9 locked for C3 and N4, clean-up: no 7 in R3C2 (step 4)
16d. R45C3 = 16,17 -> R45C7 = 15,16 contains one of 8,9 -> R6C89 contains one of 8,9 -> R6C5 = {89} (only other place for 8,9 in R6)
16e. 26(5) cage at R4C3 cannot contain all of 7,8,9, R45C3 = {79/89} -> no 7 in R6C123

17. R2C23 + R3C3 (step 9) = {189/459/468}
17a. 9 of {189/459} must be in R2C2 -> no 1,5 in R2C2

18. 14(3) cage at R3C8 must contain one of 6,7,8,9 -> R3C8 = {6789}
18a. 15(3) cage at R3C9 must contain at least one of 6,7,8,9 -> R3C9 = {6789}

[At this stage I originally analysed 29(6) cage at R1C1, which eliminated some candidates from R3C2 and R4C1. However after spotting the next step this analysis proved to be unnecessary so I’ve omitted it.]

19. 7 in R6 only in R6C689, CPE no 7 in R45C7
19a. 7 in N6 only in R6C89, locked for R6 and 25(5) cage at R6C8, no 7 in R7C89
[With hindsight I ought to have spotted that the naked quad in step 16b gives a CPE for R6C6.]

20. 7 in R6 only in R6C89 -> R6C4589 (step 7) = {4789} (only remaining combination), no 6 -> R6C4 = 4
20a. R156C4 (step 5) = {134} (only remaining combination), no 2, 3 locked for C4, CPE no 1,3 in R3C5, clean-up: no 8 in R234C4 (step 9a)
20b. 6 in N6 only in R45C7, locked for C7 and 23(4) cage at R4C7, no 6 in R6C6
[This last elimination should also have been made by CPE from the naked quad in step 16b.]

21. 6 in R6 only in R6C123, locked for N4, clean-up: no 4 in R3C2 (step 4)
21a. 26(5) cage at R4C3 contains 6,9 = {12689/13679}, 1 locked for R6 and N4

22. 23(4) cage at R4C7 contains 6 = {3569} (only remaining combination) -> R45C7 = {69}, locked for C7 and N6, R6C67 = {35}, locked for R6

23. 26(5) cage at R4C3 (step 21a) = {12689} (only remaining combination), locked for N4, 8 also locked for C3, clean-up: no 6 in R3C2 (step 4)

24. R2C23 + R3C3 (step 9) = {459/468} (cannot be {189} because 8,9 only in R2C2), no 1, 4 locked for N1

25. R234C4 (step 9a) = {279} (only remaining combination, cannot be {567} which clashes with R2C23 + R3C3), locked for C4 and 36(6) cage at R2C2, no 9 in R2C2-> R789C4 = [658]
25a. R2C23 + R3C3 (step 24) = {468} (only remaining combination) -> R2C2 = 8, R23C3 = {46}, locked for C3 and N1 -> R8C3 = 7

26. R78C4 = [65] = 11 -> R7C12 = 3 = {12}, locked for R7 and N7
26a. Naked pair {12} in R67C3, locked for C3

27. R8C3 + R9C4 = [78] = 15 -> R9C23 = 8 = {35}, locked for R9

28. 22(3) cage at R8C9 = {679} (only remaining combination, cannot be {589} because 5,8 only in R8C9), 7 locked for R9 and N9

29. R6C4 = 4 -> R678C5 = 19 = {289/379}, no 1, 9 locked for C5
29a. 1 in R8 only in R8C78, locked for N9

30. Naked quad {1249} in 16(4) cage at R8C6, 9 locked for C6 and N8 -> R7C5 = 7, R6C5 = 9 (hidden single in C5), R8C5 = 3 (step 29), R7C6 = 4, R8C6 = 9, R8C9 = 6, R8C12 = [84], R79C1 = [96]
30a. R9C7 = 4 (hidden single in R9)

31. Naked pair {78} in R6C89, locked for 25(5) cage at R6C8, no 8 in R7C89
31a. Naked pair {35} in R7C89, locked for R7 -> R7C7 = 8
31b. R6C89 = {78} = 15, R7C89 = {35} = 8 -> R8C8 = 2 (cage sum), R8C7 = 1

32. R1C2 = 9, R6C2 = 6 (hidden singles in C2)

33. Naked triple {357} in R459C2, locked for C2, 7 also locked for N4, clean-up: no 5 in R4C1 (step 4)

34. 7,9 in N1 only in 29(6) cage at R1C1 = {134579} (only remaining combination) -> R4C1 = 4, R3C2 = 2 (step 4)

35. 2 in C7 only in R12C7, locked for N3 and 17(3) cage at R1C6, no 2 in R1C6
35a. 17(3) cage at R1C6 = {278} (only remaining combination) -> R1C6 = 8, R12C7 = {27}, locked for N3

36. 14(4) cage at R1C8 = {1346} (only remaining combination), locked for N3 -> R3C7 = 5, R6C67 = [53]

37. R3C7 = 5 -> R345C6 = 16 = {367} (only remaining combination), locked for C6

38. Naked pair {89} in R3C89, locked for R3 -> R3C4 = 7

39. 14(3) cage at R3C8 = {149/158}, 1 locked for C8 and N6

40. 8 in N5 only in 16(4) cage at R3C5 = {1348} (only remaining combination) -> R3C5 = 4, R5C4 = 3, R45C5 = {18}, locked for N5

41. R4C4 = 2, R4C9 = 5, R45C8 = [14], R3C8 = 9 (cage sum)

and the rest is naked singles.
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Andrew
 Post subject: Re: nd's Killers
PostPosted: Tue Apr 17, 2012 9:46 pm 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
nd #8

Image

Solution: 
+-------+-------+-------+
| 7 8 9 | 5 4 6 | 2 3 1 |
| 5 3 1 | 2 8 9 | 7 6 4 |
| 2 4 6 | 1 3 7 | 9 5 8 |
+-------+-------+-------+
| 1 9 5 | 7 6 4 | 8 2 3 |
| 4 7 8 | 9 2 3 | 5 1 6 |
| 6 2 3 | 8 5 1 | 4 7 9 |
+-------+-------+-------+
| 8 6 7 | 3 9 5 | 1 4 2 |
| 9 5 4 | 6 1 2 | 3 8 7 |
| 3 1 2 | 4 7 8 | 6 9 5 |
+-------+-------+-------+
Code: Select, Copy & Paste into solver:
3x3::k:5636:6145:6145:6145:5890:5123:5123:5123:5893:5636:5636:5636:6145:5890:5123:5893:5893:5893:5126:5126:5636:5890:5890:5890:6919:5893:4872:5897:5126:5126:7946:6919:6919:6919:4872:4872:5897:5897:5897:7946:7946:5644:4872:4872:5643:4109:4109:5897:7946:7946:5644:5644:5643:5643:4109:7695:7695:7695:5644:5644:4878:4878:4878:7695:7695:4114:4114:5904:5904:5904:5904:4878:4114:4114:4114:6417:6417:6417:6417:5904:4878:
nd’s puzzles each featured a particular technique, often the key to getting into the puzzle. For this puzzle I used:
a “trick” based on an innie-outie difference.
Here is my 2012 walkthrough for nd#8:
Prelims

a) 22(3) cage at R5C9 = {589/679}
b) 27(4) cage at R3C7 = {3789/4689/5679}, no 1,2
c) 16(5) cage at R8C3 = {12346}

Steps resulting from Prelims
1. 22(3) cage at R5C9 = {589/679}, 9 locked for N6
1a. 16(5) cage at R8C3 = {12346}, CPE no 1,2,3,4,6 in R8C12

2. 19(5) cage at R7C7 = {12349/12358/12367/12457/13456}, 1 locked for N9

3. 45 rule on N5 4 innies R4C56 + R56C6 = 14 = {1238/1247/1256/1346/2345}, no 9

4. 27(4) cage at R3C7 = {3789/4689/5679} -> R3C7 = 9

5. R4C56 + R56C6 (step 3) = {1238/1256/1346} (cannot be {1247} because 27(4) cage at R3C7 only contains one of 4,7, cannot be {2345} because 27(4) cage at R3C7 only contains one or neither of 4,5), no 7, 1 locked for C6, N5 and 22(5) cage at R5C6, no 1 in R6C7 + R7C5
5a. 1,2 of {1238/1256} must be in R56C6, 1,3 of {1346} must be in R56C6 (R4C56 cannot be {34/36} because 27(4) cage at R3C7 doesn’t contain 3 and one of 4,6) -> R56C6 = {12/13}
5b. R4C56 = {38/46/56} -> R4C7 = {78} (step 4)
5c. Killer pair 7,8 in R4C7 and 22(3) cage at R5C9, locked for N6

6. 45 rule on R789 3 innies R7C156 = 22 = {589/679}, 9 locked for R7

7. 19(5) cage at R3C9 = {12358/12367/12457/13456}
7a. 7,8 of {12358/12367/12457} must be in R3C9, 5 or 6 of {13456} must be in R3C9 (both of 5,6 in N6 would clash with 22(3) cage at R5C9) -> R3C9 = {5678}
7b. Killer pair 5,6 in 19(5) cage at R3C9 and 22(3) cage at R5C9, locked for N6

8. 45 rule on N1234 1 remaining innie R3C9 = 1 outie R7C1, no 9 in R7C1
8a. 9 in R7 only in R7C56, locked for N8
8b. 9 in N7 only in R8C12, locked for R8

9. 22(5) cage at R5C6 contains 9
9a. Min R56C6 + R6C7 = 6 -> max R7C56 = 16, no 8 in R7C56
9b. R7C156 (step 6) = {589/679}
9c. 8 of {589} must be in R7C1, no 5 in R7C1, clean-up: no 5 in R3C9 (step 8)

10. 45 rule on R123 3 remaining innies R3C129 = 14
10a. Min R3C9 = 6 -> max R3C12 = 8, no 8 in R3C12

11. 45 rule on N4 2 innies R4C23 = 1 outie R7C1 + 6
11a. Min R7C1 = 6 -> min R4C23 =12, no 1,2

12. 45 rule on N14 2 innies R1C23 = 1 outie R7C1 + 9
12a. Min R7C1 = 6 -> min R1C23 = 15, no 1,2,3,4,5

13. 45 rule on N3 2 outies R12C6 = 1 remaining innie R3C9 + 7
13a. Min R3C9 = 6 -> min R12C6 = 13, no 2,3

14. 45 rule on N23 3(2+1) remaining innies R12C4 + R3C9 = 15
14a. Min R3C9 = 6 -> max R12C4 = 9, no 9 in R12C4

15. 9 in C4 only in R456C4, locked for N5

16. 45 rule on N3 3 remaining innies R1C78 + R3C9 = 13
16a. 13(3) cage cannot contain more than one of 6,7,8, R3C9 = {678} -> no 6,7,8 in R1C78

17. 45 rule on N7 2 outies R78C4 = 1 innie R7C1 + 1, IOU no 1 in R8C4
17a. Max R7C1 = 8 -> max R78C4 = 9, no 8 in R7C4
17b. 16(5) cage at R8C3 = {12346}, 1 locked for N7

18. 45 rule on R9 2 innies R9C89 = 2 outies R8C34 + 4
18a. 9 in R9 only in R9C89 but no 5 in R8C34 -> R9C89 cannot contain any of the numbers (x) in 16(5) cage at R8C3 because x9 in R9C89 would have to be paired with x5 in R8C34 -> no 1,2,3,4,6 in R9C89
18b. Max R8C34 = 10 -> max R9C89 = 14 contains 9 -> R9C89 = {59}, locked for R9 and N9
18c. R9C89 = {59} = 14 -> R8C34 = 10 = {46}, locked for R8 and 16(5) cage at R8C3, no 4,6 in R9C123

19. Naked triple {123} in R9C123, locked for R9 and N7

20. Naked quad {4678} in R8C4 + R9C456, locked for N8, 7,8 also locked for R9

21. Naked pair {59} in R7C56, locked for R7 and N8, R7C1 = 8 (step 6), R3C9 = 8 (step 8)

22. R78C4 = R7C1 + 1 (step 17)
22a. R7C1 = 8 -> R78C4 = 9 = [36], R8C3 = 4, R8C56 = [12]
22b. Naked pair {67} in R7C23, locked for R7 and N7
22c. R9C7 = 6 (hidden single in R9)

23. Naked pair {13} in R56C6, locked for C6 and N5
23a. R56C6 = {13} = 4, R7C56 = {59} = 14 -> R6C7 = 4 (cage sum)

24. 45 rule on N6 1 remaining innie R4C7 = 8
24a. 27(4) cage at R3C7 = {4689} (only remaining combination), 4,6 locked for R4 and N5

25. 22(3) cage at R5C9 = {679} (only remaining combination), locked for N6

26. 45 rule on N3 2 remaining innies R1C78 = 5 = [14/23/32], no 5, no 1 in R1C8
26a. 45 rule on N3 2 outies R12C6 = 15 = {69/78}, no 4,5

27. R7C1 = 8 -> R6C12 = 8 = [17]/{26/35}, no 9, no 7 in R6C1

28. R4C23 = R7C1 + 6 (step 11)
28a. R7C1 = 8 -> R4C23 = 14 = {59}, locked for R4, N4 and 20(4) cage at R3C1, no 5 in R3C12, clean-up: no 3 in R6C12 (step 27)
28b. R4C23 = 14 -> R3C12 = 6 = {24}, locked for R3 and N1

29. R1C23 = R7C1 + 9 (step 12)
29a. R7C1 = 8 -> R1C12 = 17 = {89}, locked for R1, N1 and 24(4) cage at R1C2, no 8 in R2C4, clean-up: no 6,7 in R2C6 (step 26a)
29b. R1C12 = 17 -> R12C4 = 7 = {25} (only remaining combination), locked for C4 and N2 -> R4C4 = 7, R3C4 = 1

30. Naked pair {67} in R13C6, locked for C6 and N2 -> R13C5 = [43], R4C56 = [64], clean-up: no 1 in R1C7 (step 26)

31. Naked pair {23} in R1C78, locked for R1 and N3 -> R12C4 = [52]

32. R6C5 = 5 (hidden single in R6), R7C56 = [95], R2C5 = 8, R3C6 = 7 (cage sum), R12C6 = [69], R5C5 = 2
32a. 2 in N6 only in R4C89, locked for R4

33. 45 rule on R89 2 remaining innies R89C9 = 12 = [39/75]
33a. Killer pair 7,9 in 22(3) cage at R5C9 and R89C9, locked for C9 -> R1C9 = 1, R1C1 = 7

34. R8C1 = 9 (hidden single in C1), R8C2 = 5, R4C23 = [95], R3C3 = 6, R7C23 = [67], R3C8 = 5, R9C8 = 9, R9C9 = 5, R8C9 = 7 (step 33)
34a. R6C8 = 7 (hidden single in N6), R6C2 = 2, R6C1 = 6 (step 27)

and the rest is naked singles.
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