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CDK

 
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Nasenbaer
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Joined: 20 Jul 2006
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PostPosted: Tue Feb 13, 2007 5:07 pm    Post subject: CDK Reply with quote

No, it's not a new perfume by Christian Dior and Calvin Klein, it's a
Twisted Evil Center Dot Killer! Twisted Evil

3x3::k:2048:2817:2817:3075:3075:3075:3078:1799:1799:2048:10:4619:4619:13:3854:3078:16:2833:3090:3090:4619:4619:3854:3854:2328:2833:2833:6427:6427:6427:6427:2335:2335:2328:4386:4386:3620:37:3366:3366:40:2857:2857:43:4386:3620:3620:2095:2096:2096:5426:5426:5426:5426:6198:6198:2095:5433:5433:3643:3643:1085:1085:6198:64:2113:5433:67:3643:3643:70:2887:1864:1864:2113:3915:3915:3915:3918:3918:2887:

The center cells in each nonet (r258c258) form their own 45-cage, so each number appears exactly once in these cells. (I think it's not solvable without this information).

Rating? Well, I think it qualifies as Assassin. Twisted Evil

Could someone please set a picture? Thanks. For those who don't use SumoCue here's a text version of the cages.

Code:

.--.-----.--------.--.-----.
|8 |11   |12      |12|7    |
|  :--.--'--.--.--:  :--.--:
|  |0 |18   |0 |15|  |0 |11|
:--'--:     :--'  :--+--'  |
|12   |     |     |9 |     |
:-----'-----+-----:  :-----:
|25         |9    |  |17   |
:--.--.-----+--.--'--+--.  |
|14|0 |13   |0 |11   |0 |  |
|  '--+--.--'--+-----'--'--:
|     |8 |8    |21         |
:-----:  :-----+-----.-----:
|24   |  |21   |14   |4    |
|  .--+--:  .--:     :--.--:
|  |0 |8 |  |0 |     |0 |11|
:--'--:  :--'--'--.--'--:  |
|7    |  |15      |15   |  |
'-----'--'--------'-----'--'


I hope you'll have fun! Twisted Evil

Peter

Edit: I just found out that you can form a cage (using shift-click) from the center dot cells, so you might want to use this version (it's the same puzzle, just connected the center dot cells for easier eliminations):

3x3::k:2048:2817:2817:3075:3075:3075:3078:1799:1799:2048:11530:4619:4619:11530:3854:3078:11530:2833:3090:3090:4619:4619:3854:3854:2328:2833:2833:6427:6427:6427:6427:2335:2335:2328:4386:4386:3620:11530:3366:3366:11530:2857:2857:11530:4386:3620:3620:2095:2096:2096:5426:5426:5426:5426:6198:6198:2095:5433:5433:3643:3643:1085:1085:6198:11530:2113:5433:11530:3643:3643:11530:2887:1864:1864:2113:3915:3915:3915:3918:3918:2887:

Peter
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sudokuEd
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PostPosted: Tue Feb 13, 2007 8:47 pm    Post subject: Reply with quote

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Nasenbaer
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PostPosted: Wed Feb 14, 2007 12:43 am    Post subject: Reply with quote

Thanks for setting the picture, Ed.

After redoing the puzzle (and knowing what moves to use) I might have to lower my rating to might-have-been-an-assassin-killer-six-months-ago. Wink If you find yourself doing heavy combination checkings (that's what I was doing the first time) you definetly should look for another way.

Peter
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Para
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PostPosted: Wed Feb 14, 2007 1:24 pm    Post subject: Reply with quote

Hi all

Here's the CDK-challenge. Forget this puzzle is a Center Dot Killer and solve it as a Zero Killer.
Had to check cause Peter wasn't sure it wasn't solvable without the Center Dot Properties.

3x3::k:2048:2817:2817:3075:3075:3075:3078:1799:1799:2048:10:4619:4619:13:3854:3078:16:2833:3090:3090:4619:4619:3854:3854:2328:2833:2833:6427:6427:6427:6427:2335:2335:2328:4386:4386:3620:37:3366:3366:40:2857:2857:43:4386:3620:3620:2095:2096:2096:5426:5426:5426:5426:6198:6198:2095:5433:5433:3643:3643:1085:1085:6198:64:2113:5433:67:3643:3643:70:2887:1864:1864:2113:3915:3915:3915:3918:3918:2887:

I have added my walk-through for anyone who doesn't believe it is unique without Center Dot properties. But don't check it if you like the challenge.

[Edit] Changed the walk-through to one without desperation moves. Now it is a nice walk-through

1. R12C1, R67C3, R6C45 and R89C3 = {17/26/35} -->> no 4,8,9
2. R1C23, R5C67 and R89C9 = {29/38/47/56} -->> no 1
3. R12C7 and R3C12 = {39/48/57} -->> no 1,2,6
4. R1C89 and R9C12 = {16/25/34} -->> no 7,8,9
5. 11(3) in R2C9: no 9
6. R34C7 and R4C56 = {18/27/36/45} -->> no 9
7. R5C34 = {49/58/67} -->> no 1,2,3
8. 21(3) in R7C4 = {489/579/678} -->> no 1,2,3
9. 14(4) in R7C6: no 9
10. R9C78 = {69/78}
11. 24(3) in R7C1 = {789} -->> locked for N7
11a. Clean up: R689C3 : no 1
11b. R9C12 : no {25} -->> clashes with 8(2) in R8C3
12. R7C89 = {13} -->> locked for R7 and N9
12a. Clean up: R6C3: no 5,7 ; R89C9 : no 8
12b. Naked Quad {2356} in R6789C3 -->> locked for C3
12c. Clean up: R5C4: no 7,8 ; R1C2: no 5,6,8,9
13. 45 on N7 : 2 innies: R7C3 + R8C3 = 6 = [24/51]
13a. Clean up : R6C3: no 2
14. 45 on R9: 2 outies : R8C39 = 11 = [29]/{56}
14a. Clean up: R9C3: no 5 ; R9C9: no 4,7,9
14b. 15(2) in R9C78 = {78}-->> locked for R9 and N9 : {69} clashes with 11(2) in R8C9
15. 9 locked in R9 for N8 -->> no 9 anywhere else in N8
15a. 21(3) in R7C4 = {678} -->> locked for N8
15b. 3 locked in R8 for N8
16. R7C45 can’t have both 78 (clashes with 24(3) in R7C1) -->> R8C4 = {78}
17. 6 locked in N8 for R7
18. 45 on N89 : 2 innies R8C58 = 10 = [19/46]
18a. Naked Pair {14} in R8C25 -->> locked for R8
18b. 14(4) in R7C6 = {2345} -->> R8C6 = 3
18c. Hidden Single 4 in R7C7
18d. Clean up: R12C7: no 8 ; R34C7: no 5 ; R5C6: no 7 ; R5C7: no 8; R4C5: no 6
19. 45 on N3: 2 innies: R2C8 + R3C7 = 15 = [96]/{78}
19a. Clean up: R4C7 : no 6,7,8
19b. Killer pair {79} in R123C7 + R2C8 -->> locked for N3
20. 11(3) in R2C9 = {128/146/245} : {236} clashes with 7(2) in R1C8 -->> no 3
21. 45 on N12: 2 innies : R2C25 = 14 = {59/68}
22. 45 on N1: 3 innies: R2C23 + R3C3 = 14 = [5]{18}/[6]{17}/[9]{14}
22a. R2C2: no 8 -->> Clean up R2C5: no 6; R23C3: no 9
22b. 1 locked in R23C3 for N1; C3 and 18(4) in R2C3
22c. Clean up: R12C1: no 7
23. 45 on R6: 1 innie and 1 outie: R6C3 – R5C1 = 2 -->> R5C1 = {14}
24. 45 on R4: 1 innie and 1 outie: R5C9 – R4C7 = 6 -->> R5C9 = {789}
25. 45 on R1: 2 outies: R2C17 = 5 = [23] -->> R1C17 = [69]
25a. Clean up: R3C7: no 6; R23C3: no 7 (step 22); R2C5: no 8 (step 21); R5C9: no 9 (step 24); R1C89: no 1,4
25b. 7(2) in R1C8 = {25} -->> locked for R1 and N3
25c. 11(3) in R2C9 = {146}
25d. Killer Pair {84} in 11(2) in R1C23 and R23C3 -->> locked for N1
26. Naked Pair {59} in R2C25 -->> locked for R2
27. Naked Pair {78} in R39C7 -->> locked for C7
27a. Clean up: R5C6 = {569}
28. Naked Pair {78} in R29C8 -->> locked for C8
29. 9 locked in C3 for N4 -->> no 9 anywhere else in N4
30. 14(3) in R5C1 = [1]{58/67}/[4]{28/37} -->> no 1,4 in R6C12
30a. 4 locked in R6 in 21(4) -->> 21(4) = {1479/2469/2478/3459/3468}
31. Useless step
32. 45 on R5: R5C258 = 9, 10, 12 or 13
32a. 3 locked in R5C258
32b. R5C19 = [48] -->> R5C258 = {135}
32c. R5C19 = [47] -->> R5C258 = {136}: no {235}: clashes with 11(2) in R5C6
32d. R5C19 = [18] -->> R5C258 = {237/345}
32e. R5C19 = [17] -->> R5C258 = {238}: no {246}: clashes with 11(2) in R5C6
32f. R5C258: no 9
33. Combination check on 17(3) in R4C8
33a. 17(3) needs a 7 or 8 because of R5C9
33b. 17(3) = {179/278/368/458/467}
33c. {179} not possible: 7 must be in R5C9 -->> R5C9 = 7 -->>R4C7 = 1 (step 24) -->> 2 1’s in R4 and N6
33d. 17(3) = {278/368/458/467}: no 1,9
33e. 9 locked in N6 for R6; 9 locked in 21(4) in R6C6
34. 21(4) in R6C6 = {1479/2469/3459}: no 8
34a. 8 locked in 17(3) in R4C8-->> 17(3) = {278/368/458}
34b. no combination possible with 2 in R4C9
34c. No 7 in R4C9 : R4C9 = 7 -->> R5C9 = 8 -->> R4C8 = 2 and R4C7 = 2 (step 24)
35. 8 locked in R6 for N4; 8 locked in 14(3)
35a. Clean up: R5C4: no 5
35b. 14(3) = [1]{58}/[482]: no 3,6,7
36. 6 locked in C7 for N6
36a. 17(3) in R4C8 = [287]/{45}[8]
36b. When 17(3) = {45}[8] -->> R4C7 = 2: 2 locked in R4C78 for R4 and N6
36c. Clean up: 9(2) in R4C5: no 7; R5C6: no 9
36d. 11(2) in R5C6 = {56}-->> locked for R5
36e. 13(2) in R5C3 = {49}-->> locked for R5
36f. R5C1 = 1; R6C3 = 3 (step 23); R7C3 = 5; R7C6 = 2; R8C7 = 5
37. More Singles
37a. R456C7 = [261]; R5C68 = [53]; R39C7 = [78]; R29C8 = [87]; R7C89 = [13]
37b. Hidden single 2 in R9C9; R8C89 = [69]; R89C3 = [26]; R1C89 = [25]; R3C8 = 4
37c. R4C8 = 5; R45C9 = [48]; R6C89 = [97]; R6C6 = 4
And the rest is also singles.



Greetings

Para


Last edited by Para on Sat Mar 10, 2007 1:02 pm; edited 1 time in total
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sudokuEd
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PostPosted: Sat Feb 17, 2007 12:04 pm    Post subject: Reply with quote

A very enjoyable puzzle Peter. Great to have a variant master in our midst.

Enjoyed your walk-through too Para. Really like the way you locked in the 2's for r4 in 36b. Perhaps an easier way to fix the 9 for r5 (step 32) would be 11(2) = {56} -> 13(2) = {49} -> 9 locked in 13(2), 11(2) for r5

Now. Time to have a crack at Ruud's Album Killer. SumoCue takes about 10 minutes to find a solution, so am very impressed with Peter and Richard solving it bounce Applause

OK - now here's a V3 for Peter's CDKiller - like Para's 'proof', the center dots are all mystery cells, plus, a few of the cages have been combined to toughen it up even more. Still has a unique solution and has a logical solve path.

CDK V3: contains 9 mystery cells not covered by a cage
3x3::k:2048:2817:2817:3075:3075:3075:4870:4870:4870:2048:10:4619:4619:13:3854:4870:16:2833:3090:3090:4619:4619:3854:3854:2328:2833:2833:6427:6427:6427:6427:2335:2335:2328:4386:4386:3620:37:3366:3366:40:2857:2857:43:4386:3620:3620:2095:2096:2096:5426:5426:5426:5426:6198:6198:2095:5433:5433:3643:3643:1085:1085:6198:64:3905:5433:67:3643:3643:70:6727:3905:3905:3905:3915:3915:3915:6727:6727:6727:

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Andrew
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PostPosted: Sat Feb 24, 2007 11:42 pm    Post subject: Reply with quote

Nice puzzle Peter.

Nasenbaer wrote:
After redoing the puzzle (and knowing what moves to use) I might have to lower my rating to might-have-been-an-assassin-killer-six-months-ago.

It wasn't that easy Peter, unless I missed some easy moves. More like might-have-been-an-assassin-killer-three-months-ago.

Here is my walkthrough. It's well over a week since the puzzle was first posted so it's in normal text.

Para gave me some feedback on v2 (that's in my next message) which also commented on a couple of steps below. Thanks Para.

Original version. The centre dot cells form a remote 45(9) cage.

Clean-up is used in various steps, using the combinations in steps 1 to 16 for further eliminations from these two cell cages

1. R12C1 = {17/26/35}, no 4,8,9

2. R1C23 = {29/38/47/56}, no 1

3. R12C7 = {39/48/57}, no 1,2,6

4. R1C89 = {16/25/34}, no 7,8,9

5. R3C12 = {39/48/57}, no 1,2,6

6. R34C7 = {18/27/36/45}, no 9

7. R4C56 = {18/27/36/45}, no 9

8. R5C34 = {49/58/67}, no 1,2,3

9. R5C67 = {29/38/47/56}, no 1

10. R67C3 = {17/26/35}, no 4,8,9

11. R6C45 = {17/26/35}, no 4,8,9

12. R7C89 = {13}, locked for R7 and N9, clean-up: no 5,7 in R6C3

13. R89C3 = {17/26/35}, no 4,8,9

14. R89C9 = {29/47/56}, no 8

15. R9C12 = {16/25/34}, no 7,8,9

16. R9C78 = {69/78}

17. 11(3) cage in N3, no 9

18. 24(3) cage in N7 = {789}, locked for N7, clean-up: no 1 in R6C3, no 1 in R89C3

19. 21(3) cage in N8 = {489/579/678}, no 1,2,3

20. 14(4) cage in N89, no 9
[When solving V3 I went further and eliminated 8 because 1,3 only in R8C6. Then when Ed reviewed my V3 steps, he pointed out that this gives R8C6 = {13}. I’d missed that because I hadn’t listed the combinations for the 14(4) cage.]

21. Naked quad {2356} in R6789C3, locked for C3, clean-up: no 5,6,8,9 in R1C2, no 7,8 in R5C4

22. 45 rule on R1 2 innies R1C17 = 15 = [69]/[78], clean-up: R2C1 = {12}, R2C7 = {34}
22a. No 3,4 in R1C89 because {34} would clash with R2C7
22b. R1C89 = {16/25} [1/2] -> 11(3) cage in N3 cannot be {128}, no 8

23. 45 rule on R9 2 innies R9C39 = 8 = {26}/[35], clean-up: no 3 in R8C3, no 2,4,7 in R8C9

24. 45 rule on R6 1 innie R6C3 – 2 = 1 outie R5C1 -> R6C3 = {36}, R5C1 = {14}, clean-up: no 6 in R7C3
24a. R7C3 = {25}, R89C3 = {26}/[35] [2/5], killer pair 2,5 for N7
24b. 45 rule on N7 1 outie R6C3 – 2 = 1 innie R8C2 -> R8C2 = {14}
24c. R5C1 = R8C2

25. R89C9 = [56/65/92] [6/9] -> R9C78 must be {78}, locked for R9 and N9 [Edit. Typo corrected. R89C9 had previously been given as R89C1.]
25a. 9 in N9 locked in R8C89, locked for R8

26. 9 in N7 locked in R7C12, locked for R7
26a. 9 in N8 locked in R9C456 = 9{15/24}, no 3,6

27. 21(3) cage in N8 = {678} (only remaining combination), locked for N8

28. Naked quad {6789} in R7C1245, locked for R7

29. 4 in R7 locked in R7C67, locked for 14(4) cage
29a. Only valid combination for 14(4) cage = {2345} (cannot be {1346} because 1,3 in same cell), no 1,6 -> R8C6 = 3, clean-up: no 6 in R4C5, no 8 in R5C7
[Para commented that there is now a killer pair 2/5 in R8C7 and R89C9 -> R7C7 = 4]

30. 45 rule on R4 1 outie R5C9 – 6 = 1 innie R4C7 -> R5C9 = {789}, R4C7 = {123}, clean-up: R3C7 = {678}
30a. 45 rule on N3 1 innie R2C8 – 6 = 1 outie R4C7 -> R2C8 = {789}
30b. R2C8 = R5C9

31. 1 in C7 locked in R46C7, locked for N6

32. 45 rule on N9 3 innies R7C7 + R8C78 = 15, max R78C7 = 9 -> min R8C8 = 6
[Para hinted at Alternatively R89C9 [6/9] -> R8C8 = {69}]

33. R7C7 = 4 (hidden single in N9) -> R2C7 = 3, R1C7 = 9, clean-up: no 2 in R1C2, no 6 in R3C7, no 2,7,8 in R5C6, no 9 in R5C9 (step 30b)
33a. R1C1 = 6 (step 22), R2C1 = 2, clean-up: no 1 in R9C2, no 1 in R1C89 = {25}, locked for R1 and N3 -> 11(3) cage = {146}
33b. 1 in R1 locked in R1C456, locked for N2

34. R39C7 = {78}, locked for C7, clean-up: no 4 in R5C6
34a. 6 in C7 locked in R56C7, locked for N6
34b. 17(3) cage in N6 valid combinations are {278/458} = 8{27/45}, no 3,9, 8 locked for N6
[Original step 34a moved to end of step 33, remaining sub-steps renumbered.]

35. 6 in R7 locked in R7C45, locked for N8

36. Naked quad {2569} in R8C3789, locked for R8

37. R8C25 = {14}, locked for centre dot cells
37a. {23} locked in R5C258 for centre dot cells, locked for R5, clean-up: no 9 in R5C6 -> R5C67 = {56}, locked for R5, clean-up: no 7,8 in R5C3 -> R5C34 = {49}, locked for R5

38. R5C1 = 1 (naked single), R6C12 = 13 = {49/58}/[76], clean-up: no 6 in R9C2
38a. R6C3 = 3 (step 24), R7C3 = 5, R8C2 = 1 (step 24c), R8C5 = 4, R7C6 = 2, R8C7 = 5, R5C7 = 6, R5C6 = 5, clean-up: no 7 in R4C5, no 4 in R4C6, no 6 in R89C9 = [92], R8C8 = 6, locked for centre dot cells, R89C3 = [26], R1C89 = [25]

39. R6C12 = {49/58} ([76] would clash with R6C45)
[Edit. Step 38a modified and step 38b renumbered to become step 39.]

40. {59} (hidden pair) locked in R2C25 for centre dot cells, locked for R2, no 7,8 in R2C25

41. R4C2= 6 (hidden single in N4), clean-up: no 3 in R4C5

42. R5C8 = 3 (hidden single in N6) -> R7C89 = [13], R3C8 = 4, R23C9 = {16}, clean-up: no 8 in R3C12

43. R6C8 = 9 (hidden single in N6), clean-up: no 4 in R6C12 = {58}, locked for R6 and N4

44. R4C8 = 5 (hidden single in N6, I should have spotted the naked pair R29C8 earlier!) -> R45C9 = [48] (only valid combination), R6C9 = 7, R2C8 = 8 (step 30b), R3C7 = 7, R4C7 = 2, R6C7 = 1, R6C9 = 7, R6C6 = 4, clean-up: no 7 in R4C6

45. R4C13 = {79}, locked for R4 and N4 -> R5C34 = [49], R5C25 = [27], clean-up: no 7 in R1C2

46. R4C4 = 3 (hidden single in N4)

47. R9C1 = 4 (hidden single in C1), R9C2 = 3, R1C2 = 4, R1C3 = 7, R4C3 = 9, R4C1 = 7, R8C1 = 8, R7C1 = 9, R7C2 = 7, R6C12 = [58], R3C12 = [39], R2C25 = [59], R23C3 = [18], R23C9 = [61]

and the rest is naked singles, naked pairs, simple elimination and cage sums

Hope I've got it right now. I made several silly mistakes while solving it that got me to the correct solution by quicker but flawed paths.

6 4 7 8 3 1 9 2 5
2 5 1 4 9 7 3 8 6
3 9 8 5 2 6 7 4 1
7 6 9 3 1 8 2 5 4
1 2 4 9 7 5 6 3 8
5 8 3 2 6 4 1 9 7
9 7 5 6 8 2 4 1 3
8 1 2 7 4 3 5 6 9
4 3 6 1 5 9 8 7 2


Last edited by Andrew on Wed Nov 28, 2007 9:37 pm; edited 5 times in total
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PostPosted: Sun Feb 25, 2007 9:05 pm    Post subject: Reply with quote

Having solved the original version, I then took up Para's challenge and solved it without using any special properties for the centre dot cells. Here is my walkthrough for this second version.

Version 2. The centre dot cells don’t necessarily form a remote nonet so there is no elimination between the centre dot cells except for ones in the same row/column.

Steps 1 to 36 as in the original version.

37. R29C8 = {78}, locked for C8 (I only spotted this at step 44 in the original version so have put it in here for version 2), no 2 in R4C9

38. R5C1 = {14}, if R5C1 = 4 => R8C2 = 4 (step 24c) => R9C12 => [16]
-> 1 in C1 locked in R59C1, locked for C1

39. 45 rule on R4 3 innies R4C789 = 11 = [128/245/254] (cannot be [227]) -> no 7 in R4C9
[Para added. 2 locked for R4 and N6. You missed this and would have saved you the contradiction move. Would lead to R5C67 = {56} and R5C34 = {49}] and then R5C1 = 1 which I think was the most critical cell to fix in the whole puzzle.

40. Try a contradiction move on R5C1 = {14}
If R5C1 = 4 => R5C3 = {78}, R5C4 = {56} => R5C67 = [92] (cannot be {56} which would clash with R5C4) => 17(3) cage in N6 = {458} => R5C9 = 8, R5C7 = 2 => R4C7 = 1 => R3C7 = 8 => R2C8 = 7 => R2C8 <> R5C9 which clashes with step 30b -> R5C1 cannot be 4
[See comment at the end.]

41. R5C1 = 1, R6C12 = 13 = {49/58}/[76]
41a. R6C3 = 3 (step 24), R7C3 = 5, R8C2 = 1 (step 24c), R8C5 = 4, R7C6 = 2, R8C7 = 5, clean-up: no 7 in R4C5, no 5 in R4C6, no 5 in R6C45, no 6 in R9C2
41b. R6C12 = {49/58} ([76] would clash with R6C45)

42. R8C89 = {69}, locked for R8 and N9, R9C9 = 2, R8C9 = 9, R8C8 = 6, R89C3 = [26], R1C89 = [25]

43. R5C8 = 3 (hidden single in N6) -> R7C89 = [13], R3C8 = 4, R4C8 = 5, R4C9 = 4, R5C9 = 8, R6C8 = 9, R6C9 = 7, clean-up: no 8 in R3C12, no 4 in R6C12, no 1 in R6C45, R6C6 = 4 (hidden single in R6), R6C7 = 1, R4C7 = 2, R3C7 = 7, R5C7 = 6, R5C6 = 5, R2C8 = 8, R9C78 = [87], R5C4 = 9, R5C3 = 4, clean-up: no 7 in R1C2, no 5 in R3C12, no 7 in R4C6

44. R6C45 = {26}, locked for N5, R5C5 = 7, R5C2 = 2, clean-up: no 3 in R4C5

45. R9C1 = 4 (hidden single in C1), R9C2 = 3, R1C2 = 4, R1C3 = 7, R3C2 = 9, R3C1 = 3

46. R6C1 = 5 (hidden single in C1), R6C2 = 8, R2C2 = 5, R7C2 = 7, R8C1 = 8, R7C1 = 9

47. R4C1 = 7, R4C2 = 6, R4C56 = {18}, locked for R4 -> R4C3 = 9, R4C4 = 3

48. R23C3 = [18], R23C9 = [61]

and the rest is naked singles, simple elimination and cage sums

[Para commented for step 40 You forgot to use step 39 in your chain. 17(3) = {458} cage in N6 -->> R4C89 = {45}, R4C7 = 2 -->> contradiction with R5C7.

But this could all be spared by locking the 2 in step 39. Would make it look a bit nicer.

The rest was good. You saw a few things i missed and vice-versa. I think The 45 test on N12 and N89 would have made things a bit easier for you too.]


Thanks very much Para for those comments. I agree that 45s on N12 and N89 would have made things a bit easier. They are almost certainly essential for Ed's V3.
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Afmob
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PostPosted: Sun Nov 18, 2007 10:16 pm    Post subject: Reply with quote

I saw this one on the unsolvable (unsolved? Wink) list so I took my chances and solved it.

Edit: Andrew noticed typos in my walkthrough and one of them was in my main step 6a. After going through my wt again I saw that 6a was flawed, so beginning with step 6 I rewrote my wt and it should be ok now.

Thanks Andrew!

CDK V3 Walkthrough:

1. N123
a) Innies N12 = 14(2) = {59/68}
b) Innies N3 = 15(2) = [78/87/96]
c) 9(2): R4C7 = (123)
d) Outies R1 = 5(2) = [14/23/32]
e) 8(2): R1C1 = (567)
f) Innies N1 = 14(3): R23C3 <> 9 because R2C2 >= 5

2. N789
a) 24(3) = {789} locked for N7
b) 15(4) = 36{15/24} -> 3,6 locked for N7
c) 4(2) = {13} locked for R7+N9
d) Innies N7 = 6(2) = [24/51]
e) Innies N8 = 9(3) <> 7,8,9
f) Innies N89 = 10(2) = [19/28/37/46/64]
g) 8(2) = [35/62]
h) Outies R9 = 11(2) = [29/38/47/56/65]

3. R46
a) Innies+Outies R6: -2 = R5C1 - R6C3: R5C1 = (14) because R6C3 = (36)
b) Innies R4 = 11(3) <> 9
c) Innies+Outies R4: 6 = R5C9 - R4C8: R5C9 = (789) because R4C8 = (123)

4. N8
a) 14(4) = 2{147/156/345} <> 8 because (13) only possible @ R8C6
b) 14(4) must have 1 xor 3 and it's only possible @ R8C6 -> R8C6 = (13)

5. R5789 !
a) Innies R5 = 21(5): R5C258 <> 9 because R5C9 >= 7
b) ! 4,6 in R7 can only be in 21(3) and 14(4) and none of them can have both
-> 21(3) = 8{49/67} -> 8 locked for N8
-> R8C47 <> 4,6
c) 8 locked in R9C789 for N9 @ 26(4) = 8{279/459/567}
d) Innies N89 = 10(2) <> 2
e) Outies R9 = 11(2) <> 3
f) 3 locked in R9C123 for R9
g) Killer pair (69) locked in 21(3) + 15(3) for N8
h) Naked triple (134) locked in R8C256 for R8
i) Outies R9 = 11(2) <> 7

6. N789 !
a) ! Innies = 16(1+3) = R7C3+R8C258 = 2+4[19/37] / 5+1[37/46] because R8C58 = 10(2)
- But 5+[137] blocked by R8C6 = (13)
-> 16(1+3) = 2+4[19/37]/5+[146]
-> Both combos force R6C8+R8C3 <> 6 because either R6C3 = 6 or R8C8 = 6
b) Naked pair (25) locked in R78C3 for C3+N7
c) Outies R9: R8C9 <> 5
d) 6 locked in 15(4) for R9
e) 15(3) = 9{15/24} -> 9 locked for R9+N8
f) 21(3) = {678} -> 6 locked for R7
g) 7 locked in 26(4) @ R9 = 78{29/56} -> 7 locked for N9
h) Hidden Killer pair (25) in R7C6 for N8 -> R7C6 <> 4
i) 14(4) = {2345} -> R7C7 = 4, R8C6 = 3
j) Naked pair (69) locked in R8C89 for R8

7. R123 !
a) Outies R1 = 5(2) = {23} locked for R2
b) 8(2) <> 7
c) 1 locked in Innies N1 = 14(3) = 1{49/58/67} <> 3; 1 locked for C3+18(4)
d) Innies N1 = 14(3): R23C3 <> 6 because R2C2 <> 1,7; R2C2 <> 8 because 5 only possible there
e) Innies N12 = 14(2): R2C5 <> 6
f) Innies N2 = 18(3): R23C4 <> 8 because R23C4 <> 1
g) ! 18(4): R23C3 <> 4 because {67} @ R23C4 blocked by R78C4 = (678)
h) Innies N1 = 14(3) = 1{58/67}
i) Naked pair (56) locked in R1C1+R2C2 for N1
j) 12(2) <> 7
k) 11(2): R1C2 <> 9
l) Innies N12 = 14(2): R2C5 <> 5

8. R123 !
a) 18(4) = 1{278/368/458/467} <> 9 because R23C3 = 1{7/8}; R3C4 <> 7 because 2 only possible there
b) 11(2) @ N1 <> 3,8 because (38) is a Killer pair of 12(2)
c) 12(3) <> 6 because R1C1 = (56) and (24) is a Killer pair of 11(2)
d) 12(3) <> {237} because (27) is a Killer pair of 11(2)
e) 19(4) = {1369/1378/2359/2458/3457} since R2C7 = (23) and other combos blocked by Killer pairs of Innies N3
f) ! Killer pair (15) locked in 19(4) + 12(3) for R1
g) R1C1 = 6 -> R2C1 = 2, R2C2 = 5, R2C7 = 3
h) 11(2) = {47} locked for R1+N1
i) 19(4) = {2359} -> {259} locked for R1+N3
j) Innies N12 = 14(2) = [59] -> R2C5 = 9
k) 12(3) = {138} locked for N2
l) 18(4) = 18{27/45}; R3C4 <> 4

9. R456 !
a) 9 locked in R45C3 @ C3 for N4
b) ! 21(4) <> {1389} because together with R7C3 it would build a Killer triple (136) for 8(2) @ N5
c) Hidden Killer pair (89) in 14(3) + 21(4) for R6
-> 14(3) = 8{15/24} -> 8 locked for R6+N4; R6C12 <> 1,4
d) 13(2) <> 5,8
e) 11(2) = {29/56} because (47) is a Killer pair of 13(2)
f) Killer pair (69) locked in 13(2) + 11(2) for R5
g) 9(2) @ N6 <> 6
h) 1,6 locked in R456C7 @ C7 for N6
i) 17(3) = 8{27/45} -> 8 locked for N6
j) ! Hidden Killer pair (36) in 9(2) + 25(4) for R4 -> One of them must have both (because of 9(2))
-> 25(4) = 79{18/36/45} <> 2; 7 locked for R4
k) 9(2) @ N5 <> 2; R4C5 <> 6

10. C6789 !
a) 2 locked in R4C789 @ R4 for N6
b) 11(2): R5C6 <> 9
c) 15(3) @ R2C6 = 6{27/45}
d) ! Killer triple (256) locked in 15(3) + R57C6 for C6
e) 9(2) @ N5: R4C5 <> 3,4

11. R456+N2
a) 3 locked in 25(4) @ R4 = {3679}
b) Hidden Single: R6C1 = 5 @ N4 -> R5C1+R6C2 = 9(2) = [18] -> R5C1 = 1, R6C2 = 8
c) Hidden Single: R5C2 = 2 @ N4, R5C3 = 4 @ N4 -> R5C4 = 9
d) 21(4) = {1479} locked for R6
e) 8(2) @ N5 = {26} locked for R6+N5
f) R5C6 = 5
g) 9(2) @ N5 = {18} locked for R4+N5
h) R4C7 = 2 -> R3C7 = 7
i) Hidden Single: R6C6 = 4 @ N5
j) 15(3) = {267} because (45) only possible @ R3C5 -> R2C6 = 7, {26} locked for R3

12. Rest is singles.

Rating: (Easy) 1.75. I used lots of Killer pairs and triples and a small forcing chain (step 6a).
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Andrew
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PostPosted: Mon Nov 03, 2008 12:03 am    Post subject: Reply with quote

This was completed some months ago. I'm posting it now for completeness.

Warning! Some of the combination work is heavy going, involving hypotheticals. If anyone wants to know how to solve V3, I'd recommend Afmob's walkthrough.

I first tried Ed's challenging CDK V3 soon after it first appeared in early 2007. At times I was struggling and without Ed's help, as discussed below, I would probably have given up. I thought I had finished it early last year and had intended to post my walkthrough but on checking I found that I'd incorrectly eliminated a combination in step 29. I've recently reworked the later steps because of that.

It's good to see that Afmob solved it using very different methods that I did.

It's hard to rate this puzzle the way I did it, particularly since some of the steps were done over a year ago, so I'll just agree with Afmob's rating of 1.75.

Here is my walkthrough, including Ed's hints and some discussion between us. A couple of steps use detailed analysis of remaining combinations. If you don't want to work through this analysis, I've provided summaries after these steps. I won't say enjoy, some of it is heavy going. Wink

As with V2, the centre dot cells don’t necessarily form a remote nonet so there is no elimination between the centre dot cells except for ones in the same row/column.

Many thanks to Ed for his feedback on my earlier steps and the hint he gave me after step 27. In a couple of cases, steps 14a and 25, the feedback has been included and forms part of the walkthrough.

1. R12C1 = {17/26/35}, no 4,8,9

2. R1C23 = {29/38/47/56}, no 1

3. R3C12 = {39/48/57}, no 1,2,6

4. R34C7 = {18/27/36/45}, no 9

5. R4C56 = {18/27/36/45}, no 9

6. R5C34 = {49/58/67}, no 1,2,3

7. R5C67 = {29/38/47/56}, no 1

8. R67C3 = {17/26/35}, no 4,8,9

9. R6C45 = {17/26/35}, no 4,8,9

10. R7C89 = {13}, locked for R7 and N9, clean-up: no 5,7 in R6C3

11. 11(3) cage in N3, no 9

12. 24(3) cage in N7 = {789}, locked for N7, clean-up: no 1 in R6C3

13. 21(3) cage in N8 = {489/579/678}, no 1,2,3

14. 14(4) cage in N89, no 9; only remaining 1,3 in same cell -> no 8
14a. 14(4) must have 1/3 -> R8C6 = {13} (thanks Ed)
[I’d only got “Min R7C67 + R8C7 = 11 -> max R8C6 = 3”. I’d missed Ed’s better move because I hadn’t listed the combinations for the 14(4) cage.]


15. 45 rule on R1 1 innie R1C1 = 1 outie R2C7 + 3 -> R1C1 = {567}, R2C7 = {234}, clean-up: R2C1 = {123}

16. 45 rule on R9 2 outies R8C39 = 11 = [29/38/47/56/65], no 1 in R8C3, no 2,4 in R8C9

17. 45 rule on R4 1 outie R5C9 = 1 innie R4C7 + 6 -> R4C7 = {123}, R5C9 = {789}, clean-up: R3C7 = {678}
17a. 45 rule on N3 1 innie R2C8 = 1 outie R4C7 + 6 -> R2C8 = R5C9 = {789}
[Alternatively 17b. 45 rule on N3 2 innies R2C8 + R3C7 = 15]

18. 45 rule on R6 1 innie R6C3 = 1 outie R5C1 + 2 -> R6C3 = {36}, R5C1 = {14}, clean-up: no 6 in R7C3
18a. 45 rule on N7 1 outie R6C3 = 1 innie R8C2 + 2 -> R8C2 = R5C1 = {14}
[Alternatively 18b. 45 rule on N7 2 innies R7C3 + R8C2 = 6]

19. 45 rule on N8 3 innies R7C6 + R8C56 = 9 = {126/135/234}, no 7,8,9
19a. 5 of {135} must be in R7C6 -> no 5 in R8C5

20. 21(3) cage cannot have 4,5,6 in R8C4 because {89/79/78} would clash with R7C12 -> R8C4 = {789}
20a. Killer triple 7,8,9 in R7C1245, locked for R7

21. Only valid combinations for 15(4) cage in N7 are {1356/2346} = 36{15/24}

22. 45 rule on N12 2 innies R2C25 = 14 = {59/68}

23. 45 rule on N89 2 innies R8C58 = 10 = [19/28/37/46/64]

24. 45 rule on N1 3 innies R2C23 + R3C3 = 14, min R2C2 = 5 -> max R23C3 = 9, no 9 in R23C3

25. 45 rule on N4 3 innies R5C23 + R6C3 – 6 = 1 outie R4C4, min R5C23 + R6C3 = 9 (cannot be {124} because R6C3 only contains 3,6, cannot be {134} which would clash with R5C1, thanks Ed) -> min R4C4 = 3

26. 45 rule on R4 3 innies R4C789 = 11 -> no 9 in R4C89
26a. 9 in R4 locked in 25(4) cage = 9{178/268/358/367/457}

[While reviewing the early steps, Ed commented
Just noticed a nice elim from this. I'll put it into tt since it ends up being potentially very helpful.

9 in r4c4 -> 9 cannot be in r5c23. Here's how.
a. 9 in r4c4 -> from step 25: R5C23 + R6C3 = 15.
i. 3 in r6c3 -> r5c23 = 12 but cannot be {39} -> no 9 in r5c23
ii. 6 in r6c3 -> r5c23 = 9 -> cannot have 9
b. 9 elsewhere in 25(4) must be in n4 -> no 9 in r5c23
]


27. 45 rule on N47 2 outies R45C4 – 9 = 2 innies R58C2, max R45C4 = 17 -> max R58C2 = 8 -> max R5C2 = 7

At this stage I was struggling. Ed reviewed my earlier steps, including the first part of step 28, and then added
“Now, in case these things above don't unlock it, here's a big hint. The way to unlock this puzzle is combining steps 15 and 17b and seeing what this means for R1. Easy. Wink If you want a harder way, do a similar thing for R9! If you want to make it a really easy puzzle, do both.”

Many thanks for the hint. A typical hint from Ed, just enough to provide help but still make one work to make progress. That’s how good hints ought to be! Not sure about the last sentence. There was still a lot of hard work.


28. 19(4) cage in N3 must contain {234} in R2C7, valid combinations at this stage are {1279/1369/1378/1459/1468/2359/2368/2458/2467/3457}
[When Ed reviewed this he told me that I had too many combinations, leaving me to work out which ones weren’t valid. I found that was because I hadn’t been looking at the effect of steps 15 and 17b.]
28a. There cannot be any combinations with {67}, {68} or {79} which would clash with R2C8 + R3C7, eliminating {1279/1468/2368/2467}
28b. There cannot be any combinations with 5,6,7 in R1 when 2,3,4 (respectively) are in R2C7 (step 15), eliminating {1369} and also limiting three of the other combinations to having a specific value in R2C7
28c. The remaining valid combinations, with [] indicating the value in R2C7, are {159[4]/178[3]/258[4]/259[3]/457[3]} -> no 3,6 in R1C789, R2C7 = {34}, R1C1 = {67} (step 15), clean-up: R2C1 = {12}

29. Consider each of these combinations and their effect on R1
For {1378}, R2C7 = 3, R1C789 = {178}, R1C1 = 6, R1C23 = {29), R1C456 = {345}
For {1459}, R2C7 = 4, R1C789 = {159}, R1C1 = 7, R1C23 = {38}, R1C456 = {246}
For {2359}, R2C7 = 3, R1C789 = {259}, R1C1 = 6, R1C23 = {38/47}, R1C456 = {138/147}
For {2458}, R2C7 = 4, R1C789 = {258}, R1C1 = 7, R1C23 is blocked
For {3457}, R2C7 = 3, R1C789 = {457}, R1C1 = 6, R1C23 = {29/38}, R1C456 = {138/129}

Summary of step 29: no {2458} combination in 19(4) cage in N3, no 5,6 in R1C23

[Ed said that he’d done similar analysis of hypotheticals but using 4 innies in R1, R1C1789 = 22 together with the restrictions from steps 15 and 17b.]

30. If R2C7 = 3, R34C7 <> [63] => R2C8 + R3C7 => {78} -> 19(4) cage in N3 cannot have 178[3] or 457[3] combinations.
Remaining valid combinations are {159[4]/259[3]} -> R1C789 = {159/259}, no 4,7,8 -> 5,9 locked for R1 and N3, clean-up: no 2 in R1C23, no 6 in R3C7 (step 17b), no 3 in R4C7
30a. R2C8 + R3C7 = {78}, locked for N3
30b. R5C9 = {78} (step 17a)

31. 17(3) cage in N6 must have R5C9 = {78}, valid combinations {278/368/458/467}, no 1
31a. R4C789 = 11 (step 26), R4C7 = {12} -> 17(3) cage combination {278} can only have 7 in R5C9 (cannot have [227] in R4C789) -> no 7 in R4C89

32. R3C12 = {39/57} (cannot be {48} which clashes with R1C23)

33. Killer pair 3,7 in R1C23 and R3C12, locked for N1 -> R1C1 = 6, R2C1 = 2, clean-up: no 8 in R2C5 (step 22)
33a. 1 in N1 locked in R23C3, locked for C3 and 18(4) cage -> no 1 in R23C4

34. R2C7 = 3 (step 15) -> R1C789 = {259} (step 30), locked for R1 and N3, clean-up: no 8 in R5C6
34a. 1 in R1 locked in R1C456, locked for N2
34b. 1 in C7 locked in R46C7, locked for N6

35. 18(4) cage in N12 must contain 1 = 1{278/359/368/458/467} (cannot be {1269} because no 2,6,9 in R23C3)
35a. 3 of {1359} must be in R3C4 -> no 9 in R3C4

36. 9 in C3 locked in R45C3, locked for N4
36a. 14(3) cage in N4 must have R5C1 = {14}, valid combinations are {158/167/248/347}, no 1,4 in R6C12

37. 4,9 in R6 locked in 21(4) cage = 49{17/26/35}, no 8

38. 8 in R6 locked in R6C12, locked for N4 -> 14(3) cage = 8{15/24}, no 3,6,7, clean-up: no 5 in R5C4
38a. R6C123 = 8{26/35} (step 18)

39. 25(4) cage = 9{178/268/358/367/457} (step 26a), any combinations with 8 must have R4C3 = 9, R4C4 = 8 -> cannot be {2689} because no 2,6 in R4C1 -> no 2 in 25(4) cage
39a. 25(4) cage = 9{178/358/367/457}

40. 2 in N4 locked in R56C2, locked for C2

41. Consider 14(4) cage in N89 = {1247/1256/2345} (only combinations because 1,3 only in R8C6)
If R8C6 = 1 => R8C2 = 4 => R7C3 = 2 (step 18b) => R7C67 = {456} => only valid combination for 14(4) cage in N89 = {1256}, no 7
If R8C6 = 3 => only valid combination for 14(4) cage in N89 = {2345}, no 7
-> 14(4) cage in N89 = 25{16/34}, no 7
41a. 2 of {1256} must be in R8C7 -> no 6 in R8C7

42. 45 rule on N9 1 innie R8C8 = 2 outies R78C6 + 1
42a. Min R78C6 = 5 (from combinations in step 41) -> min R8C8 = 6, clean-up: no 6 in R8C5 (step 23)

43. R7C6 + R8C56 (step 19) = {126/135/234}
43a. If {126} => R7C6 = 6, R8C56 = [21], R8C2 = 4, R7C3 = 2 (step 18b), R78C7 = [45] which gives wrong cage total in R78C67 -> R7C6 + R8C56 cannot be {126}
43b. R7C6 + R8C56 = {135/234}, no 6, 3 locked in R8C56 for R8 and N8, clean-up: no 8 in R8C9 (step 16)

44. R8C1 = {789}, R8C4 = {789} -> R8C89 must contain one of 7,8,9 -> R9C789 must contain two of 7,8,9. Combinations for the 26(4) cage in N9 are {2789/4589/4679/5678}, in the case of {5678} either 5 or 6 must be in R8C9

Here’s a discussion with Ed relating to the next step
Andrew “I looked at R9 but could only see how to make progress by doing hypotheticals on the five pairs of values for R8C39 (the discussion took place before I found step 43 which eliminated one pair of values for R8C39). This did provide progress by eliminating at least one of those pairs. Did you use hypotheticals in that way or did you have a more direct way to use r9?”
Ed “Yeah, I used the hypo's you've mentioned, not including the 15(3) in R9”
Andrew “but there is also interaction with R67C3 = [35/62] and of course with R8C39.”
Ed “True.”
Andrew “Since sending yesterday's message I haven't made any more progress and can't see how to proceed except to use those hypotheticals. However some of the steps that I made after doing R1/N3 should help to make the hypotheticals a bit simpler. I had a look at doing two hypotheticals for R7C3 rather than more of them for R8C39 but that looks very messy and appears that it doesn't produce as much useful "output information". ”


The second part of Ed’s hint suggests that a similar approach is needed for R9. The interactions between R8C2 + R7C3 and the 15(4) cage are already built into the latter which must be 36{15/24} (step 21). Other useful interactions are provided by R8C39 = 11 (step 16) and by R67C3 = [35/62]. Values for R9C789 must be consistent with step 44.

45. Consider the combinations for R8C39 and their effect on R9
For R8C39 = [29], R9C123 = {346}, R9C789 = {278}, R9C456 = {159}
For R8C39 = [47], R9C123 = [362], R9C789 is blocked
For R8C39 = [56], R9C123 = [163] (6 cannot be in R9C3 because R67C3 = [62] when R8C3 = 5), R9C789 = {479/578}, R9C456 = {258/249}
For R8C39 = [65], R9C123 = {234} (cannot be {135} because R67C3 = [35] when R8C3 = 6), R9C789 = {678}, R9C456 = {159}

46. Summarising the results of step 45
R8C39 = [29/56/65], no 4 in R8C3, no 7 in R8C9
R9C123 = [163]/{234}/{346}, no 5, no 1 in R9C2
R9C456 = {159/249/258} -> no 6,7 in R9C456
R9C789 = {278/479/578/678}
46a. 5 in N7 locked in R78C3, locked for C3, clean-up: no 8 in R5C4

47. 6 in N8 locked in R7C45, locked for R7
47a. 21(3) cage in N8 (step 13) = {678}, locked for N8
47b. 8 in R9 locked in R9C789, locked for N9, clean-up: no 2 in R8C5 (step 23)

48. 14(4) cage in N89 (step 41) = {2345} (only remaining combination) -> R8C6 = 3, clean-up: no 6 in R4C5, no 8 in R5C7, no 7 in R8C8 (step 23)
48a. Naked pair {14} in R8C25, locked for R8

49. 8 in R9 locked in R9C789 (step 46) = {278/578/678} (cannot be {479} which doesn’t contain 8), no 4,9
49a. 9 in N9 locked in R8C89, locked for R8

50. R7C7 = 4 (hidden single in N9), clean-up: no 7 in R5C6

51. Combined cage R5C3467 = 24 = {2679/4569}, 6,9 locked for R5
51a. R5C67 = {29/56} (cannot be [47] which clashes with combined cage), no 4,7

52. 15(3) cage in N2 = {249/258/267/456} (cannot be {348/357} which clash with R1C456), no 3
52a. 8 of {258} must be in R23C6 (R23C6 cannot be [52] which clashes with R7C6), no 8 in R3C5

53. 45 rule on N2 3 innies R2C45 + R3C4 = 18 = {279/369/459/567} (cannot be {378/468} which clashes with R1C456), no 8
53a. 2,3 of {279/369} must be in R3C4
53b. 6 of {567} must be in R2C5 (R23C4 cannot be {67} which clashes with R78C4)
53c. Combining steps 53a and 53b -> no 6 in R3C4

54. 18(4) cage in N12 (step 35) = 1{278/368/458} (cannot be {1359} because 3,5,9 only in R23C4, cannot be {1467} which clashes with R78C4), no 9
54a. 1,8 of {1458} must be in R23C3 -> no 4 in R23C3
54b. 2 of {1278} must be in R3C4 -> no 7 in R3C4

55. R2C23 + R3C3 = 14 (step 24) = {158} (only remaining combination) -> R2C2 = 5, R2C5 = 9 (step 22), R23C3 = {18}, locked for N1, clean-up: 3 in R1C23, no 7 in R3C12
55a. Naked pair {47} in R1C23, locked for R1
55b. Naked pair {39} in R3C12, locked for R3
55c. Naked triple {138} in R1C456, locked for N2

56. 18(4) cage in N12 (step 54) = 1{278/458}, no 6
56a. 5 of {1458} must be in R3C4 -> no 4 in R3C4

57. 45 rule on R789 4 innies R7C3 + R8C258 = 16, R8C25 = {14} = 5 -> R7C3 + R8C8 = 11 = [29/56]
57a. If R7C3 = 2 => R6C3 = 6 -> no 6 in R8C3
57b. If R7C3 = 5 => R8C8 = 6 -> no 6 in R8C3
57c. -> no 6 in R8C3

58. Naked pair {25} in R78C3, locked for N7
58a. Naked pair {25} in R8C37, locked for R8
58b. 6 in N7 locked in R9C23, locked for R9

59. 6 in C7 locked in R56C7, locked for N6
59a. 17(3) cage in N6 (step 31) = {278/458}, no 3, 8 locked for N6
59b. R4C789 = {128/245}, 2 locked for R4 and N6, clean-up: no 7 in R4C56, no 9 in R5C6

60. 45 rule on N36 2 innies R25C8 = 2 outies R56C6 + 2
60a. Min R25C8 = 10 -> min R56C6 = 8, no 1 in R6C6

61. 7 in R4 locked in R4C1234
61a. 25(4) cage (step 39a) = 9{178/367/457} (cannot be {3589} which doesn’t contain 7)
61b. 8 of {1789} must be in R4C4
61c. 3,6 of {3679} must be in R4C4 (R4C123 cannot contain both 3,6 which would clash with R6C3)
61d. 4,5 of {4579} must be in R4C4 (R4C123 cannot contain both 4,5 which would clash with 14(3) cage in N4)
61e. -> no 7,9 in R4C4

62. R4C3 = 9 (hidden single in R4), clean-up: no 4 in R5C4
62a. 7 in R4 locked in R4C12, locked for N4, clean-up: no 6 in R5C4

63. R1C3 = 7 (hidden single in C3), R1C2 = 4, R8C25 = [14], R5C1 = 1 (step 18a), R6C3 = 3 (step 18), R7C3 = 5, R7C6 = 2, R8C37 = [25], R56C2 = [28], R6C1 = 5, clean-up: no 5 in R4C6, no 6 in R5C6, no 9 in R5C7
63a. R5C67 = [56], R5C3 = 4, R4C4 = 9, R4C12 = [76], R4C4 = 3 (step 61a), R78C1 = [98], R7C2 = 7, R8C4 = 7, R3C12 = [39], R9C123 = [436], R8C8 = 6 (step 57), R8C9 = 9, R2C4 = 4, R3C4 = 5 (step 54), clean-up: no 1 in R6C5

64. R9C5 = 5 (hidden single in R9), clean-up: no 4 in R4C6

65. Naked pair {18} in R4C56, locked for R4 and N5 -> R4C7 = 2, R5C5 = 7, R5C89 = [38], R1C7 = 9, R2C8 = 8 (step 17a), R3C7 = 7, R3C56 = [26], R2C6 = 7, R23C3 = [18], R2C9 = 6, R6C7 = 1, R9C7 = 8, R6C45 = [26], R7C45 = [68], R4C56 = [18], R1C456 = [831], R9C46 = [19], R6C6 = 4, R7C89 = [13], R3C89 = [41], R4C89 = [54], R1C89 = [25], R9C89 = [72], R6C89 = [97]
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