SudokuSolver Forum

3x3::k:2304:3329:3329:3331:3331:3331:2822:2822:5128:2304:4362:4875:4875:3331:4110:4110:3856:5128:2304:4362:4875:2581:1814:4119:4110:3856:5128:3867:4362:2581:2581:1814:4119:4119:3856:3363:3867:3621:3621:4391:4391:4391:2858:2858:3363:3867:3886:2351:2351:3377:2610:2610:4404:3363:5430:3886:3128:2351:3377:2610:5180:4404:3134:5430:3886:3128:3128:6467:5180:5180:4404:3134:5430:1353:1353:6467:6467:6467:2638:2638:3134:

This solution is not the elegant version nd is looking for. Uses some contradictions/hypotheticals between nonets - though seemed pretty straight-forward ones. They made the puzzle quick first time through.

Please let me know if anything is not clear, accurate or valid.

Assassin 33
1. "45" r5 -> r5c19 = 3 = {12} locked for r5 -> no 9 11(2)r5

2. 14(2) r5 = {59/68} = [5/6.5/8..]
2a.-> 11(2)r5 = {38/47} ({56} blocked by 14(2))
2b.17(3) = {359/368/467} ({458} blocked by 14(2))

3. 15(3)c1 must have 1 or 2 but no combo can have both -> no 1,2 r46c1
3a. 9(3)c1 {126} blocked -> = 3{15/24} (no 6): 3 locked for c1,n1

4.21(3)c1 = {489/579/678} = [5/8] -> 15(3)c1 {258} blocked

5. 15(3)n4 = {159/168/249/267}, 14(2)n4 = {59/68}
5a. -> when 14(2) = {68}, 15(3) = 9{15/24} -> 9 locked for n4 in 14(2) or 15(3).
5b. -> when 14(2) = {59} -> 15(3) = 6{18/27} -> 6 locked for n4 in 14(2) or 15(3)

6. "45" n1 -> r2c4 + r4c2 = 13 -> r4c2 = {4578}, r2c4 = {5689}

7. "45" n3 -> r2c6 + r4c8 = 17 = {89} -> no {89} in r4c6.
7a. both the 16(3)n23 and 15(3)n36 must have [8/9] and cannot have both (8+9=17)-> no {89} in r23c78

8. "45" n9 -> r8c6 + r6c8 = 14 -> min 5 in both outies

9. "45" n7 -> r6c2 + r8c4 = 8 -> max 7 in both outies

10. "45" r1234 -> r4c19 = 14 = {59/68}

11. "45" r6789 -> r6c19 = 11 -> r6c1 = {456789}, r6c9 = {234567}

Now for the hypotheticals that break this puzzle.

12. Putting steps 1,5,10 and 11 together we can easily see what the 4 combinations for 15(3)n4 mean:

15(3)n4[r456c1] -> 14(2) = {r5c23} and 13(3)=[r456c9] -> 11(2) = {r5c78}

12a.15(3)n4 = {159} ->

-15(3)[519] -> 14(2)={68} and 13(3)=[922] Blocked (2 2's)
-> no [519] 15(3)

-15(3)[915] -> 14(2)={68} and 13(3)=[526] -> 11(2)r5={47}

12b.15(3)n4 = {168} ->

-15(3)[618] -> 14(2)={59} and 13(3)=[823] -> 11(2)r5={47}

-15(3)[816] -> 14(2)={59} and 13(3)=[625] -> 11(2)r5={38/47}

12c.15(3) = {249} -

-15(3)[924] only -> 14(2) {68} and 13(3) = [517] -> 11(2)r5={38}:Blocked, (2 8's r5)
-> no {249} 15(3) -> no 4 15(3)

12d. 15(3) = {267} ->

-15(3)[627]only -> 14(2)={59} and 13(3)=[814] -> 11(2)r5 Blocked, (no 4 or 8 available)
-> no {267} 15(3)

13. In summary: 15(3)n4 = [915/618/816]= 1{59/68} (no 4,7)
13a.-> {59/68} locked for n4 in 15(3) and 14(2) -> no 5,8 r2c4 (step 6)
13a. r5c1 = 1, r4c1 = {689}, r6c1 = {568}

14. In summary:13(3)n6 = [526/823/625] = 2{38/56}
14a. r5c9 = 2, r4c9 = {568}, r6c9 = {356}

15. 9(3)c1 = {234}:locked for c1,n1
15a.-> 1 in n1 only in 17(3) = {179}
15b.-> r4c2 = 7, r2c4 = 6 (step 6)

16.r23c2 = {19}:locked for n1,c2

17.r23c3 = {58}:locked for n1,c3

18.13(2)n1 = [67]

19. 1 in r1 locked in r1c456->1 locked for n2 -> 7(2)n25 = {25/34} (no 1,6)

20.15(3)c2 = 8{25/34}:8 locked for n7,c2

21.21(3)n7 = {579}:locked n7c1

22.15(3)n47 = {348}:locked for c2 -> r9c23 = [23] -> r6c2= 3,r8c4 = 5 (step 9)

23.r78c2 = {48}:locked for n7, r78c3 = {16}

24.9(3)n458 = {234} -> r7c4 = 3, r6c34 = {24}:locked for r6

25.10(3)n254 = {127} -> r4c34 = [21]

the rest is straightforward

I got stuck on this puzzle when I first tried it and came back to it yesterday after getting a hint from Peter.

“Got stuck? Here is a vague hint in tiny font.
Hint: Forget about innies/outies (for now). The key is in another approach.
Not enough? Here's a more specific hint.
Hint: Look at N4 and N6 and there at c1 and c9. Look at the possible combinations and possible killer pairs.”
That solved it. The extra work prompted by these hints is mainly in steps 24 and 31. I don’t think I’d been making full use of R4C19 and R6C19 before. Thanks Peter!

I think the key step that I used to break it open is sufficiently different from Ed's method so here is my walkthrough.

Clean-up is used in various steps, using the combinations in steps 1 to 10 for further eliminations from these two cell cages; it is also used for the two cell split sub-cages that are produced by applying the 45 rule to R4, R6, N1, N3, N7 and N9. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. R1C23 = {49/58/67}, no 1,2,3

2. R1C78 = {29/38/47/56}, no 1

3. R34C5 = {16/25/34}, no 7,8,9

4. R5C23 = {59/68}

5. R5C78 = {29/38/47}(not {56} which would clash with R5C23), no 1,5,6

6. R67C5 = {49/58/67}, no 1,2,3

7. R9C23 = {14/23}

8. R9C78 = 10(2), no 5

9. 9(3) cage in N1 = {126/135/234}, no 7,8,9

10. 19(3) cage in N12, no 1

11. 20(3) cage in N3 = {389/479/569/578}, no 1,2

12. 10(3) cage in N254 = {127/136/145/235}, no 8,9

13. 9(3) cage in N458 {126/135/234}, no 7,8,9

14. 10(3) cage in N658 = {127/136/145/235}, no 8,9

15. 21(3) cage in N7 = {489/579/678}, no 1,2,3

16. 20(3) cage in N89 = {389/479/569/578}, no 1,2

17. 13(4) cage in N2 = 1{237/246/345}, no 8,9, 1 locked for N2, clean-up no 6 in R4C5

18. 25(4) cage in N8 = {1789/2689/3589/3679/4579/4678}, must contain at least two of 7,8,9 and contains three of 5,6,7,8,9

19. 45 rule on R5 2 innies R5C19 = 3 = {12}, locked for R5, clean-up: no 9 in R5C78

20. 45 rule on R1234 2 innies R4C19 = 14 = {59/68}

21. Valid combinations for R456C1 with R5C1 = {12} are {159/168/249/267} (cannot be {258} which would clash with R5C23), no 1,2,3 in R6C1, R456C1 must contain one of 4,5,6 and one of 7,8,9

22. Valid combinations for R456C9 with R5C9 = {12} are {139/157/238/247/256} (cannot be {148} which would clash with R5C78), no 1,2 in R6C9, R456C9 must contain one of 3,4,5 and one of 6,7,8,9

23. 12(3) in R789C9 must have 1 or 2, valid combinations are {138/147/156/237/246} ({129} not valid because it has both 1 and 2), no 9, must contain one of 3,4,5 and one of 6,7,8

24. 45 rule on R6789 2 innies R6C19 = 11 = {38/47/56}, clean-up: no 9 in R6C1, no 8,9 in R6C9
24a. Only combination in R456C1 with 5 is {159} -> no 5 in R4C1, clean-up: no 9 in R4C9 -> no {139} in R456C9
24b. From step 22, no {247} in R456C9 because 4,7 both in same cell -> no 4 in R6C9, clean-up: no 7 in R6C1
24c. R456C1 = {159/168/249} [8/9], R5C23 = {59/68} [8/9], killer pair 8,9 for N4
24d. 45 rule on N4 4 innies R4C23 + R6C23 = 16, must contain 1 or 2 and must contain 3,7 -> no 6 in R4C23 + R6C23 [Edited. 3 added]

25. 45 rule on N1 2 outies R2C4 + R4C2 = 13 = {49/58/67}, no 1,2,3
25a. Clean-up: no 4,5,7 in R2C4

26. 45 rule on N3 2 outies R2C6 + R4C8 = 17 = {89} -> no 8,9 in R2C8 and R4C6
26a. Killer pair 8,9 in R4C189 for R4

27. 45 rule on N7 2 outies R6C2 + R8C4 = 8 = {17/26/35/44}, no 8,9, clean-up: no 2 in R8C4

28. 45 rule on N9 2 outies R6C8 + R8C6 = 14 = {59/68/77}, no 1,2,3,4

29. 9 in C9 locked in R123C9 = 9{38/47/56}, clean-up: no 2 in R1C78

30. 9 in N6 locked in R46C8, locked for C8, clean-up: no 1 in R9C7

31. Remaining combinations for R456C9 are {157/238/256}. It looks like the 3 cages in C9 are meant to be effectively 9-11, 2-11 and 1-11 so let’s look at a contradiction move to see if {157} can be eliminated.
If R456C9 = {157}, R4C9 = 5, R5C9 = 1, R6C9 = 7 => R4C1 = 9, R5C1 = 2, R6C1 = 4 => R5C23 = {68} which clashes with R5C78 = {38} so R456C9 cannot be {157}
31a. R5C9 = 2, R5C1 = 1
31b. R46C1 = {68}/[95]
31c. R46C9 = {56}/[83]
31d. R123C9 = 9{38/47/56}
31e. R789C9 = 1{38/47/56}, 1 locked for N9, clean-up: no 9 in R9C7

32. R123C1 = {234}, locked for C1 and N1, clean-up: no 9 in R1C23

33. 7 in C1 locked in R789C1, locked for N7, R789C1 = 7{59/68}

34. 1 in N1 locked in R23C2, locked for C2, clean-up: no 4 in R9C3
34a. Only valid combination for R234C2 = {179} -> R4C2 = 7, R23C2 = {19}, locked for C2 and N1, clean-up: no 6 in R1C3, no 5 in R5C3

35. R9C2 = {234} -> R678C2 must contain two of 2,3,4 -> only possible combination = {348}, locked for C2 with 8 in R78C2, locked for N7 -> R9C2 = 2, R9C3 = 3, clean-up: no 5 in R1C3, no 6 in R5C3, no 6 in R789C1 = {579}, locked for C1 and N7, no 7,8 in R9C78 -> R9C78 = {46}, locked for R9 and N9, clean-up: no 5,7 in R789C9

36. R46C1 = {68}, locked for N4 -> R5C23 = [59], R1C23 = [67], clean-up: no 4,5 in R1C78 -> R1C78 = {38}, locked for R1 and N3

37. 1 in N2 locked in R1C456

38. R1C9 = 9 (hidden single in R1)

39. R789C9 = {138}, locked for C9 and N9 -> R46C9 = {56}, locked for C9 and N6 -> R23C9 = {47}, locked for N3

40. R78C2 = {48}, locked for C2 and N7 -> R6C2 = 3
40a. R78C3 = {16}, locked for C3 -> R8C4 = 5, clean-up: no 8 in R6C5

41. 6 in N5 locked in R456C5 = 6{38/47}, 6 locked for N5, clean-up: no 7 in R7C5
[Alternatively could have used X-wing with 6s in R46C19]

42. R23C3 = {58}, locked for C3 -> R2C4 = 6, clean-up: no 1 in R4C5 [Edited. "locked for C3" added for clarity"]

43. Only remaining combination for 9(3) cage in N458 = {234} -> R6C34 = {24}, locked for R6, R7C4 = 3, clean-up: no 9 in R7C5

44. Only remaining combination for 10(3) cage in N254 is R3C4 = 7, R4C34 = [21]
-> R6C34 = [42], R23C9 = [74], R1C4 = 4, clean-up: no 5 in R3C5, no 3 in R4C5

45. R1C1 = 2, R3C1 = 3, R2C1 = 4, R1C56 = {15}, locked for N2 -> R2C5 = 3, R3C5 = 2, R4C5 = 5, R1C56 = [15], R46C9 = [65], R46C1 = [86], R4C8 = 9, R23C6 = {89}, locked for C6 [Edited. "locked for N2" added for clarity"]

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



Was the intended solving path to use one or more contradiction moves or is there something else that we've all missed?

Hi guys,

I decided to tackle this one after Ruud's call for assistance in getting some player ratings for earlier Assassins. This puzzle was the active Assassin when I first discovered the sudocue.net site. I couldn't have imagined doing such a puzzle back then, but fortunately I've learnt a lot around here since. Yes, it can be solved elegantly, and without any hypotheticals or "contradiction moves". In fact, it's probably the shortest walkthrough I've ever written! Therefore, although this puzzle has had a fearful reputation in the past, the optimum route through the puzzle probably only deserves a rating of around 1.25.


Assassin 33 Walkthrough

Prelims:

a) 9(3) at R1C1 and R6C3 = {126/135/234} (no 7..9)
b) 13(2) at R1C2 and R6C5 = {49/58/67} (no 1..3)
c) 13(4) at R1C4 = {1237/1246/1345} (no 8,9); 1 locked for N2
d) 11(2) at R1C7 = {29/38/47/56} (no 1)
e) 20(3) at R1C9 and R7C7 = {389/479/569/578} (no 1,2)
f) 19(3) at R2C3 = {289/379/469/478/568} (no 1)
g) 10(3) at R3C4 and R6C6 = {127/136/145/235} (no 8,9)
h) 7(2) at R3C5 = {16/25/34} (no 7..9); no 6 in R4C5
i) 14(2) at R5C2 = {59/68}
j) 11(2) at R5C7 = {29/38/47} (no 1,5,6) (Note: {56} blocked by 14(2) at R5C2 (prelim i))
k) 21(3) at R7C1 = {489/579/678} (no 1..3)
l) 5(2) at R9C2 = {14/23}
m) 10(2) at R9C7 = {19/28/37/46} (no 5)

1. Innies R5: R5C19 = 3(2) = {12}, locked for R5
1a. cleanup: no 9 in R5C78

2. Outies N1: R2C4+R4C2 = 13(2) = {49/58/67} (no 1..3)

3. Outies N3: R2C6+R4C8 = 17(2) = {89}
3a. no 8,9 in R2C8 and R4C6 (CPE)

4. Outies N7: R6C2+R8C4 = 8(2) = {17/26/35/44} (no 8,9)

5. Outies N9: R6C8+R8C6 = 14(2) = {59/68/77} (no 1..4)

6. Innies R1234: R4C19 = 14(2) = {59/68}
6a. -> R4C19 and R4C8 form killer pair on {89} -> no 8,9 elsewhere in R4
6b. cleanup: no 4,5 in R2C4 (step 2)

7. Innies R6789: R6C19 = 11(2) = {29/38/47/56} (no 1)

8. 17(3) at R5C4 and R6C5 form hidden killer pair in N5 on {89}
8a. -> R6C5 = {89}; 17(3) at R5C4 = {359/368} (no 4,7)
(Note: {458} blocked by 14(2) at R5C2 (prelim i))
8b. 3 locked in 17(3) for R5 and N5
8c. cleanup: no 4 in R3C5; no 8 in R5C78; R7C5 = {45}

9. Naked pair (NP) at R5C78 = {47}, locked for N6
9a. cleanup: no 7 in R8C6 (step 5); no 4,7 in R6C1 (step 7)

10. 7 in C1 locked in 21(3) at R7C1 (prelim k) = {579/678} (no 4)
10a. 7 locked for N7

11. 4 in C1 locked in 9(3) at R1C1 (prelim a) = {234}, locked for C1 and N1
11a. cleanup: no 9 in R1C23; no 8,9 in R6C9 (step 7)

12. R5C19 = [12]
12a. cleanup: no 7 in R8C4 (step 4); no 9 in R6C1 (step 7)

13. 1 in N1 locked in R23C2
13a. -> 17(3) at R2C2 = {179}
13b. -> R4C2 = 7; R23C2 = {19}, locked for C2 and N1
13c. -> R2C4 = 6 (step 2)
13d. cleanup: no 6 in R1C3; no 5 in R5C3; no 4 in R9C3; no 1 in R4C5; no 2 in R6C2 (step 4); no 1 in R8C4 (step 4)

14. Hidden single (HS) in R6 at R6C6 = 7
14a. -> split 3(2) at R6C7+R7C6 = [12]
14b. cleanup: no 9 in R9C8; no 6 in R6C2 (step 4)

15. HS in N1 at R1C2 = 6
15a. -> R1C3 = 7
15b. cleanup: no 4,5 in R1C78; no 8 in R5C3

16. Innies C2: R59C2 = 7(2) = [52] (last combo/permutation)
16a. -> R59C3 = [93]
16b. cleanup: no 7,8 in R9C78

17. HS in C2 at R6C2 = 3
17a. -> R8C4 = 5 (step 4)
17b. cleanup: no 9 in R6C8 (step 5)

18. HS in C4 at R9C4 = 9
18a. cleanup: no 1 in R9C8; no 5 in R6C8 (step 5)

The rest is now just singles and cage sums

3x3::k:5888:2817:2817:4355:4355:4355:3590:3590:4872:5888:5888:5888:5644:4355:3598:4872:4872:4872:3858:5888:5644:5644:2838:3598:3598:4872:4634:3858:3858:3101:3101:2838:7456:7456:4634:4634:4132:4132:3101:3101:2838:7456:7456:2859:2859:4132:2606:3631:6448:6448:6448:4659:2612:2859:5430:2606:3631:3129:6448:827:4659:2612:2878:5430:4416:3631:3129:2883:827:4659:5446:2878:5430:4416:4416:4416:2883:5446:5446:5446:2878:

Am I looking at the right one - Assassin 34. I get the following steps - not too bad I thought - certainly no T&E required [but quite a few limited what if's on permutations - I must admit ]

I start numbering after I've set all marks based on internal cage combos

1. Cannot have combination {56} in cage 11(2) at r1c2 - conflicts with 14(2) at r1c7

2. Cannot have combination {29} in cage 11(2) at r8c5 - conflicts with cages 11(2) at r8c5 & 3(2) at r7c6

3. Naked pair {12} at r7c6 r8c6

4. Only combinations {347} {356} {239} {248} {149} {257} {158} {167} allowed in cage 14(3) at r2c6 - cannot have 8/9 in r3c7

5. Must use 1 in cage 19(5) at r1c9 -> 1 cannot be in r3c79

6. Must use 9 in cage 22(3) at r2c4 -> 9 cannot be in r3c6

7. 45 rule on column 5. Innies r6c5 r7c5 minus outies r1c4 r1c6 equals 6
7a. Max of r6c5+r7c5 is 17 so max of r1c4 r1c6 is 11.

9. 45 rule on column 7. Innies cells r1234597 equal 27 -> {124578} {123579} {124569} {123678} {134568} {234567}
9a. No valid combination with 5/6/7/8/9 in r2c7
9b. No valid combination with 5/6/7 in r3c7
9c. No valid combination with 5/6/7/8/9 in r9c7

10. 45 rule on column 9. Innies cells r1c9 r2c9 minus outies r4c8 r5c8 equals 5
10a Max of r1c9+r2c9 is 13 -> max of r4c8+r5c8 is 8.

11. 18(3) at r3c9 only combinations with 1 is {189} - not possible for 1 to be in r4c9

12. 45 rule on column 9. Outies r2c7 r2345c8 equal 14
12a Cage 10(2) at r6c8 restricts possible values in r2345c8 {1236} & {1234} not possible
12b r2c7 cannot be 4
12c r2c8 can't now be 8/9
12d r3c8 can't now be 8/9

13. No possible combinations left with 8 in 19(5) at r1c9

14. Must use 2 in cage 19(5) at r1c9 -> cannot have 2 in r3c79

15. No possible combinations left in 14(3) at r2c6 with 8 or 9

16. Must use 3 in cage 14(3) at r2c6 -> cannot have 3 in r3c5

17. 45 rule on N1. Innies r3c1 r3c3 equal 11
17a Only possibilities are {56}/[47]/[38]/[29]
17b r3c1 can't be 1/7/8/9

18. 45 rule on N2. Innies r3c5 minus outies r3c3 r3c7 equals -8
18a Max val in r3c5 is 5 so max of r3x3+r3c7 is 13
18b No valid combination with 8
19c Remembering that r3c13 total 11 r3c1 cannot now be 3

20. 45 rule on N2. Innies r2c4 r3c4 r2c6 r3c6 r3c5 equal 28
20a r3c13 limits possibilities in r3c456 - no possible combo with 6 at r3c4
20b.this combo now means 9 cannot be in 17(4) at r1c5, r1c6 or r2c5

21. 22(3) at r2c4 can now only be {679} or [5]{89} with 5 at r4c4

22. 9 locked in column 4 of N2 - nowhere else in the column or in r3c3

23. 12(2) at r7c4 can now only be {48} {57} - no 3
23a. 11(2) at r8c5 cannot now be {47} otherwise no valid combo left in 12(2)

24. 45 rule on column 5. r1c5 r2c5 r6c5 r7c5 total 23.
24a 11(3) at r3c5 doesn't allow a combo with {15}
24b 11(2) at r8c5 doesn't allow a combo with {35}
24c so no calid combo with 1/5 at r6c5 or 5 at r7c5

25. 17(4) at r1c4 {1457} only combo with 7 - but no valid pattern with a 7 at r1c4 or r1c6

26. h11(2) found at step 17 cannot now be {29} (no 9) - so remove the 2 from r3c1

27. h28(5) found at step 20 can't now have a combo with 2&7 because of 17(4) at r1c4. So no valid pattern with 6 @ r2c4
27a. 22(3) at r2c4 now cannot have 7 at r3c3
27b h11(2) found at step 17 can now only be {56}

28. Naked pair {56} @ r3c13 - nowhere else in n1 or row 3

29. Only combination left @ 14(3) at r2c6 is {347} - no 5 or 6

30. h28(5) found at step 20 can't now have a combo with 7 @ r2c4 or r3c4 or a 4 at r2c6 or r3c6. 1 must be in r3c5

31. 17(4) at r1c4 must now be {2456} (step 30 combos have either 3/8, so 17(4) can't have both).
31a no 2/3/8 in r1c4
31b no 3/6/7/8 in r1c5 or r2c5
31c no 3/8 in r1c6

32 Only possibilty in 22(3) at r2c4 is [5]{89} with 5 in r3c3

33. Only possibility in 14(3) ar r2c6 is [4]{37} with 4 locked in r3c7

34. 19(5) at r1c9 can only be {12367} - no 5 or 9

35. 14(2) at r1c7 can now only be {59} - no 6 or 8
35a.11(2) at r1c2 can't now be {29}

36. 15(3) at r3c1 can only nw be [6]{18}/{27}/{45} - no 3 or 9

37. 11(3) at r3c5 can only now be [1]{28}/{37}/{46} - no 5

38. No combo left in 18(3) at r6c7 with a 5

39. Innies found at step 24 can only now be {2489}/{2579}
39a no 2/3/4/6 in r6c5
39b no 3/4/6 in r7c5
39c 11(3) at r3c5 must now be [1]{37}/{46} - 2 and 8 locked - no 2 or 8

40. 25(4) at r6c4 - limited permutations left - no valid pattern with 7/8 at r6c4 or 9 at r6c6

41. Innies found at step 9 can only now be {124578} -> 5 only at r1c7, no 9 no 3
41a. 18(3) at r6c7 can now only be {369} - no 1/2/7/8
41b 10(2) at r6c8 - {19} not now possible so no 1

42. 17(4) at r1c4 now must be [25]{46} - 2 at r1c5 and 5 @ r2c5
42a 11(2) at r8c5 can't now be {56} - so no 6
42b 11(3) at r3c5 can't now be {137} (3 used in 42a) - so no 3/7
42c 12(2) at r7c4 can't now be {48} (8 used in 42a) - so no 4/8
42d. innies found at step 24 now has to be [2][5]{79} - no 8

43. 18(3) at r3c9 - only combo with 2 is {279} and can't have the 2 at r4c9

44. 29(4) at r4c6 - only possibility is {5789} with r45c6 {59} and r45c7{78} - no 7/8 in r45c6

45. 21(4) at r8c8 now has 1/2 in r9c7. no 1/2 anywhere else in the cage and r9c6 cannot be 3

46 25(4) at r6c4 has r6c5 {7/9} and r7c5 {7/9} - only combos with 7&9 are {1789}/{3679}/{4579} - no 2, 7 can't be in r6c6
. . . the rest falls out fairly easily from here

Haven't proofed this yet but here's a walkthrough.

Step 1. 12(4) cage at R4C3 = {1236|1245}. 29(4) cage at R4C6 = {5789}. R78C6 = {12}. R1C78 = {59|68}. 22(3) cage at R2C4+R3C34 = {589|679}.

Step 2. 18(3) cage at R6C7 must have 2 cells containing {56789}. (If it had only one, then the max cage-sum would be 3+4+9 = 16.) So the 18(2) cage forms a hidden quint on {56789} in conjunction with R145C7 => R239C7 = {1..4}.

Step 3. 45 rule on N3 => R3C79 = 12 = [39|48] => R23C6 = {47|56|37}. This means that R45C6 cannot be {57}. 45 rule on C6789 => R16C6 = 12 = {39|48} (again, this cannot be {57} because of R23C6). This means that R45C6 cannot be {89}. So R45C6 must contain exactly one of {57} and one of {89}. This means we have a hidden quad on {5789} within R123456C6!!! So R9C6 = {346}.

Step 4. 45 rule on N1 => R3C13 = 11 = {56} or [29] or [47] ([38] is blocked because R3C79 = [39|48], see step 3).

Step 5. 45 rule on N2 => R3C37 = R3C5 + 8 => maximum value of R3C5 = 5 (since max val of R3C37 = 13).

Step 6. 45 rule on N2 => R2C46 + R3C456 = 28(5). Obviously if the 28(5) cage has a 1 or 2 in it, then R3C5 = {12}. If it does not, the only possible combos are {34579} or {34678}, i.e. it must have both 3 and 4 in it. But this is impossible, because R3C7 = {34} "sees" all three cells R2C6 + R3C56. Therefore R3C5 = {12}. All combinations of 28(5) with 1 or 2 in them include a 9 => the 9 is locked in R23C4 within N2/C4/the 22(3) cage.

Step 7. So at this point there is only one square in N8 where the 9 can go (since 11(2) = {29} is blocked by the 3(2) cage). So R7C5 = 9 => 9 is locked in N5/the 29(4) cage in R45C6 => R16C6 = {48}, R45C6 = {(5|7)9}, R45C7 = {(5|7)8}.

Step 8. R78C4 = {48|57}, R89C5 = {38|56} ({47} is blocked by the 12(2) cage). 45 rule on N8 => R9C46 = 10 = [73|46]. Thus R78C4 + R9C4 contain a hidden {47} pair => {47} is excluded in R1..6C4 => R23C4 = {(6|8)9}, R3C3 = {57}, R3C1 = {46}.

Step 9. 45 rule on C1234 => R16C4 = 7 = {16|25}. Therefore the only spot left in C4 for the 3 is R45C4 => the 12(4) cage is {1236} with the 3 in R45C4.

Step 10. This leaves only two possible combinations for the 25(4) cage at R6C4: {1789} or {2689} => R6C6 = 8, R1C6 = 4, and the 4 is locked in C5/N5/the 11(3) cage within R45C5 => R45C5 = {4(6|7)}. This in turn forms a hidden {67} pair with the 25(4) cage => R45C6 = {59}!

Step 11. The rest is mop-up. R45C7 = {78}. R23C6 = {37} (see step 3), R3C7 = 4, R3C9 = 8, R9C6 = 6, R89C5 = {38}, R78C4 = {57}, R9C4 = 4, R23C4 = [89] (because this is the only spot for 8 in C4), R3C3 = 5, R3C1 = 6, R16C4 = [61], R45C4 = {23}, R45C3 = {16}. 45 rule on N2 => R3C5 = 1. R1C78 = {59}, R12C5 = [25], R6C5 = 7, R45C5 = {46}.

Step 12. In C9 the only spot for the 9 is R4C9 (can't be in a 11(3) cage) => R4C89 = [19], R45C6 = [59], R4C12 = {27}, R45C3 = [61], R45C4 = [32], R45C7 = [87], R45C5 = [46]. 45 rule on N4 => R6C23 = 13 = {49}, 45 rule on N6 => R6C78 = {36}, R6C19 = [52], R789C1 = {489}, R5C12 = [38], R1C23 = [38] (only place for 8 to go in N1).

Step 13. 11(3) cage in N9 must have a 1 in it (since it must have 1 and/or 2, & 2 is blocked) => R2C7 = 1, R1C1 = 1. 45 rule on C7 => R19C7 = 7 = [52], R1C8 = 9, R1C9 = 7, R89C8 = {58}. R89C58 form an x-wing on 8 => R789C1 = [849], and you carry on........

As I commented in the A34 and A71 threads on Ruud's site, I didn't manage to solve three Assassins when they first appeared. Now having caught up with my backlog of other walkthroughs I'm having another go at them.

I originally used a contradiction move for the key breakthrough but, after I'd finished I had another look and found an alternative more direct way; I've given both in step 17.

I was going to rate A34 at Easy 1.5 using my original step 17 but have reduced my rating to Hard 1.25 after finding the alternative.

Here is my walkthrough.

Prelims

a) R1C78 = {59/68}
b) R1C23 = {29/38/47} (cannot be {56} which clashes with R1C78), no 1,5,6
c) R67C2 = {19/28/37/46}, no 5
d) R67C8 = {19/28/37/46}, no 5
e) R78C4 = {39/48/57}, no 1,2,6
f) R78C6 = {12}, locked for C6 and N8
g) R89C5 = {38/47/56}, no 1,9
h) 22(3) cage at R2C4 = {589/679}, CPE no 9 in R3C56
i) R345C5 = {128/137/146/236/245}, no 9
j) 11(3) cage in N6 = {128/137/146/236/245}, no 9
k) R789C1 = {489/579/678}, no 1,2,3
l) R789C9 = {128/137/146/236/245}, no 9
m) 12(4) cage at R4C3 = {1236/1245}, no 7,8,9
n) 29(4) cage at R4C6 = {5789}
o) 19(5) cage in N3 = {12349/12358/12367/12457/13456}, 1 locked for N3

1. 45 rule on N1 2 innies R3C13 = 11 = [29/38/47/56/65], no 1,7,8,9 in R3C1

2. Min R23C6 = 7 -> max R3C7 = 7
2a. 45 rule on N3 2 innies R3C79 = 12 = [39/48/57/75], no 2,6, no 3,4 in R3C9

3. 45 rule on R6789 2 innies R6C19 = 7 = {16/25/34}, no 7,8,9

4. 45 rule on C1234 2 innies R16C4 = 7 = {16/25/34}, no 7,8,9

5. 45 rule on C6789 2 innies R16C6 = 12 = {39/48/57}, no 6

6. R678C7 = {189/279/369/378/459/468/567}, must contain at least two of 5,6,7,8,9
6a. Killer quint 5,6,7,8,9 in R145C7 and 18(3) cage in N69, locked for C7, clean-up: no 5,7 in R3C9 (step 2a)
6b. R678C7 can only contain two of 5,6,7 = {189/279/369/378/459/468}

7. Killer pair 8,9 in R1C78 and R3C9, locked for N3

8. 14(3) cage at R2C6 = {347/356}, no 8,9, CPE no 3 in R3C5
8a. R16C6 (step 6) = {39/48} (cannot be {57} which clashes with R23C6), no 5,7
8b. Killer quad 5,7,8,9 in R16C6, R23C6 and R45C6, locked for C6

9. Hidden killer pair 8,9 in R16C6 and R45C6 for C6 -> R45C6 must contain one of 8,9 -> R45C7 must contain one of 8,9
9a. R678C7 (step 6b) = {279/369/378/459/468} (cannot be {189} which clashes with R45C7), no 1
9b. Killer pair 8,9 in R45C7 and R678C7, locked for C7, clean-up: no 5,6 in R1C8
9c. 45 rule on N3 2 remaining innies R13C7 = 9 = [54/63]
9d. R678C7 = {279/369/378/459} (cannot be {468} which clashes with R13C7)

10. 45 rule on N2 2 outies R3C37 = 1 innie R3C5 + 8, IOU no 8 in R3C3, clean-up: no 3 in R3C1 (step 1)
10a. Max R3C37 = 13 -> max R3C5 = 5

11. 22(3) cage at R2C4 = {589/679}
11a. 5 of {589} must be in R3C34 (R3C34 cannot be {89} which clashes with R3C9), no 5 in R2C4

12. 45 rule on C89 3 innies R189C8 = 1 outie R2C7 + 21
12a. Min R189C8 = 22, no 1,2,3,4, 9 locked for C8, clean-up: no 1 in R67C8
12b. Max R189C8 = 24 -> max R2C7 = 3

13. 45 rule on C9 2 innies R12C9 = 2 outies R45C8 + 5
13a. Max R12C9 = 13 -> max R45C8 = 8, no 8
13b. Max R3C9 + R4C8 = 16 -> min R4C9 = 2

14. 45 rule on R123 3 innies R3C159 = 15 = {159/168/249/258} (cannot be {267/456} because R3C9 only contains 8,9)
14a. R3C13 = 11 (step 1) -> R3C19 cannot be 11 (I’ll call that an overlap IOU) -> no 4 in R3C5

15. 45 rule on N8 3 innies R7C5 + R9C46 = 19 = {379/469/478/568}
15a. 3 of {379} must be in R9C6 -> no 3 in R7C5 + R9C4

16. 25(4) cage at R6C4 = {1789/2689/3589/3679/4579/4678}
16a. 1 of {1789} must be in R6C4 -> no 1 in R6C5

My original step 17
17. 17(4) cage in N2 = {1268/1349/1358/2348/2456} (cannot be {1259} which clashes with R3C5, cannot be {1367/1457/2357} which clash with R23C6), no 7
17a. Cannot be {1268}, here’s how
17(4) cage = {1268} => 14(3) cage at R2C6 = {347} => R23C4 = [95] is impossible because no 8 in R3C3
17b. 17(4) cage = {1349/1358/2348/2456}
17c. Hidden killer pair 1,2 in 17(4) cage and R3C5 for N2 -> R3C5 = {12}

Alternative step 17 which avoids using a contradiction move
17. 17(4) cage in N2 = {1268/1349/1358/2348/2456} (cannot be {1259} which clashes with R3C5, cannot be {1367/1457/2357} which clash with R23C6), no 7
17a. 45 rule on N2 (from result of step 17) 5 innies R2C46 + R3C456 = 28 = {13789/15679/24679/25678} (cannot be {34579} which clashes with 14(3) cage at R2C6)
17b. 1,2 only in R3C5 -> R3C5 = {12}
17c. 17(4) cage = {1349/1358/2348/2456}

18. R3C159 (step 14) = {159/168/249/258}
18a. R3C5 = {12} -> no 2 in R3C1, clean-up: no 9 in R3C3 (step 1)
18b. 9 in 22(3) cage at R2C4 locked in R23C4, locked for C4 and N2, clean-up: no 3 in R6C6 (step 5), no 3 in R78C4

19. R7C5 = 9 (hidden single in N8), clean-up: no 3 in R1C6 (step 5), no 1 in R6C2
19a. 45 rule on N8 2 remaining innies R9C46 = 10 = [46/64/73], no 5,8 in R9C4
19b. R89C5 = {38/56} (cannot be {47} which clashes with R78C4), no 4,7

20. Naked pair {48} in R16C6, locked for C6, clean-up: no 6 in R9C4 (step 19a)

21. 9 in C6 locked in R45C6, locked for 29(4) cage at R4C6
21a. 8 in 29(4) cage at R4C6 locked in R45C7, locked for C7 and N6, clean-up: no 2 in R7C8

22. 22(3) cage at R2C4 = {589/679}
22a. 5 of {589} must be in R3C3 -> no 5 in R3C4

23. 17(4) cage (original step 17b or alternative step 17c) = {1358/2348/2456}
23a. 1 of {1358} must be in R1C45 (R1C456 cannot be {358} which clashes with R1C78), no 1 in R2C5

24. 45 rule on C5 3 remaining innies R126C5 = 14 = {167/248/257/347} (cannot be {158/356} which clash with R89C5)
24a. 1 of {167} must be in R1C5 -> no 6 in R1C5
24b. 7 of {167/257/347} must be in R6C5 -> no 3,5,6 in R6C5

25. 25(4) cage at R6C4 = {1789/2689} (cannot be {3589/3679} because 3,5,6 only in R6C4, cannot be {4579} which clashes with R45C6), no 3,4,5 -> R6C6 = 8, R1C6 = 4, clean-up: no 7 in R1C23, no 2,3 in R1C4 (step 4), no 1 in R1C45 (step 23), no 6 in R6C4 (step 4)
25a. 25(4) cage at R6C4 = {1789} (only remaining combination) -> R6C45 = [17], R1C4 = 6 (step 4), R1C7 = 5, R1C8 = 9, R3C9 = 8, R3C7 = 4 (step 2a), clean-up: no 2 in R1C23, no 3,8 in R12C5 (step 23), no 7 in R3C3 (step 1), no 6 in R6C19 (step 3), no 2,3 in R7C2, no 3 in R7C8
25b. R123C5 = [251], clean-up: no 6 in R89C5
25c. R3C37 = R3C5 + 8 (step 10), R3C5 = 1, R3C7 = 4 -> R3C3 = 5, R3C1 = 6

26. R9C6 = 6 (hidden single in C6), R9C4 = 4 (step 19a), clean-up: no 8 in R78C4
26a. Naked pair {57} in R78C4, locked for C4 -> R23C4 = [89]
26b. Naked pair {23} in R45C4, locked for N5 and 12(4) cage at R4C3
26c. 12(4) cage at R4C3 = {1236} (only remaining combination), no 4
26d. Naked pair {16} in R45C3, locked for C3 and N4, clean-up: no 4 in R7C2
26e. Naked pair {78} in R45C7, locked for C7 and N6

27. R678C7 (step 9d) = {369} (only remaining combination), locked for C7

28. 21(4) cage at R8C8 = {2568} (only remaining combination) -> R9C7 = 2, R2C7 = 1, R89C8 = {58}, locked for C8 and N9, clean-up: no 2 in R6C8

29. R4C9 = 9 (hidden single in C9), R4C8 = 1 (cage sum), R45C3 = [61], R45C5 = [46], R45C6 = [59]

30. R3C9 = 6 -> R4C12 = 9 = {27}, locked for R4 and N4 -> R45C4 = [32], R45C7 = [87], clean-up: no 8 in R7C2

31. R6C9 = 2 (hidden single in R6), R5C89 = 9 = [45], R6C1 = 5 (step 3), clean-up: no 6 in R67C8
31a. R67C8 = [37], R678C7 = [639], R78C4 = [57], R9C9 = 1

32. Naked pair {48} in R78C1, locked for C1 and N7, R7C3 = 2, R7C6 = 1, R7C2 = 6, R6C2 = 4

and the rest is naked singles

3x3:d:k:5889:2562:5889:2307:2307:5124:5381:5381:5382:2562:5889:3335:5889:5124:5381:5382:5382:5382:2562:2824:3335:3335:5124:5124:2057:5382:4618:6411:2824:6411:3340:2057:2057:2829:2829:4618:0000:6411:2831:3340:3340:3088:3088:4881:4618:0000:0000:6411:2831:2578:2578:3088:4881:4881:0000:0000:0000:2580:2580:3605:3605:5654:4119:0000:0000:0000:2584:2574:5654:5654:2067:4119:0000:0000:0000:2584:5654:2574:2067:4119:4119:

Congratulations guys for finishing it off - great way for Peter to bring up his 100 and Para's 50 =D> .

PANIV is the hardest of the tag Killers - at least according to the amount of grunt time for SumoCue to find a solution- just piping UTA. However, still way short of nd's #9 (still one of my all time favs) - takes SumoCue over twice as long to get it out. (BTW - good luck nd on your new job.)

Here's a condensed walk-through for PANIV, with just the essential steps (just about all of them!). I've taken the liberty of slightly changing a couple to make them clearer and added some at the end just to finish.

PANIV - condensed walkthrough : (1-9 cannot repeat on diagonals)
Preliminary steps (easy way, without writing down all poss. combinations)
0a. 10(2) at r6c5, r7c4, r8c4, r8c5 : no 5
0b. 10(3) at r1c2 : no 8,9
0c. 11(2) at r3c2, r5c3, r4c7 : no 1
0d. 9(2) at r1c4 : no 9
0e. 21(3) at r1c7 : no 1,2,3
0f. 8(3) at r4c5 : no 6,7,8,9
0g. 8(2) at r8c8 : no 4,8,9
0h. 14(2) at r7c6 : no 1,2,3,4,7
0i. 19(3) at r5c8 : no 1

1. 45 on r123456 : r56c1 + r6c2 = 15(3)
1a. N7 is 45(9) cage

2. 45 on N4 : r4c2 + r5c3 = 5(2) = {23} -> 2,3 locked in N4

3. 45 on N6 : r3c9 + r5c6 = 15(2) = {69|78}

4. 11(2) at r3c2 : r3c2 = {89}
4a. "45" on n4: 2 outies r3c2 + r6c4 = 17 = {89} only
4b. -> no 8,9 at r3c4 and r6c2 (CPE)
4c. no 8,9 for R8C2 (and theoretically for R3C7), over D/ (CPE)

5. 8(3) at r4c5 : 8(3) = 1{25|34} -> 1 locked in 8(3) -> no 1 in r5c5 because of D/

6. 45 on r123 : r3c279 = 18(3) -> min. r3c29 = 14 -> r3c7 = {1234}

7. 12(3) at r5c6 : r56c7 = {12345}

9. deleted

10. N4 : 25(4) = 7{189|459|468} -> 7 locked in 25(4)

11. 45 on N5 (2 innies, 1 outie) : r3c7 + 14 = r5c6 + r6c4
11a. r5c6 + r6c4 : minmax 15..17
11b. r3c7 : minmax 1..3 -> no 4 in r3c7

12. 45 on N5 (4 outies) : r5c3 + r356c7 = 9(4) (doubles allowed!) -> r5c3 = {23} -> r356c7 = 6 or 7 -> 1,2 locked in r356c7 for c7, no 5 in r56c7, no 3 in r5c7

13. Cleanup : no 6,7 in r8c8, no 9 in r4c8

15. 45 on N9 : r7c78 + r8c7 = 21(3) = {489|579|678} -> no 1,2,3

16. 45 on N8 : r78c6 + r9c5 = 15(3) = 5{19|28|37|46} -> 5 locked in 15(3) -> no 5 in r7c8 (seen by all poss. for 5 in N8)

17. 45 on N12 : r3c2 + r2c6 = 15(2) = [87]|[96] -> r2c6 = {67}

18. 21(3) at r1c7 : 21(3) = 7{59|68} (no 4)-> 7 locked in 21(3) -> no 7 in r2c789 and r1c456

19. no 2 in r1c45

20. N3 : 4 locked in 21(5) = 4{1259|1268|1358|1367|2357}

22. N2 : combination check {67} in r2c6 : 20(4) : {2567|3467} not poss.

23. N9 : combination check between 16(4) and 8(2) : {1258|1456|2347} not poss. for 16(4)

25. 45 n3 -> r2c6 + 3 = r3c79
25a. r2c6 = 6 -> r1c78 = {78} and r3c79 = 9 = [36] ([18/27] blocked by r1c78)
25b. r2c6 = 7 -> r1c78 = {59} and r3c79 = 10 = [28/37] ([19] blocked by r1c78) [postscript: [28] also blocked by step 17!]
....................................= {68} and r3c79 = 10 = [19/37] ([28] blocked by r1c78)

26. from step 3 r5c6 + r3c9 = 15(2) = {69|78}. combining this with steps 25ab ->
26a. r5c6 + r3c9 = [69] -> r3c79 = [19] -> r2c6 = 7
26b....................= [96] -> r3c79 = [36] -> r2c6 = 6
26c....................= [78] -> r3c79 = [28] -> r2c6 = 7 -> blocked - 2 7's in c6
26d....................= [87] -> r3c79 = [37] -> blocked - r56c7 = {13} with 8 in r5c6

steps 27-31 come from steps 26 - 26d.
27.-> r5c6 = {69}
27a. r56c7 = 2{1/4}: 2 locked for c7,n6
27b. no 9 r4c7

28. -> r3c9 = {69}

29. -> r25c6 = [76/69] = 6{7/9}: 6 locked for c6
29a. no 4 r6c5
29b. no 8 r7c7
29c. no 4 r8c5

30. -> combined with step 25ab -> r1c78 = {68/78} = 8{6/7}:8 locked for r1, n3
30a. -> no 1 r1c45

31. -> 4 innies n3 = {1689/3678} = 68{19/37}: 6 locked for n3

32. 21(3)n23 = {678}
32a. -> no 6 r1c45

33. 9(2)n2 = {45}:locked for n2, r1

34. 20(4)n2 = {1289/1379/2369/2378}

37. no 6,9 in r5c9 and no 9 in r3c6 because r3c9 and r5c6 are a naked pair {69} (from step 3) (CPE)

38. 45 on N5 (1 outie, 2 innies) : r3c7 + 14 = r5c6 + r6c4
38a. if r3c7 = 1 then r5c6 + r6c4 = 15(2) = [96]
38b. if r3c7 = 3 then r5c6 + r6c4 = 17(2) = [89]
38c. -> 9 locked in r5c6 and r6c4 for N5

39. no 1 in r6c56

42. {47} not poss. in r4c78
Explanation: 1 in r3c7 means {24} in r56c7, 3 in r3c7 means {14} in r4c56 -> no 4 in r4c78

43. N6 : 19(3) : {568} not poss. because of 11(2) -> no 5 in 19(3)

44. 8(3) at r3c7 is {134} -> 4 locked for N5 and r4 and no 3 in r5c5 and r4c7
Explanation: 8(3) = {125} means r3c7 = 1, r4c56 = {25}, r4c2 = 3 -> no poss. combination left for r4c78

45. no 8 in r4c8, no 6 in r6c5

46. 5 in N5 in 13(3) = 5{17|26} no 3,8
46a. -> 8 locked in r6c456 for r6

47. 5 locked in r78c6 for N8

48. 5 locked in r6c123 for N4

50. R3C7 = 3
50a.R4C56 is not 3. R4C56=3 -->> R4C2 = 2 -->> R3C2 = 9 -->> R3c9 = 6 -->> R3C7=3 (step 26b) -->> contradiction 2 3's in 8(3) R3C7
50b R4C56={14} locked for N5 and R4

51. r3c9 = 6 (step 26b), r1c67 = {78}:locked for r1,n3 and r2c6 = 6, 13(3)n5 = {256}..the rest unravels pretty quickly - so when you get stuck, try these:
{56} is blocked from 11(2)r4 by r4c5,
hidden single 9 in r1c9
{1357} is blocked from 16(4)n9
hidden single 4 on D\ in r3c3,
that horrible zig-zag cage 23(4)n1 = {1679}only, then its singles

Thanks Ed for PANIV, another of your interesting series of interestingly-named "tags" to go with ULURU, UTA and UTA2.

When I looked at this puzzle again last month I found that my files only included the first 24 steps of the original "tag" from Nasenbaer, plus a very short addition from Para. I must have been too busy then to continue looking at the "tag". Therefore I set up new files and started again from the beginning.

Ed introduced this puzzle as a "tag" so that's why it was solved as one at the time. Still I thought I'd have a try at it because I hadn't taken part in the original "tag" and because the condensed "tag" was the only walkthrough posted.

I did some fairly difficult analysis but it didn't feel as heavy as some steps in the "tag" because I didn't try to do so much in any step. My final breakthrough (step 31) was interesting and very different from anything in the "tag".

Here is my walkthrough for PANIV. Thanks Ed for pointing out a minor error, which I've corrected, and an alternative way to do step 13.

Prelims

a) R1C45 = {18/27/36/45}, no 9
b) R34C2 = {29/38/47/56}, no 1
c) R4C78 = {29/38/47/56}, no 1
d) 11(2) cage at R5C3 = {29/38/47/56}, no 1
e) R6C56 = {19/28/37/46}, no 5
f) R7C45 = {19/28/37/46}, no 5
g) R7C67 = {59/68}
h) R89C4 = {19/28/37/46}, no 5
i) 10(2) cage at R8C5 = {19/28/37/46}, no 5
j) 8(2) cage in N9 = {17/26/35}, no 4,8,9
k) 10(3) cage in N1 = {127/136/145/235}, no 8,9
l) 21(3) cage at R1C7 = {489/579/678}, no 1,2,3
m) 8(3) cage at R3C7 = {125/134}
n) 19(3) cage in N6 = {289/379/469/478/568}, no 1

1. N7 must be a 45(9) cage
1a. 45 rule on R789 3 outies R5C1 + R6C12 = 15

2. 45 rule on N5 2 innies R5C6 + R6C4 = 1 outie R3C7 + 14
2a. Min R5C6 + R6C4 = 15, no 1,2,3,4,5 in R5C6 + R6C4, clean-up: no 6,7,8,9 in R5C3
2b. Max R5C6 + R6C4 = 17 -> max R3C7 = 3

3. 45 rule on N12356 2 outies R4C2 + R5C3 = 5 = {23}, locked for N4
3a. 45 rule on N12356 2 innies R3C2 + R6C4 = 17 = {89}, CPE no 8,9 in R3C4 + R6C2, no 8,9 in R8C2 using D/
[Alternatively 45 rule on N4 (using step 1a) 2 remaining innies for step 3 and 2 outies for step 3a.]

4. 8(3) cage at R3C7 = {125/134}, CPE no 1 in R5C5 using D/

5. 45 rule on N6 2(1+1) outies R3C9 + R5C6 = 15 = {69/78}

6. 45 rule on N12 1 innie R2C6 = 1 outie R4C2 + 4, R4C2 = {23} -> R2C6 = {67}
6a. 21(3) cage at R1C7 = {579/678} (cannot be {489} because R2C6 only contains 6,7), no 4, CPE no 7 in R1C456 + R2C789, clean-up: no 2 in R1C45

[I originally used 45 rule on N36 1 outie R2C6 = 3 innies R356C7 here but replaced it by the following step which gives the same result plus an extra elimination.]
7. 45 rule on N5 4(3+1) outies R5C3 + R356C7 = 9 = 2{124}/3{123}, no 5,6,7,8,9 in R56C7, 1,2 locked for C7, clean-up: no 9 in R4C8, no 6,7 in R8C8
7a. 3{123} -> no 3 in R5C7

8. 45 rule on N9 3 innies R7C78 + R8C7 = 21 = {489/579/678}, no 1,2,3
8a. 7 of {579/678} must be in R7C8 (R7C78 cannot be {59/68} which clash with R7C67, CCC) -> no 7 in R8C7

9. 25(4) cage in N4 = {1789/4579/4678}, 7 locked for N4

10. 45 rule on N1 2 outies R23C4 = 1 innie R3C2 + 1, R3C2 = {89} -> R23C4 = 9,10, no 1 in R2C4

11. 5 in N8 only in R78C6 + R9C5, CPE no 5 in R7C8

12. 45 rule on N2 3 innies R2C46 + R3C4 = 16, max R2C6 + R3C4 = 13 -> min R2C4 = 3

13. 45 rule on R123 3 innies R3C279 = 18
13a. 45 rule on N3 2 innies R3C79 = 1 outie R2C6 + 3
13b. R2C6 = {67} -> R3C79 = 9,10 = [18/36/19/37] (cannot be [27] which clashes with 21(3) cage at R1C7 = {78}6, cannot be [28] because R3C279 cannot be [828]) -> no 2 in R3C7
[While adding this walkthrough to the archive Ed pointed out the alternative
21(3) cage at R1C7 (step 6a) = {579/678} must contain 7 in R2C6 and/or 8 in R1C78 -> R3C9 + R5C6 (step 5) = [69/78/96] (cannot be [87] which clashes with 21(3) cage
R5C6 + R6C4 = R3C7 + 14 (step 2), R5C6 + R6C4 = {69/89} = 15,17 -> R3C7 = {13}, also 9 locked in R5C6 + R6C4 for N5.]

14. 2 in C7 only in R56C7, locked for N6, clean-up: no 9 in R4C7
14a. 12(3) cage at R5C6 = {129/237/246}, no 8, clean-up: no 7 in R3C9 (step 5)

15. R3C279 = 18 (step 13) = {189/369}, 9 locked for R3
15a. 13(3) cage at R2C3 = {148/157/238/247/256/346} (cannot be {139} which clashes with R3C7), no 9

16. 21(5) cage in N3 must contain 4 = {12459/23457} (cannot be {13458/13467} which clash with R3C7, cannot be {12468} which clashes with R3C279 which must have 1 or 6 in R3C79), no 6,8, 5 locked for N3

17. 21(3) cage at R1C7 (step 6a) = {678} (only remaining combination), 8 locked for R1 and N3, CPE no 6 in R1C456, clean-up: no 1,3 in R1C45, no 7 in R5C6 (step 5)
17a. Naked pair {69} in R3C9 + R5C6, CPE no 6,9 in R3C6 + R5C9

18. Naked pair {45} in R1C45, locked for R1 and N2

19. 12(3) cage at R5C6 (step 14a) = {129/246}, no 3

20. R2C6 = {67} -> R3C79 = 9,10 = [36/19] (step 13b) -> R2C6 + R3C79 = [636/719]
20a. R2C46 + R3C4 = 16 (step 12) = {169/268/367} (cannot be {178} which clashes with R2C6 + R3C79), 6 locked for N2, CPE no 6 in R2C3

21. 18(3) cage at R3C9 = {189/369/459/468/567} (cannot be {378} because R3C9 only contains 6,9)
21a. 3 of {369} must be in R5C9 -> no 3 in R4C9

22. R2C46 + R3C4 = 16 (step 20a) = {169/367} (cannot be {268} = [862] which clashes with R5C6 + R6C4, ALS combo blocker), no 2,8

23. R2C6 + R3C79 = [636/719] (step 20), R3C9 + R5C6 = {69} (step 17a) -> R25C6 = [69/76], 6 locked for C6, clean-up: no 4 in R6C5, no 8 in R7C7, no 4 in R8C5

24. R7C78 + R8C7 (step 8) = {489/579/678}
24a. 6 of {678} must be in R7C7 -> no 6 in R7C8 + R8C7
24b. 9 of {489} must be in R7C7, 7 of {579} must be in R7C8 -> no 9 in R7C8

25. 1,2 in N6 only in R45C9 + R56C7
25a. 45 rule on N6 4 innies R45C9 + R56C7 = 15 = {1239/1248/1257}, no 6

26. 13(3) cage at R2C3 (step 15a) = {148/157/247/256/346} (cannot be {238} which clashes with R3C279 = [819/936], step 15)
26a. 1 of {148/157} must be in R3C4 (R23C3 cannot be {15} which clashes with 10(3) cage in N1) -> no 1 in R23C3

[I’d forgotten about this 45, it’s so long since I originally used it.]
27. R5C6 + R6C4 = R3C7 + 14 (step 2)
27a. R3C7 = {13} -> R5C6 + R6C4 = 15,17 = [69/98], 9 locked for N5, clean-up: no 1 in R6C56

28. 9 in N5 only in R5C6 + R6C5
28a. 45 rule on N5 4 innies R4C56 + R5C6 + R6C4 = 22 = {1489/2569/3469} (cannot be {2389} because 8(3) cage at R3C7 cannot contain both of 2,3)
28b. R6C56 = {28/37} (cannot be [64] which clashes with R4C56 + R5C6 + R6C4), no 4,6
28c. 13(3) cage in N5 = {148/157/247/256} (cannot be {238/346} which clash with R4C56 + R5C6 + R6C4), no 3

29. R2C6 + R3C79 = [636/719] (step 23), R2C46 + R3C4 (step 22) = {169/367} -> R23C4 + R2C6 + R3C79 = [37][636] / [91][636] / [36][719] / [63][719], no 7 in R2C4, CPE no 3 in R3C56

30. 19(3) cage in N6 = {379/469/478/568}
30a. 3 of {379} must be in R6C89 (R6C89 cannot be {79} which clashes with R6C4 + R6C56, Killer ALS block), no 3 in R5C8

[When I did steps 28 and 30 I didn’t realise how important those eliminations of 3 from R5 were; then I spotted the next step which cracks this puzzle.
With hindsight step 29 wasn’t needed but I’ve left it in anyway.]

31. 45 rule on N12356 1 innie R3C2 = 1 outie R5C3 + 6 -> R3C2 + R5C3 = [82/93]
[Alternatively 45 rule on N4 (using step 1a) ...]
31a. 9 in R3 only in R3C29, 3 in R5 only in R5C39, R3C2 + R5C3 contains both or neither of 3,9 -> R35C9 must contain both or neither of 3,9
31b. R3C2 + R5C3 = [93] (R35C9 cannot be [93] because no 6 in R4C9), R4C2 = 2, R6C4 = 8, placed for D/, R3C9 = 6, R3C7 = 3 (step 15), placed for D/, R5C6 = 9 (step 5)
31c. R2C6 = 6 (hidden single in C6), R1C9 = 9 (hidden single in R1), placed for D/, R6C8 = 9 (hidden single in N6), R56C7 = {12} (hidden pair in C7), locked for N6
31d. Naked pair {78} in R1C78, locked for R1 and N3
31e. Clean-ups: no 8 in R4C8, no 2 in R6C56, no 2 in R7C5, no 5 in R7C7, no 2 in R89C4, no 1 in R8C5, no 5 in R8C8

32. 8(3) cage at R3C7 = {134} (only remaining combination) -> R4C56 = {14}, locked for R4 and N5, clean-up: no 7 in R4C78

33. 9 in N9 only in R78C7 -> R7C78 + R8C7 (step 8) = {489/579}, no 6 -> R7C7 = 9, R7C6 = 5, clean-up: no 1 in R7C45

34. Naked pair {37} in R6C56, locked for R6 and N5
34a. R4C8 = 3 (hidden single in R4), R4C7 = 8, R1C78 = [78]

35. R7C78 + R8C7 (step 33) = {579} (only remaining combination) -> R7C8 = 7, R8C7 = 5, R2C7 = 4, R9C7 = 6, R8C8 = 2, placed for D\, clean-up: no 3 in R7C45, no 4 in R8C4, no 8 in R9C6

36. Naked pair {15} in R23C8, locked for C8 and N3 -> R2C9 = 2, R9C8 = 4, R5C8 = 6, R6C9 = 4 (step 30), R5C5 = 5, placed for both diagonals, R4C4 = 6, placed for D\, R5C4 = 2, R45C9 = [57], R56C7 = [12], R7C4 = 4, R7C5 = 6, R2C8 = 1, placed for D/, R3C8 = 5, R4C6 = 4, placed for D/, R4C5 = 1, R7C3 = 2, R9C1 = 7, R8C2 = 6, R4C13 = [97], clean-up: no 3 in R8C4, no 3 in R8C5

37. Naked pair {13} in R1C12, locked for R1 and N1 -> R1C3 = 6, R1C6 = 2, R2C1 = 5, R2C3 = 8, R2C2 = 7, placed for D\, R6C6 = 3, placed for D\, R1C1 = 1, placed for D\, R9C9 = 8, R1C2 = 3, R3C3 = 4, R3C4 = 1 (step 26), R2C4 = 9 (cage sum)

and the rest is naked singles, not using the diagonals.

Rating comment. It was hard to know how to rate my walkthrough for PANIV; I've decided on 1.5 based on steps 13, 20, 22, 23 and 31. Step 29 was also a hard one but, as I've commented, I found that it wasn't needed for the breakthrough so I haven't taken it into account.

Many thanks to Andrew for trying this puzzle and rekindling my interest. Some really nice moves by Andrew. He found some nice "45"s and made the key elimination quite early (2 from r3c7). Step 31 is a ripper!

This time, I found a very different way again from after Andrew's step 12. Have included the whole start to optimise. What a difference for Andrew and I 4 years after PANIV first appeared!

Prelims:
i. 10(3)n1: no 8,9
ii. 9(2)n2: no 9
iii. 21(3)r1c7: no 1,2,3
iv. 11(2)r3c2: no 1
v. 8(3)r3c7: no 6,,9
vi. 11(2)n6: no 1
vii. 11(2)r5c3: no 1
viii. 19(3)n6: no 1
ix. 10(2)n5: no 5
x. 3 10(2) cages n8: no 5
xi. 14(2)r7c6: no 1..4,7
xii. 8(2)n9: no 4,8,9

1. "45" on n12356: 2 outies r4c2 + r5c3 = 5
1a. = {23} only: both locked for n4
1b. r3c2 = (89) (cage sum)
1c. r6c4 = (89) (cage sum)

2. "45" on n12: 2 innies r2c6+r3c2 = 15 = [78/69]
2a. r2c6 = (67)

3. 21(3)r1c7 must have 6/7 for r2c6 = {579/678}(no 4) = [8 in r1c78 and/or 7 in r2c6 (important for next step)]
3a. must have 7 -> no 7 in r1c456 nor r2c789
3b. no 2 in 9(2)n2

4. "45" on n6: 2 outies r3c9+r5c6 = 15
4a. but [87] clashes with 21(3)r1c7 (step 3)
4b. r3c9+r5c6 = {69}/[78]
4c. r3c9 = (679), r5c6 = (689)

5. "45" on n5: 1 outie r3c7 + 14 = 2 innies r5c6+r6c4
5a. -> min. 2 innies = 15 = [69]/{89} = 15/17
5b. -> r3c7 = (13)
5c. innies must have 9: 9 locked for n5
5d. no 1 in 10(2)n5

6. "45" on n123: 1 outie r4c2 + 7 = 2 innies r3c79
6a. = [2][36]/[3][19/37]
6b. from step 6a, must have 3 -> no 3 in common peers in r4c567, r8c2 (from D/)
6c. no 8 in r4c8

7. 8(3)r3c7 = {125/134} -> r4c56 = {25/14} = [1/5,2/4..](no eliminations yet)

8. 13(3)n5: {157/247} blocked by r4c56
8a. = {148/238/256/346}(no 7)

9. 7 in n5 only in 10(2) = {37}: both locked for r6, n5

10. "45" on n36: 1 outie r2c6 = 3 innies r356c7
10a. -> r356c7 = 6/7 = {123/124}(no 5..9)
10b. must have 1 & 2: both locked for c7
10c. & 2 locked for n6

11. r6c7 = 2 (hsingle r6)
11a. r5c67 = 10 = [64/91] (no 3,8)
11b. no 7 in r3c9 (outies n6 = 15)
11c. no 9 in 11(2)n6

12. 19(3)n6: can't be {379} since 3,7 only in r5c8
12a = {469/478/568}(no 3) = [4/8,4/6..]

13. "45" on n6: 3 remaining innies r45c9 + r5c7 = 13
13a. must have 1/4 for r7c6
13b. {148/346} blocked by 19(3)n6
13c. = {139/157}(no 4,6,8)
13d. -> r5c7 = 1
13e. r5c6 = 9 (cage sum)

cracked

3x3::k:2304:2304:4866:4866:2820:2820:6662:6662:6662:3593:1290:1290:4866:4109:4109:4109:2320:6662:3593:4371:2324:2324:4118:4118:3864:2320:6662:3593:4371:6429:3102:3102:4118:3864:2320:3875:2340:4371:6429:3623:1832:1832:3864:1835:3875:2340:4398:6429:3623:3623:3634:3634:1835:2357:6198:4398:6429:6429:2362:2362:2620:2620:2357:6198:4398:2369:2369:4163:4163:3653:1862:1862:6198:6198:6198:1611:1611:3653:3653:3919:3919:

Pretty straightforward after the past couple of weeks' puzzles.

Step 1. R56C8 = {16|34} (cannot be {25} as this would block all combos in the 9(3) cage above) => R234C8 = {2(16|34)}. The two cages together block {12346} in the rest of C8 => R8C89 = [52], R179C8 = {789} => R7C78 = [19|37].

Step 2. R8C56 = {79}. R7C56 = {18|36} (can't be {45} because it conflicts with the 6(2) = {15|24} in N8), forming a hidden {13} pair with R7C7 => {13} is eliminated in the rest of R7.

Step 3. 45 rule on N9 => R6C9 + R9C6 = 10 => R67C9 = [54], R9C6 = 5 (only combination no longer blocked!!), R9C45 = {24}.

Step 4. 45 rule on N5 => R46C6 = 12 => R6C67 = [86], R4C6 = 4, R56C8 = {34}, R234C8 = {126}, R45C9 = {78}, R89C9 = [96], R7C78 = [37], R89C7 = {18}, R4C8 = 1, R1C8 = 8, R123C9 = {139}, R1C7 = 5, R45C7 = {29}, R23C7 = [74].

Step 5. 45 rule on C1234 => R469C5 = 9 = [324] (only combo not blocked). R4C4 = 9, R9C4 = 2, , R45C7 = [29], R7C56 = [81], R78C4 = [63], R8C3 = 6, R5C56 = [16], R56C4 = [57].

Step 5. The rest is mop-up. R3C34 = [81] (only place in C4 for the 1 to go), R12C4 = [48], R1C3 = 7, R1C12 = {36}, R2C23 = {14}, R1C56 = [92], R1C9 = 1, R3C56 = [57], R2C56 = [63], R23C8 = [26], R23C9 = [93], R2C1 = 5, R34C1 = [27], R3C2 = 9, R45C9 = [87], R4C23 = [65], R5C2 = 2, and you carry on........

3x3:d:k:2816:3585:3585:5379:5379:5379:1030:1030:3336:2569:2816:2059:2059:5379:3342:3342:3336:3601:2569:2835:2816:4373:4373:4373:3336:2585:3601:4635:2835:6685:6686:4373:6686:5153:2585:5667:4635:4635:6685:6685:6686:5153:5153:5667:5667:4635:814:6685:6686:5169:6686:5153:2868:5667:2870:814:4920:5169:5169:5169:5180:2868:2110:2870:4920:3137:3137:4931:836:836:5180:2110:4920:2633:2633:4931:4931:4931:2894:2894:5180:

A nice puzzle, Ruud. I like Killer-X. Thanks a lot.

But the cages lying directly on the diagonals and minimal 45-checks made it too easy.

OK, here is a walkthrough

Edit: Preliminary steps included and typo corrected

Walkthrough SKX 4

0. Preliminary steps (as requested by Andrew because I'm too lazy )
0a. 11(3) at r1c1 : no 9
0b. 10(2) at r2c1, r3c8, r9c2 : no 5
0c. 14(2) at r1c2, r2c9 : no 1,2,3,4,7
0d. 11(2) at r3c2, r7c1, r6c8, r9c7 : no 1
0e. 8(2) at r2c3, r7c9 : no 4,8,9
0f. 19(3) at r9c1 : no 1
0g. 26(4) at r4c3 : no 1
0h. 13(2) at r2c6 : no 1,2,3
0i. 12(2) at r8c3 : no 1,2,6
0j. 20(3) at r7c7 : no 1,2
1. r67c2 = {12} -> 1,2 locked for c2
1a. -> no 9 in r34c2
2. r8c67 = {12} -> 1,2 locked for r8
3. r1c78 = {13} -> 1,3 locked for r1 and N3
4. 45 on N7 : r7c2 + r8c3 = 5(2) = [14|23]
5. 45 on N9 : r8c7 + r7c8 = 6(2) = [15|24]
6. 45 on N3 : r2c7 + r3c8 = 14(2) = [59]|{68}
7. 45 on N1 : r2c3 + r3c2 = 10(2) = [28|64]|{37}
8. Cleanup: r8c4 = {89}, r7c9 = {1235}, r6c8 = {67}, r4c8 = {124}, r2c6 = {578}, r2c4 = {1256}, r4c2 = {3478}
9. 5,6,8,9 locked in r2c79 and r3c89 for N3
10. N3 : 13(3) = {247} -> 2,4,7 locked for N3 and D/
11. N7 : 19(3) = {568} -> 5,6,8 locked for N7 and D/
12. 45 on D/ : r6c4 + r5c5 + r4c6 = 13(3) = {139} -> 1,3,9 locked for N5 and D/
13. 45 on D\ : r4c4 + r5c5 + r6c6 = 14(3) -> N5 : 26(5) = 139{58|67} -> r5c5 = 1, r4c4 = {5678}, r6c6 = {5678} (OK, I now know that this is the long way around, thanks Ed. Sometimes I'm a little thickheaded )
14. 2 locked in r1c1 and r3c3 for N1 and D\
15. no 8 in r23c1, no 6 in r2c4, no 8 in r3c2, no 3 inr4c2
16. N1 : 11(3) = 2{36|45}
17. (step 7) [64] not poss., conflict with 11(3) -> 3,7 locked in r2c3 and r3c2 -> 11(3) = {245} -> 2,4,5 locked for N1 and D\
18. 6,7 locked in r4c4 and r6c6 for N5 and D\
19. -> N9 : 20(3) = {389} -> 3,8,9 locked for N9 and D\
20. r1c23 = {68} -> 6,8 locked for r1 and N1
21. r23c1 = {19 -> 1,9 locked for c1 and N1
22. r78c1 = {47} -> 4,7 locked for c1 and N7
Now its only cleanup
23. r8c34 = [39], r9c23 = [91], r67c2 = [12], r2c34 = [71], r23c1 = [91], r34c2 = [38], r1c23 = [68], r8c2 = 5, r7c3 = 6, r9c1 = 8, r25c2 = [47], r28c8 = [28], r9c9 = 3, r7c7 = 9, r78c9 = [17], r8c67 = [12], r78c1 = [74], r8c5 = 6, r6c4 = 3, r4c6 = 9, r1c9 = 4, r3c7 = 7
I leave the rest to you.

Comments and/or corrections are appreciated.

Yes, i enjoyed it.
It was a smooth puzzle. But i didn't use any 45-tests. Wasn't really any need for it. I've posted my walkthrough too in tiny text. Tell me what you think of it.
Just hope i didn't make any mistakes this time.


1. R1C78 = {13} 1,3 locked for R1 and N3
2. R67C2 and R8C67 = {12} 1,2 locked for C2 and R8
3. R1C23 and R23C9 = {59/86} : no 2,4,7
4. R23C1, R34C8 and R9C23 = {19/28/37/46}: no 5
5. R2C34 and R78C9 = {17/26/35}: no 4,8,9
6. R2C67 = {49/58/67}: no1,2,3
7. R34C2, R67C8, R78C1 and R9C78 = {29/38/48/56}: no 1
8. R8C34 = {39/48/57} : no 1,2,6
9. 26(4) in R4C3: no 1
10. 20(3) in R7C7: no 1,2
11. sum 5 cages on both diagonals = 89 -->>R5C5 = 90-89 = 1
12. 17(3) in R1C9 = {247} : {256} not possible because of 14(2) in R2C9 -->> 2,4,7 locked in D/ and N3
13. 19 (3) in R7C3 = {568} -->> 5,6,8 in D/ and N7
14. R4C6 and R6C4 = {3,9} -->> 3,9 locked for N5
15. R4C4 and R6C6 = {58/67}
16. 20(3) in R7C7 = {389/479}: {578} and {569} clash with step 15 -->> 9 locked in N9 and D\
17. 11(3) in R1C1 = {236/245} -->> 2 locked in N1
18. 10(2) in R9C2 = {37}/[91] -->> no 4 in R9C2 and no 2,4,9 in R9C3
19. 2 locked in N7 for R7 -->> no 2 anywhere else
20. 11(2) in R7C1 = [29]/{47} -->> killer pair {7,9} in N7 in R78C1 and R9C23 -->> no 7,9 anywhere else
21. Time for some clean up
21a. R23C1: no8
21b. R2C4: no6
21c. R2C6: no 6,9
21d. R34C2: no 9
21e. R4C8: no 3,6,7,8,9
21f. R6C8: no 2,9
21g. R7C1: no 9
21h. R7C9: no 6,7
21i. R8C4: no 3,4,5,7
21j. R8C9: no 6
21k. R9C78: no 2
22. hidden single 2 in R8C7 -->> R8C6 = 1
23. hidden single 1 in R7C9 and R9C3
24. R7C2 = 2 -->> R6C2 = 1
25. R8C9 = 7
26. R9C2 = 9
27. R78C1 = [74]
28. R8C34 = [39]
29. R6C4 = 3
30. R4C6 = 9
31. 20(3) in R7C7= {983} -->> R7C7 = 9; R8C8 = 8; R9C9= 3
32. R4C4 and R6C6 = {67} -->> locked for N5 and D\
33. 11(3) in R1C1 = {245} -->> 2,4,5 locked for N1
34. 14(2) in R1C2 = {68} -->> locked in N1 and R1 -->> naked single 7 in R2C3 -->> R2C4 = 1
35. R23C1 = {19} = [91]
36. R9C78 = {56} -->> locked for N9 -->> single 4 in R7C8
37. R2C67 = {58} -->> locked for R2
And the rest is singles.


Just edited it a little to make it more clear, after a few comments.

Para

Ed suggested to me that I move my walkthrough to a separate message because it was originally posted as an edit to my previous message. As he pointed out, editing can be missed by those looking for new messages.

I've also made a few corrections and added some comments based on Ed's feedback. Thanks Ed.

Here is my walkthrough. I'll convert it to normal text next week.

I took a fairly similar solving route to Peter.

Clean-up is used in various steps, using the combinations in preliminary steps 1 to 16 for further eliminations from these two cell cages. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. R1C23 = {59/68}

2. R1C78 = {13}, locked for R1 and N3

3. R23C1 = 10(2), no 5

4. R2C34 = {17/26/35}, no 4,8,9

5. R2C67 = {49/58/67}, no 1,2,3

6. R23C9 = {59/68}

7. R34C2 = {29/38/47/56}, no 1

8. R34C8 = 10(2), no 5, no 7,9 in R4C8

9. R67C2 = {12}, locked for C2; clean-up: no 9 in R34C2 (R9C3 cleaned-up in step 15)

10. R67C8 = {29/38/47/56}, no 1

11. R78C1 = {29/38/47/56}, no 1

12. R78C9 = {17/26/35}, no 4,8,9

13. R8C34 = {39/48/57}, no 1,2,6

14. R8C67 = {12}, locked for R8, clean-up: no 9 in R7C1, no 6,7 in R7C9

15. R9C23 = 10(2), no 5, no 8,9 in R9C3

16. R9C78 = {29/38/47/56}, no 1

17. 11(3) in N1, no 9

18. 19(3) in N7, no 1

19. 20(3) in N9, no 1,2

There’s one that I missed! 26(4) in N45, no 1. Fortunately it didn’t matter.

20. 45 rule on N1 2 innies R2C3 + R3C2 = 10 -> R2C3 = {2367}, R3C2 = {3478}, clean-up: R2C4 = {1256}, R4C2 = {3478}

21. 45 rule on N3 2 innies R2C7 + R3C8 = 14 = [59]/{68}, killer quad 5/6/8/9 in R2C7 + R3C8 and R23C9 for N3 -> 13(3) cage in N3 = {247}, locked for D/
21a. Clean-up: R2C6 = {578}, R4C8 = {124}

22. Only valid combination for 19(3) cage in N7 = {568}, locked for N7 and D/
22a. Clean-up: no 3 in R78C1, no 2,4 in R9C23, no 4,7 in R8C4

23. R4C6 + R5C5 + R6C4 = 13 = {139}, locked for N5

24. 45 rule on D\ 3 innies R4C4 + R5C5 + R6C6 = 14

25. Crossed 13(3) and 14(3) cages in 26(5) cage -> R5C5 = 1, locked for diagonals
25a. R4C4 + R6C6 = {58/67}

26. R23C1 = {19} (only remaining 1s in N1), locked for C1 and N1, clean-up: no 5 in R1C23, no 2 in R7C1
26a. R1C23 = {68}, locked for R1 and N1, clean-up: no 2 in R2C4, no 3 in R4C2 [Ed has pointed out no 2 in R3C2 and no 4 in R2C3 (step 20). I wish I’d seen that. It would lock 3,7 for N1 and therefore give quicker eliminations for D\]
26b. R78C1 = {47}, locked for C1 and N7, clean-up: no 5,8 in R8C4, no 3 in R9C23 -> R9C23 = [91], more clean-up: no 2 in R9C78
26c. R8C34 = [39], clean-up: no 5 in R2C4, no 5 in R7C9
26d. R67C2 = [12], clean-up: no 6 in R8C9

27. 3 in C1 locked in R456C1, locked for N4

28. 45 rule on N9 2 innies R7C8 + R8C7 = 6 = [42]/[51], clean-up: R6C8 = {67}

29. R7C7 = 9 (hidden single on D\), 20(3) cage in N9 = 9{38/47/56} -> no 8 in R9C9

30. 2 on D\ locked in 11(3) cage in N1 = {245}, locked for D\ and N1, clean-up: no 6 in R2C4, no 7 in R4C2
30a. R2C34 = [71], clean-up: no 6 in R2C7
30b. R34C2 = [38], R1C23 = [68]
30c. R8C2 = 5, R7C3 = 6, R9C1 = 8, clean-up: no 3 in R9C78
30d. R78C9 = [17], clean-up: no 4 in R9C78
30e. R23C1 = [91]

31. 14(3) diagonal sub-cage in N5 = {167}, locked for D\ and N5
31b. 20(3) cage in N9 = [983], clean-up: no 2 in R4C8

32. R3C7 = 7 (hidden single on D/)

33. R9C78 = {56}, locked for R9 and N9
33a. R67C8 = [74], R78C1 = [74]
33b. R8C5 = 6 (naked single) [corrected], R9C456 = {247}, locked for N8
33c. R8C67 = [12]

34. R2C8 = 2, R1C9 = 4, R2C2 = 4, R6C6 = 6, R4C4 = 7 (naked singles)

35. R5C2 = 7 (naked single), R456C1 = {236} (subtraction combo), locked for C1 and N4 -> R1C1 = 5, R3C3 = 2

36. R1C4 = 2 (naked single), R1C56 = {79}, locked for N2 -> R2C5 = 3

37. R2C67 = {58}, locked for R2 -> R23C9 = [68] (naked single in R2C9) -> R2C67 = [85] [Thanks Ed for correcting me on this step.]
37a. R34C8 = [91] -> R1C78 = [13]

38. R3C56 = {45} -> R3C4 = 6, R4C5 = 2

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

3x3::k:4608:4608:3074:2051:5636:3333:4870:4359:4359:4608:3074:5636:5636:2051:3333:2319:4870:4359:4608:5636:788:788:4630:4630:4630:2319:4870:2075:2075:3613:2334:2334:2848:4897:3874:4870:5412:2075:3613:3613:4136:4136:2848:4897:3874:5412:2350:2350:3888:3888:4136:4897:2612:4897:5412:7735:3888:5945:5945:3643:3643:2612:2622:7735:7735:7735:3650:5945:5444:3643:5444:2622:7735:3650:3650:5945:1868:1868:5444:2622:5444:

I’ve said on this forum that all Assassins, except for the very early ones, and other puzzles posted on this forum ought to have walkthroughs posted.

SampuZ4 V2 was solved as a tag solution with an excellent condensed walkthrough from Ed. Having gone through both the tag solution and the condensed walkthrough, after completing the original puzzle, I felt that there were actually three "money" moves in the condensed walkthrough. Step 22a, which I don’t remember being in the tag solution, was a really powerful typical Ed move. Nice one!

The original puzzle did have "a couple of front doors" but was still a very hard puzzle needing a lot of the moves that were used for V2. If you want a slightly easier puzzle with crossover cages, I recommend SampuZ5 which is another excellent puzzle from Ed; it is in its own thread on this forum.

No walkthrough was ever posted for the original SampuZ4 puzzle so here’s my one. I must thank Ed for his encouragement when I was stuck, feedback on my partial walkthrough and a few corrections. Without him I would never have finished his excellent puzzle. Very many thanks Ed!

I wonder if I found any moves that Ed suspected he had missed?

Steps 1 to 12 set up Ed’s “cell population” diagram and provide some extra eliminations.

1. R3C34 = {12}, locked for R3

2. 7(2) cage in N8 = {16/25/34}, no 7,8,9

3. 8(2) cage in N2 = {17/26/35}, no 4,8,9

4. 9(2) cages in N3, N4 and N5 = {18/27/36/45}, no 9
4a. Clean-up: no 7,8 in R2C7

5. 10(2) cage in N69 = {19/28/37/46}, no 5

6. 11(2) cage in N56 = {29/38/47/56}, no 1

7. 12(2) cage in N1 = {39/48/57}, no 1,2,6

8. 13(2) cage in N2 = {49/58/67}, no 1,2,3

9. 15(2) cage in N6 = {69/78}

10. 8(3) cage in N4 = 1{25/34} [2/3, 4/5], 1 locked for N4, clean-up: no 8 in R6C23
10a. No 4,5 in R6C23 (these clash with the 8(3) cage)
10b. Killer pair 2,3 in 8(3) and 9(2) cages for N4

11. 10(3) cage in N9 = {127/136/145/235}, no 8,9

12. 21(3) cage in N47 = {489/579/678}, no 1,2,3

13. 45 rule on R123 1 outie R4C9 = 4, clean-up: no 5 in R4C45, no 7 in 11(2) cage in N56, no 6 in R7C8
13a. 8(3) cage in N4 = 1{25/34} (step 10), 4 only in R5C2 -> no 3 in R5C2

14. 45 rule on N12 1 outie R3C7 = 4 -> R3C56 = 14 = {59/68}, clean-up: no 5 in 9(2) cage in N3

15. 45 rule on N6 2 remaining innies R5C7 + R6C8 = 7 = [52/61], clean-up: R7C8 = {89}, R4C6 = {56} -> no 5,6 in R4C78 and R5C456 (all the cells “seen” by both R4C6 and R5C7), clean-up: no 9 in R5C9

16. 45 rule on N4 2 outies R5C4 + R7C1 = 7 = [16/25/34]

17. R7C1 = {456} -> R56C1 = {789} (from combinations in step 12)

18. 45 rule on N1 3 outies R1C5 + R23C4 = 10 = {127/136/145/235}, no 8,9

19. R3C56 contains 8/9, only other 8,9 in N2 is in R12C6 -> no 6,7 in R12C6
19a. R3C56 = {68} (cannot be {59} which clashes with R12C6), locked for R3 and N2
19b. R12C6 = {49}, locked for C6 and N2
19c. Clean-up: no 2 in 8(2) cage in N2, no 1,3 in R2C7, no 3 in R9C5
19d. R1C5 + R23C4 (step 18) = 2{17/35}

20. 3 in N6 locked in 19(4) cage, valid combinations are 3{169/178/259} (cannot be {2368} which clashes with the 15(2) cage)

21. 17(3) cage in N3 and 19(4) cage in N36 must each contain 8 or 9 (neither can contain both)
21a. Only valid combinations for the 19(4) cage are 4{159/258} (cannot be {1468} because no 1,4,6,8 in R3C9) = 45{19/28}, [1/2], no 3,6,7, 5 locked for N3
21b. Only valid combinations for the 17(3) cage are {179/278/368}(cannot be {269} which clashes with R2C7)

22. 45 rule on N1 3 innies R2C3 + R3C23 = 15, R3C3 = {12} -> R2C3 + R3C2 = 13 or 14, no 1,2,3
22a. Only valid combinations with R3C2 = {579} and R3C3 = {12} are {159/249/258/267}
22b. 6 only in R2C3 -> no 7 in R2C3

23. 45 rule on N7 4 innies R7C13 + R9C23 = 15, min. R7C1 = 4 -> max. R7C3 + R9C23 = 11, no 9 -> 9 locked in 30(5) cage

[This is the alternative to Para’s neat move]


23a. Valid combinations for 30(5) cage are 9{1578/2478/2568/3468/3567}

24. 45 rule on N5 2 innies R4C6 + R5C4 – 5 = 1 outie R7C3, max R4C6 + R5C4 = 9 -> max R7C3 = 4

25. 45 rule on N7 2 innies R7C13 – 1 = 1 outie R8C4, min R7C13 = 5 -> min R8C4 = 4

26. 45 rule on N9 2 outies R78C6 = 1 innie R7C8, R78C6 = 8 or 9

27. 45 rule on N8 3 innies R8C4 + R78C6 = 15, R78C6 = 8 or 9 (step 26) -> R8C4 = {67}
[Alternatively 45 rule on N89 2 innies R7C8 + R8C4 = 15 -> R8C4 = {67}]
27a. Valid combinations for 14(3) cage in N78 with R8C4 = {67} are {167/257/347/356} [1/2/3], no 8
27b. 8 locked in 30(5) cage, valid combinations 89{157/247/256/346} [1/2/3]
27c. 30(5) contains [1/2/3], R9C23 contains [1/2/3] -> R7C3 = {123}
27d. 15(3) cage in N57 = {159/168/249/258/267/348/357}, R7C3 = {123} -> no 1,2,3 in R6C45

28. Valid combinations for 14(3) cage in N45 are {149/158/248/257/347/356} (cannot be {167} which would clash with R6C23, cannot be {239} because 2,3 only in R5C4)
28a. 4 only in R5C3 -> no 9 in R5C3

29. R3C4 = {12}, 8(2) cage in N2 must contain 1/3 and 5/7 -> 22(4) cage must contain 1/2/3 and 5/7 in N2, valid combinations {1579/2479/2578} (cannot be {2569/3478} because R1C5 + R23C4 must total 10 (step 18), cannot be {3568} because 6,8 only in R2C3) = 7{159/249/258}, no 3,6
29a. {1579} can only have 1,7 in R1C5 + R2C4 (1,5 in R1C5 + R2C4 clashes with the 8(2) cage)
29b. {2578} can only have 2,7 in R1C5 + R2C4 (2,5 in R1C5 + R2C4 -> R3C4 = 1 clashes with the 8(2) cage)
[Alternatively steps 29a and 29b are clashes with step 18 for the invalid cases.]
29c. From steps 29a and 29b no 5 in R1C5 + R2C4 -> 8(2) cage in N2 = {35} (hidden pair)
29d. From steps 29a and 29b no 7 in R3C2
29e. Naked pair {59} in R3C29, locked for R3

[I originally had “29. ...-> 22(4) cage must contain 5/7 in N2, valid combinations {1579/2479/2569/2578/3478/3568}” and Ed commented “Hint: Not all of these combinations are valid”. He added in a later message that it was deliberately phrased that way after I’d assumed he meant one combination was invalid. After looking in more detail, I managed to eliminate 3 of the 6 listed combinations; the 3 explained in step 29.]

30. 6 in N1 locked in 18(4) cage
30a. R3C3 = {12} -> 18(4) cage in N1 must contain 1/2, combinations with R3C1 = {37} are 6{138/147/237}, no 5,9

31. 4 in R6 locked in R6C45, locked for N5
31a. 15(3) cage in N57 = 4{29/38} -> R7C3 = {23}, R6C45 = {489}

32. 5 in N5 locked in R46C6, locked for C6, clean-up: no 2 in R9C5

33. R5C4 = {123}, R4C45 contains [1/2/3] -> 16(3) cage in N5 must contain [1/2/3], valid combinations {169/178/259/358/367} (cannot be {268} which clashes with R4C45)
33a. 5,6 only in R6C6 -> no 2,3 in R6C6

34. 9 in N8 locked in 23(4) cage, valid combinations 9{158/248/257/347/356} (cannot be {1679} which clashes with R8C4) [1/2/3], R9C56 contains [1/2/3] -> R78C6 must contain [1/2/3]

35. 16(3) in N5 contains [1/2/3] (step 33), 7(2) cage in N8 contains [1/2/3] and R78C6 must contain [1/2/3] (step 34) -> the [1/2/3] in the 16(3) cage must be in C6 -> no 1,2,3 in R5C5, the [1/2/3] in 7(2) cage must be in C6 -> R9C6 = {123}, R9C5 = {456}

36. 45 rule on C6789 3 outies R359C5 = 19 = [685/694/874], no 6 in R9C5, clean-up: no 1 in R9C6

37. 30(5) cage in N7 = 89{157/247/256/346} (step 27b) -> R7C13 + R9C23 = {1257/1347/1356/2346}
37a. R7C1 = {456}, R7C3 = {23} and R9C23 = 7 or 8 (because R8C4 = {67}) -> R9C23 = {16/17/26/34} (cannot be {35} because {1356} has 1 in R9C23), no 5
37b. 14(3) cage in N78 (step 27a) = {167/347}, no 2 -> R9C23 = {16/17/34}

[At this stage Ed suggested Hint "look at the combo's in the 18(4)n1 and 21(3)n4" after I’d showed him my changes to step 29. I’m sure I would have found this but the hint focussed my thoughts.]

38. 18(4) cage in N1 (step 30a) = 6{138/147/237}
38a. R123C1 must contain 6, 7 and/or 8 -> R567C1 cannot be {678} -> no 6 in R7C1, no 1 in R5C4 (step 16)
38b. R567C1 = {489/579} = 9{48/57}, 9 locked for C1 and N4
38c. For the {1467} combination, R3C1 = 7 -> R567C1 = {489} -> R1C2 = 4 -> no 4 in R12C1

39. 9 in R4 locked in R4C78, locked for N6
39a. 19(4) cage (step 20) = 3{169/178/259}, 9 only in R4C7 -> no 2 in R4C7

40. 4 in C1 locked in R789C1, locked for N7

41. R7C13 + R9C23 (step 37) = {1257/1347/1356} (cannot be {2346} because R7C13 = [42] clashes with step 25) = 1{257/347/356}, 1 locked in R9C23 for R9 and N7
41a. 14(3) cage in N78 (step 37b) = {167}, no 3, no 6,7 in R8C123 and R9C4
41b. 30(5) cage (step 37) = 89{247/256/346}

42. 14(3) cage in N45 (step 28) = {248/257/347/356}
42a. 45 rule on N4 4 innies R45C3 + R56C1 = 28 = {4789/5689}, R56C1 = {79/89} (step 38b) -> R45C3 = {47/48/56} -> 14(3) cage = {248/347/356}
42b. 4 only in R5C3 -> no 7,8 in R5C3

[I said to Ed "Looks like I'm now at the next hurdle." He replied "Couple more yet. Very Vague Hint: Richard took the next hurdle" (in the V2 tag solution)]

43. 5 in R6 locked in R6C679
43a. If R6C6 = 5 => R4C6 = 6 => R5C7 = 5 => no 5 in R5C8
43b. If R6C79 = 5 => no 5 in R5C8
43c. No 5 in R5C8

[Step 44 is fairly heavy innie/outie and combination work with a summary after sub-step 44e. If you want to skip this, go to the comment after step 47 and then look at step 47.]

44. 45 rule on R6789 3 innies R6C679 – 6 = 1 outie R5C1 -> R6C679 = 13, 14 or 15 = 5{17/18/27/28/36} (cannot be 5{26/37} which clash with R6C23)
Examining each of these combinations separately, noting that 19(4) cage in N6 (step 39a) = 3{169/178/259}
44a. {157} can only have 5 in R6C6 (1,7 in R6C6 don’t give valid combinations for 19(4) cage) => R6C79 = {17}, R67C8 = [28], R4C7 = 8, R5C8 = 3
44b. {158} can only have 5 in R6C6 (1,8 in R6C6 don’t give valid combinations for 19(4) cage) => R6C79 = {18}, R67C8 = [28], R4C7 = {37}, R5C8 = {37}
44c. {257} can only have 7 in R6C6 (5 in R6C6 doesn’t give valid combination for 19(4) cage) => R6C79 = {25}, R67C8 = [19], R4C7 = 9, R5C8 = 3
44d. {258} can only have 8 in R6C6 (5 in R6C6 doesn’t give valid combination for 19(4) cage) => R6C79 = {25}, R67C8 = [19], R4C7 = 9, R5C8 = 3
44e. {356} can have 5 or 6 in R6C6
44ea. If 5 in R6C6 => R6C79 = {36}, R4C7 = 9, R5C8 = 1
44eb. If 6 in R6C6 => R6C79 = {35}, R4C7 = 9, R5C8 = 2

Summary of sub-steps 44a to 44e, R6C6 = {5678}, R4C7 = {3789}, R5C8 = {1237}, R6C79 unchanged

44f. 1 in R6 locked in R6C789, locked for N6 -> no 1 in R5C8 -> R6C79 cannot be {36} (step 44ea) -> no 6 in R6C79 -> 6 in N6 locked in R5C79, locked for R5
44g. 19(4) cage in N6 = 3{178/259}

45. 14(3) cage in N45 (step 42a) = {248/347/356}
45a. 6 only in R4C3 -> no 5 in R4C3

46. 16(3) cage in N5 (step 33) = {169/178/259/358/367}
46a. 1,2,3 only in R5C6 -> no 7,8 in R5C6
[Alternatively R4C45 contains 1/2/3, R5C4 = {23} -> R5C6 = {123}]

47. 16(3) cage in N5 (step 33) = {169/178/259/358/367}
47a. {259} => R4C6 = 6 => R4C45 = {18} clashes with R6C45
47b. {358} => R4C6 = 6 => R4C45 = {27} clashes with R5C4

Summary 16(3) cage = {169/178/367}, no 2,5

[Ed said “This is the key move.” He added that “big step 44” isn’t really necessary. I’ve checked that and he’s correct. Step 47 can be done directly after step 43. In that case there will be detail changes to the remaining steps to remove the candidates eliminated in steps 44, 45 and 46.

Maybe I sometimes suffer from "Assassin 42V2 Syndrome", as in step 44, but that's better than not having learned how to do those sort of steps.]

48. R4C6 = 5 (hidden single in N5) -> R5C7 = 6, R2C7 = 2, R3C8 = 7, R3C1 = 3, clean-up: no 9 in 12(2) in N1, no 9 in R4C8 -> R4C8 = 8, R5C9 = 7, R7C8 = 9, R6C8 = 1, R2C8 = 5, R3C9 = 9, R1C7 = 1 (cage sum), R3C2 = 5, R2C5 = 3, R1C4 = 5, clean-up: no 7 in 12(2) cage in N1, no 1 in R4C45, no 6 in R4C4
48a. Naked pair {48} in 12(2) cage in N1, locked for N1 -> R2C3 = 9, R12C6 = [94], R2C2 = 8, R1C3 = 4, R5C3 = 5

49. R4C7 = 9 (hidden single in N6)

50. Naked pair {23} in R5C48, locked for R5 -> R5C6 = 1, R5C2 = 4
50a. R5C7 + R6C8 = 15 = [87/96], no 8 in R6C8
50b. R5C2 = 4 -> R4C12 = 4 = [13], clean-up: no 6 in R4C5, no 6 in R6C23
50c. Naked pair {27} in R4C45, locked for R4 and N5 -> R4C3 = 6, R5C4 = 3, R5C8 = 2
50d. Naked pair {27} in R6C23, locked for R6 -> R6C6 = 6, R5C5 = 9 (cage sum), R56C1 = [89], R7C1 = 4 (cage sum), R3C56 = [68]
50e. R6C45 = {48} -> R7C3 = 3 (cage sum)

51. 2,3,7 in C6 locked in R789C6, locked for N8-> R8C4 = 6

52. R78C6 = R7C8 (step 26) = 9 = {27} (only remaining combination) -> R9C56 = [43], R9C8 = 6, R1C8 = 3, R8C8 = 4
52a. R9C8 = 6 -> R78C9 = 4 = [13], R6C79 = [35], R9C9 = 2 (hidden single in C9), R8C6 = 7, R9C7 = 8 (cage sum), R7C6 = 2, R78C7 = [75], R8C1 = 2, R8C23 = [98], R8C5 = 1, R7C45 = [85], R6C45 = [48], R9C4 = 9

and the rest is naked and hidden singles

Thanks again Ed. You were a great help for reviewing my partial walkthroughs and providing good cryptic hints. All the moves are my own so any errors in them are mine.

3x3::k:4608:4608:3074:2051:5636:3333:3846:4359:4359:4608:3074:5636:5636:2051:3333:3343:3846:4359:4608:5636:788:788:3606:3606:3343:3343:3846:2075:2075:3613:2334:2334:2848:4897:4898:4898:5412:2075:3613:3613:4136:4136:2848:4897:4898:5412:2350:2350:3888:3888:4136:4897:2612:4897:5412:7735:3888:5945:5945:3643:3643:2612:2622:7735:7735:7735:3650:5945:5444:3643:5444:2622:7735:3650:3650:5945:1868:1868:5444:2622:5444:

Here's a condensed walk-through for SampuZ4 V2 - though many of the key steps are needed for V1 also.

All the steps have been reordered and renumbered, and in some cases reworked to make the shortest possible solution with clarity. If there are any suggestions for improvements, please let me know.

There are many contradiction/ short chain moves - with 2 "money" moves that are very clever. These make this a very, very difficult puzzle. So, it was really great to work as a team. Thanks a lot.

SampuZ4 V2
1. Naked pair {12} r3c34:locked for r3

2. 14(2)n2 = {59/68} = [5/8,6/9..]

3. 13(2)n2 = {49/67} ({58}Blocked by 14(2)) = [6/9..]

4. Killer pair {69} in 14(2) and 13(2): locked for n2

5. 8(2)n2 = {17/35}(no 2)
5a. In n2:8(2) = {17} -> 13(2) = {49} -> 14(2) = {68}
5b. In n2:8(2) = {35} -> 14(2) = {68} -> 13(2) = {49}

6. ->13(2) = {49} locked for n2 and c6
6a. no 2 or 7 r5c7
6b. no 3 r9c5

7. 14(2)n2 = {68}:locked for n2,r3

8. "45" n1 -> r3c3 + {r3c2 & r2c3} = 15 = h15(3)
8a. r3c3 = {12} -> other 2 = 13/14 -> min 4 in those two cells
8b. h15(3) = {159/249/258}(no 6,7) ({168} blocked:6,8 only in r2c3;
.........................................................{267} blocked:no {18/45} in rest of 22(4) in n2)

9. "45" n1 -> [r3c4] + {r2c4 + r1c5} = 10 = h10(3) = [2]{17}/[1]{27} = {127} ({235} blocked: rest of 22(4) in n1 can only be {68}:but no 6 available)
9a. {127} locked for n2
9b. 8(2)n2 = {35}
9c. 22(4)n21 = 7{159/249/258}

10. 6 in n1 only in 18(4) = 6{138/147/237}(no 5,9) ({1269} blocked by r3c3): {3456} blocked by h15(3)(step 8b)

11. 8(3)n4 must use a 1 = 1{25/34} = [2/3,4/5..]
11a. 1 locked for n4
11b. 8(3) = [4/5] -> {45} blocked from 9(2)
11c. 9(2)n4 = {27/36}(no 8) = [2/3..]
11d. Killer pair {2/3} in 8(3), 9(2) for n4

12. 14(3) in n45: r5c4 can now only be 1/2/3

13. "45" n4: r5c4 + r7c1 = 7
13a. r7c1 = {456}
13b. -> no 4/5/6 in r56c1

This is one of the "money" moves to crack this puzzle
14. 21(3)c1:{678} blocked by 18(4)n1
14a.Explanation: from step 10, each combo in 18(4)n1 ({1368/1467/2367}) shares two common digits with {678}, but only 1 of the common digits can be in r1c2
-> {678} blocked
14b.21(3)c1 = 9{48/57} (no 6)
14c. 9 locked in r56 for c1 and n4
14d. no 6 in r7c1 -> no 1 in r5c4 (step 13)

15. "45" n6 -> r5c7 + r6c8 = 7 = [34/43/52/61]
15a. r5c7 = {3456}, r6c8 = {1234}

16. r4c6 = {5678}
16a. r7c8 = {6789}

17. "45" n789 -> r7c138 = h16(3)
17a. min r7c18 = {46} = 10 -> max r7c3 = 6 but 2 6's in r7
17b. min r7c18 = {47} = 11 -> max r7c3 = 5
17c. h16(3) = 16 = [439/529/538] ([457/547]Blocked: 30(5)n7 needs [4/5])
17d. r7c3 = {23}, r7c8 = {89}
17e. r6c8 = {12} -> r4c6 = {56} (step 13) -> r4c6 = {56}

18. 11(2)n56 = {56} only
18a. no 5 or 6 r5c56
18b. no 5 or 6 r4c789

19. "45" c6789 -> r359c5 = 19
19a. Max r39c5 = [86] = 14 -> min r5c5 = 5
19b. Max r35c5 = [89] = 17 -> min r9c5 = 2
19c. no 6 r9c6

20. 16(3)n5 = {169/178/259/367} ({268} -> r4c6 = 5 but {2568} is blocked by 9(2)n5;{358}-> r5c4 = 2 but{2358} blocked by 9(2)n5)

21. 15(4)n57: must use one of 2,3 but not both -> no 123 in r6c45
21a. r6c45 = {49/58/48}(no 6,7) ({67} blocked by 9(2)r6;{57} ->16(3) = {169} but {5..6.} blocked by r4c6)
21b. r6c45 = {4589}

22. "45"n5 -> 4 innies = 20
22a. r6c45(step 21a) + r5c4 + r4c6 = 20 = [{49}25/{48}35] ([{58}25] 2 5's;[{48}26] blocked by 9(2)n5)
22b. -> r4c6 = 5
22c. r6c45 = {48/49} = 4{8/9}:4 locked for n5,r6

23. r5c7 = 6, r6c8 = 1, r7c8 = 9

24. "45" n89 ->r8c4 = 6

Now the second "money" move
25. "45" n7 -> r7c13 = 7; 45 n4 -> r5c4 + r7c1 = 7
25a. -> r7c3 = r5c4
25b. only 2 2's in r3, r3c34
.... 2 in r3c4 -> r5c4 = 3 -> r7c3 = 3
.... 2 in r3c3 -> r7c3 = 3 -> r5c4 = 3

26. -> r7c3 = 3; r5c4 = 3; r7c1 = 4

27. r6c45 = {48}:locked for n5, r6

28. r56c1 = [89]

29. 9(2)n5 = {27}:locked for n5. r4

30. r6c6 = 6; r5c56 = [91]

31. r3c56 = [68]

32. Hidden single 6 r4c3, r5c3 = 5 -> 9(2)n4 = {27}:locked for n4, r6

33. r5c2 = 4, r4c12 = {13}:locked for r4

34. 8 locked in 19(3) in n6 (all remaining combinations have 8), no 8 in r4c7

35. r9c23 = {17}(only remaining combination):locked for n7, r9

36. 7 in c1 only in n1 and 18(4):7 locked for n1
36a. 18(4) must have 6 and 7 = {2367}: locked for N1
36b. r3c34 = [12]

37. 12(2) in n1 = {48} -> r1c3 = 4; r2c2 = 8

38. r2c3 = 9; r3c2 = 5

And the rest are singles (naked and hidden) or cage sums
Thanks Andrew for some corrections & clarifications

Andrew in 2011: I tried this puzzle because I hadn't taken part in the original 'tag' solution. I mistakenly thought that I had found a simple solution to this puzzle and went through 'Ed's condensed walkthrough' before checking my own walkthrough and finding the error. I then re-worked from that stage, deliberately avoiding the key moves which had been used in the 'tag' and included in the 'condensed WT', so possibly found a solving path closer to the way that SudokuSolver would solve it.

This was originally solved as a “tag” by Richard and Ed, with Para joining in later.

Since I wasn’t involved in the “tag”, I had a go at this puzzle in 2011 without looking at how I’d solved the V1 back in 2007.

Prelims

a) 12(2) cage at R1C3 = {39/48/57}, no 1,2,6
b) 8(2) cage at R1C4 = {17/26/35}, no 4,8,9
c) R12C6 = {49/58/67}, no 1,2,3
d) R3C34 = {12}
e) R3C56 = {59/68}
f) R4C45 = {18/27/36/45}, no 9
g) 11(2) cage at R4C6 = {29/38/47/56}, no 1
h) R6C23 = {18/27/36/45}, no 9
i) R67C8 = {19/28/37/46}, no 5
j) R9C56 = {16/25/34}, no 7,8,9
k) 8(3) cage at R4C1 = {125/134}
l) 19(3) cage at R4C8 = {289/379/469/478/568}, no 1
m) 21(3) cage at R5C1 = {489/579/678}, no 1,2,3
n) 10(3) cage at R7C9 = {127/136/145/235}, no 8,9

1. Naked pair {12} in R3C34, locked for R3
1a. Min R3C78 = 7 -> max R2C7 = 6

2. 8(3) cage at R4C1 = {125/134}, 1 locked for N4, clean-up: no 8 in R6C23
2a. R6C23 = {27/36} (cannot be {45} which clashes with 8(3) cage), no 4,5
2b. Killer pair 2,3 in 8(3) cage and R6C23, locked for N4
2c. 14(3) cage at R4C3 must contain one of 1,2,3 -> R5C4 = {123}

3. 45 rule on N1 3 innies R2C3 + R3C23 = 15
3a. Max R3C3 = 2 -> min R2C3 + R3C2 = 13, no 1,2,3 in R2C3 + R3C2
3b. Max R2C3 + R3C2 = 14

4. Hidden killer triple 1,2,3 in 8(2) cage at R1C4 and R1C5 + R23C4 for N2, 8(2) cage contains one of 1,2,3 -> R1C5 + R23C4 must contain two of 1,2,3
4a. 45 rule on N2 3 innies R1C5 + R23C4 = 10 = {127/235} (cannot be {136} which clashes with 8(2) cage at R1C4, cannot be {145} which only contains one of 1,2,3), 2 locked for N2, clean-up: no 6 in 8(2) cage at R1C4

5. 4 in N2 only in R12C6 = {49}, locked for C6 and N2, clean-up: no 5 in R3C56, no 2,7 in R5C7, no 3 in R9C5

6. Naked pair {68} in R3C56, locked for R3

7. 22(4) cage at R1C5 = {1579/2479/2578} (cannot be {1489/2389/2569} because max R2C3 + R3C2 = 14, step 3b, cannot be {1678/3568} because 6,8 only in R2C3, cannot be {3469} because 4,6,9 only in R2C3 + R3C2, cannot be {3478} because R1C5 + R2C4 cannot contain both of 3,7, cannot be {4567} because R1C5 + R2C4 cannot contain both of 5,7), no 3,6

8. R1C5 + R23C4 (step 4a) = {127} (only remaining combination), locked for N2, 7 locked for 22(4) cage at R1C5, no 7 in R2C3 + R3C2

9. 13(3) cage at R2C7 = {139/157/247/346} (cannot be {256} because 2,6 only in R2C7)
9a. 1,2,6 only in R2C7 -> R2C7 = {126}

10. 45 rule on N4 2(1+1) outies R5C4 + R7C1 = 7 = [16/25/34] -> R7C1 = {456}

11. 21(3) cage at R5C1 = {489/579/678}
11a. R7C1 = {456} -> no 4,5,6 in R56C1

12. 45 rule on N6 2 innies R5C7 + R6C8 = 7 = [34/43/52/61], R5C7 = {3456}, R6C8 = {1234}, clean-up: no 2,3 in R4C6, no 1,2,3,4 in R7C8

13. 45 rule on N89 2(1+1) innies R7C8 + R8C4 = 15 = {69/78}
13a. Min R8C4 = 6 -> max R9C23 = 8, no 8,9 in R9C23

14. 45 rule on R789 3 innies R7C138 = 16 = {169/259/268/349/358/367/457} (cannot be {178} because R7C1 only contains 4,5,6)
14a. 4,5 of {457} must be in R7C13, 1,2,3 of the other combinations must be in R7C3 -> R7C3 = {12345}
14b. R7C138 must contain one of 7,8,9 -> R7C8 = {789}, clean-up: no 4 in R6C8, no 3 in R5C7 (step 12), no 8 in R4C6, no 9 in R8C4 (step 13)

15. 45 rule on N7 4 innies R7C13 + R9C23 = 15 = {1257/1347/1356/2346}
15a. R7C138 (step 14) = {169/259/268/349/358/367} (cannot be {457} because R7C13 + R9C23 only contains one of 4,5)
15b. 1,2,3 only in R7C3 -> R7C3 = {123}
15c. 15(3) cage at R6C4 cannot contain more than one of 1,2,3 -> no 1,2,3 in R6C45

16. 16(3) cage at R5C5 cannot be {349} because 4,9 only in R5C5 -> 16(3) cage only contains one of 1,2,3,4
16a. Hidden killer quad 1,2,3,4 in R4C45, R5C4, 16(3) cage and R6C45, R4C45 contains one of 1,2,3,4, R5C4 = {123}, 16(3) cage contains one of 1,2,3,4 -> R6C45 must contain 4 (doesn’t contain any of 1,2,3), locked for R6 and N5, clean-up: no 5 in R4C45

17. 15(3) cage at R6C4 contains 4 = {249/348} (cannot be {456} because R7C3 only contains 1,2,3) -> R6C45 = {48/49}, R7C3 = {23}

18. 16(3) cage at R5C5 = {169/178/259/358/367} (cannot be {268} which clashes with R4C45)
18a. 9 of {259} must be in R5C5 -> no 2 in R5C5

19. 6 in N1 only in 18(4) cage at R1C1 = {1368/1467/2367} (cannot be {1269} which clashes with R3C3, cannot be {3456} which clashes with 22(4) cage at R1C5), no 5,9
[Ed followed this with a nice step. 21(3) cage at R5C1 cannot be {678} which clashes with 18(4) cage at R1C1 which contains two of 6,7,8 so at least one of them must be in C1.]

20. Consider combinations for 21(3) cage at R5C1 = {489/579/678}
21(3) cage = {489} = {89}4 -> R7C138 (step 14) = {349} = [439]
or 21(3) cage = {579/678}, 7 locked for N4, R6C23 = {36}, locked for R6 => no 3 in R6C8 => no 7 in R7C8
-> R67C8 = [19/28], no 3 in R6C8, no 7 in R7C8, clean-up: no 4 in R5C7 (step 12), no 7 in R4C6, no 8 in R8C4 (step 13)
[Ed pointed out that there is now
45 rule on N689 1 innie R8C4 = 1 outie R4C6 + 1 -> R4C6 + R8C4 = [56/67], CPE no 6 in R4C6 and more importantly no 6 in R789C6. Then step 28 would be simplified to a placement R8C4 = 6, rather than just locking 6 for N8, giving several more immediate placements and should simplify the rest of the solving path.
Thanks Ed for showing me a neat step! It’s not surprising that I continued as I did with the naked pair being so obvious.]

21. Naked pair {56} in 11(2) cage at R4C6, CPE no 5,6 in R4C789 + R5C56
21a. 5 in N5 only in R46C6, locked for C6, clean-up: no 2 in R9C5

22. 5 in N4 only in R4C123 + R5C23
22a. Grouped X-Wing for 5 in R4C123 + R5C23 and 11(2) cage at R4C6 in R45, no other 5 in R5

23. 45 rule on N5 4 innies R4C6 + R5C4 + R6C45 = 20 with R6C45 = {48/49} (step 17) = {1469/2459/3458} (cannot be {2468} which clashes with 16(3) cage at R5C5)

24. 16(3) cage at R5C5 (step 18) = {169/178/358/367} (cannot be {259} which clashes with R4C6 + R5C4 + R6C45), no 2
24a. 5,6 of {358/367} only in R6C6 -> no 3 in R6C6
24b. 2 in C6 only in R789C6, locked for N8

25. 45 rule on N7 2 innies R7C13 = 1 outie R8C4 + 1
25a. R8C4 = {67} -> R7C13 = 7,8 = [43/52/53/62]
25b. R7C13 + R9C23 (step 15) = {1257/1347/1356/2346}
25c. 2,5 of {1257} must be in R7C13, 2,6 of {2346} must be in R7C13 (R9C23 cannot be {26} because 14(3) cage at R8C4 cannot be {266}) -> no 2 in R9C23
25d. 2,5 of {1257} and 3,5 of {1356} must be in R7C13 -> no 5 in R9C23

26. 14(3) cage at R8C4 = {167/347}, CPE no 7 in R8C123 + R9C4

[Just spotted]
27. 45 rule on C6789 3 outies R359C5 = 19 = {469/478/568} (cannot be {379} because 3,7,9 only in R5C5), no 1,3, clean-up: no 6 in R9C6
27a. 6,8 of {469/478} must be in R3C5, 5 of {568} must be in R9C5 -> no 6 in R9C5, clean-up: no 1 in R9C6

28. 45 rule on N8 3 innies R78C6 + R8C4 = 15 = {168/267}, no 3, 6 locked for N8

[While checking my walkthrough, I found that I’d made a careless mistake in step 27. I’m now re-working from there, trying to avoid the key steps used in Ed’s 'condensed walkthrough', which I worked through before starting to check my walkthrough.]
29. 45 rule on N9 2(1+1) outies R6C8 + R78C6 = 10
29a. Consider placements for R6C8
R6C8 = 1 => R78C6 = 9 = {18/27}
or R6C8 = 2 => R5C7 = 5 (step 12), R4C6 = 6, R78C6 = 8 = {17}
-> R78C6 = {17/18/27}, no 6

30. R8C4 = 6 (hidden single in N8), R7C8 = 9 (step 13), R6C8 = 1, R5C7 = 6 (step 12), R4C6 = 5, clean-up: no 3 in R4C5

31. R8C4 = 6 -> 14(3) cage at R8C4 (step 26) = {167} (only remaining combination) -> R9C23 = {17}, locked for R9 and N7

32. R7C138 (step 15a) = {259/349}, no 6, clean-up: no 1 in R5C4 (step 10)

33. 21(3) cage at R5C1 = {489/579}, 9 locked for C1 and N4
[This has achieved the same result as Ed’s neat block by 18(4) cage at R1C1.]

34. R9C4 = 9 (hidden single in R9)

35. 13(3) cage at R2C7 (step 9) = {139/157/247}
35a. 9 of {139} must be in R3C7 -> no 3 in R3C7

36. 17(3) cage at R1C8 cannot be {179} which clashes with 13(3) cage at R2C7, no 1 in R12C9
36a. 1 in N3 only in R12C7, locked for C7

[After step 32 Para’s neat move, the final key move in the condensed walkthrough, cracks this puzzle.
45 rule on N5 1 innie R5C4 = 1 outie R7C3
R5C4 and R7C3 cannot be 2, which would block R3C34 -> R5C4 = R7C3 = 3.
I’ll try to find an alternative breakthrough.]

37. Consider placements for R9C56 = [43/52]
R9C56 = [43], 3 locked for N8
or R9C56 = [52] => R2C5 = 3
-> no 3 in R78C5

38. R2C5 = 3 (hidden single in C5), R1C4 = 5, clean-up: no 9 in R1C3, no 7 in R2C2

[A bit of cleaning up]
39. 6 in C3 only in R46C3, locked for N4, clean-up: no 3 in R6C3
39a. 16(3) cage at R5C5 (step 24) = {169/178/367}
39b. 1,3 only in R5C6 -> R5C6 = {13}
39c. 23(4) cage at R7C4 contains 9 = {1589/3479}
39d. 3 of {3479} only in R7C4 -> no 4,7 in R7C4

40. R6C4 = 4 (hidden single in C4)

41. Killer pair 8,9 in R359C5 and R6C5, locked for C5, clean-up: no 1 in R4C4
41a. 23(4) cage at R7C4 (step 39c) = {1589/3479}
41b. 3,8 only in R7C4 -> R7C4 = {38}

42. 1 in C4 only in R23C4, locked for N2

43. Killer quad 6,7,8,9 in R6C1, R6C23, R6C5 and R6C6, locked for R6
[Alternatively hidden killer pair 2,3 in R6C23 and R6C79 for R6, R6C23 contains one of 2,3 -> R6C79 must contain one of 2,3 -> R6C79 = {235}.]

44. 19(4) cage at R4C7 = {2359/2458/3457}
44a. 9 of {2359} must be in R4C7, 2,3,5 of {2458/3457} must be in R6C79 -> no 2,3 in R4C7

45. 45 rule on N8 2 outies R78C7 = 1 remaining innie R8C6 + 5, IOU no 5 in R7C7
45a. R8C6 = {1278} -> R78C7 = 6,7,12,13 = {24/25/34/48/57/58}
45b. 5 of {57} must be in R8C7 -> no 7 in R8C7

46. 21(3) cage at R5C1 = {489/579} (step 33), 15(3) cage at R6C4 (step 17) = {249/348}, R7C138 (step 32) = {259/349} -> R7C13 = [43/52]
46a. Consider placements for R6C5
R6C5 = 8 => R7C3 = 3, R7C1 = 4, R56C1 = {89} = [89]
or R6C5 = 9 => R7C3 = 2, R7C1 = 5, R56C1 = {79} = [97]
-> R5C1 = {89}, R6C1 = {79}

47. 8 in R6 only in R6C56, locked for N5, clean-up: no 1 in R4C5

48. R5C6 = 1 (hidden single in N5)

49. R7C4 = 8 (hidden single in C4)
49a. 23(4) cage at R7C4 (step 39c) = {1589} (only remaining combination) -> R78C5 = {15}, locked for N8 -> R9C5 = 4, R9C6 = 3

50. Naked pair {27} in R78C6, locked for C6, CPE no 2 in R8C7

51. 14(3) cage at R7C6 = {257/347} (cannot be {248} = [248] which clashes with R7C13), no 8, 7 locked for R7
51a. 8 in N9 only in 21(4) cage at R8C6 = {2478/2568} (cannot be {3468} because R8C6 only contains 2,7), no 3
51b. 4 of {2478} must be in R8C8 -> no 7 in R8C8
51c. Killer pair 4,5 in 14(3) cage and 21(4) cage, locked for N9

52. 1 in N9 only in 10(3) cage at R7C9 = {127/136}
52a. R9C8 = {26} -> no 2,6 in R78C9

53. R7C2 = 6 (hidden single in R7)

54. 14(3) cage at R7C6 (step 51) = {257/347} -> R78C7 = [25/34/43/75]
54a. 45 rule on N9 5 remaining innies R7C7 + R8C78 + R9C79 = 26 = {24578/34568}
54b. [74] of {24578} must be in R7C7 + R8C8 (because R78C7 cannot be [74]) -> no 2 in R7C7 + R8C8
54c. 6 of {34568} must be in R9C9 -> no 5 in R9C9

[And now a longer forcing chain which cracks this puzzle.]
55. R7C7 + R8C78 + R9C79 (step 54) = {24578/34568}
55a. Consider the combinations for R7C7 + R8C78 + R9C79
R7C7 + R8C78 + R9C79 = {24578} => R7C7 = 7, R8C7 = 5 (step 54), R8C8 = 4, R9C79 = {28}, locked for R9 => R9C1 = 5, R7C1 = 4, R7C3 = 3 (step 46) => R7C9 = 1
or R7C7 + R8C78 + R9C79 = {34568}, locked for N9 => R7C9 = 1
-> R7C9 = 1, R78C5 = [51], R7C1 = 4, R7C3 = 3 (step 46), R6C5 (step 33) = 8, R3C56 = [68], R6C6 = 6, R7C78 = [27], R8C7 = 5 (step 54), R8C6 = 7, R8C9 = 3, R9C8 = 6 (step 52), clean-up: no 9 in R2C2, no 3 in R4C4, no 3 in R6C2
55b. R6C1 = 9 (hidden single in R6), R5C1 = 8, R8C1 = 2, R9C1 = 5, R8C8 = 4 (hidden single in N9)

56. Naked pair {27} in R6C23, locked for R6 and N4 -> R6C7 = 3, R6C9 = 5

57. Naked pair {27} in R4C45, locked for R4 and N5 -> R4C8 = 8, R5C45 = [39]

58. 8(3) cage at R4C1 = {134} (only remaining combination) -> R5C2 = 4, R45C3 = [65], clean-up: no 8 in R1C3

59. 13(3) cage at R2C7 (step 35) = {139/247} (cannot be {157} because 5,7 only in R3C8), no 5

60. R2C8 = 5 (hidden single in N3), R2C2 = 8, R1C3 = 4, R12C6 = [94], R2C3 + R3C2 = [95], R8C23 = [98]

61. R2C8 = 5 -> 15(3) cage at R1C7 = {159} (only remaining combination, cannot be {258} because 2,8 only in R1C7) -> R1C7 = 1, R3C9 = 9, R23C7 = [24], R3C8 = 7 (step 59)

and the rest is naked singles.

Rating Comment. The SSv3.2.1 score looks about right for the way I solved this puzzle avoiding key moves that were used in the 'tag'.

3x3::k:3073:3074:3074:0000:2307:0000:2564:2564:4613:3073:3074:0000:4358:2307:2311:0000:2564:4613:3073:0000:2824:4358:0000:2311:3849:0000:4613:0000:3850:2824:0000:3339:0000:3849:3596:0000:3597:3850:0000:1550:3339:3855:0000:3596:2320:3597:0000:2321:1550:4626:3855:3347:0000:2320:0000:2580:2321:4626:4626:4626:3347:3861:0000:3350:2580:1815:1815:3352:2073:2073:3861:1818:3350:2331:2331:2331:3352:4124:4124:4124:1818:

Hi

Very nicely crafted puzzle. But it goes pretty smoothly when you get it to it's bare essentials.

Para

[edit]

ok here is a walkthrough. I am sorry about not cleaning up the grid properly, but i made this walkthrough quickly and following these steps will get you to the correct solution without cleaning up much.

Walkthrough Chevron Killer

1. R34C7, R45C2, R56C6 and R78C8 = {69/78}
2. R45C5, R67C7, R89C1 and R89C5 = {49/58/67} : no 1,2,3
3. R12C5, R23C6, R56C9 and R67C3 ={18/27/36/45}: no 9
4. R34C3 = {29/38/47/56}: no 1
5. R8C34 and R89C9 = {16/25/34}: no 7,8,9
6. R8C67 = {17/26/35}: no 4,8,9
7. 10(3) in R1C7: no 8,9
8. 9(3) in R9C2: no 7,8,9
9. R23C4 = {89} -->> locked for N2 and C4
10. R45C8 = {59/68}-->> {59} : {68} clashes with R78C8 -->> 59 locked for C8 and N6
11. R7C8 = {78}-->> {78} locked for C8 and N9
12. R56C1 = {59/68} -->> {59} : {68} clashes with R45C2 -->> 59 locked for C1 and N4
13. R45C2 = {78} -->> locked for N4 and C2
14. Naked pair {59} in R5C18; no 5 or 9 anywhere else in R5
15. 45 on R89: 2 innies = 17 -->> R8C28 = [98]
16. R7C28 = [17]
17. 45 on C1: 2 innies = 6 -->> R47C1 = {24} -->> locked for C1
18. R89C1 = {67} -->> locked for C1 and N7
19. 12(3) in R1C1 = {138} -->> locked for N1
20. Hidden single 8 in R7C3 -->> R6C3 = 1
21. 9(3) in R9C2 = {135/234} : {126} not possible 1 and 6 only possible in 1 cell so not both possible in 9(3) -->> no 6 in R4C9; -->> 3 locked in R9
22. 16(3) in R9C6 = {169/259/268}-->> {178} not possible 7,8 only in one cell; {457} not possible, clashes with 9(3) in R9C2, no 7 in R9C6
23. Killer Pair {1,2} in 9(3) and 16 (3) in R9
24. Hidden Killer Pair in R9C5 and 16(3) -->> R9C5 = {89} -->> Clean up: R8C5 = {45}
25. 12(3) in R1C2 = {246} -->> locked for N1
26. Naked Single: R3C2 = 5
27. Hidden Single 9 in R1C9
28. R34C7 = {78} -->> locked for C7
29. R67C7 = [49]
30. Hidden Pair {89} in R9 and N8 -->> R9C56 = {89}
31. Killer Pair {89} in R569C6 -->> no {89} anywhere else in C6
32. Naked Pair {78} R4C27 -->> locked for R4
33. Hidden single 7 in R8C6
34. R56C6 = [69]
35. Clean up: No 1 in R12C5
there is a lot more clean up, but this is all you need to get all naked or hidden singles from now on

A nice puzzle Ruud! I don't think I've done one like this before. It's interesting how one can sometimes make use of the blank cells, even though they don't form parts of cages. At other times one has to look for other ways to make progress.


It was fairly easy to find what I thought was the key move which was step 15 in my walkthrough, steps 13, 14 and 18 were also very useful.

I decided to post my walkthrough because, although I followed roughly the same solving path as Para, in several places we made moves for different reasons. I hope people find this walkthrough interesting.

Clean-up is used in various steps, using the combinations in preliminary steps 1 to 12 for further eliminations from these two cell cages. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. 9(2) cages R12C5, R23C6, R56C9 and R67C3 = {18/27/36/45}, no 9

2. R23C4 = {89}, locked for C4 and N2, clean-up: no 1 in R12C5 and R23C6

3. R56C4 = {15/24}

4. 7(2) cages R8C34 and R89C9 = {16/25/34}

5. 8(2) cage R8C67 = {17/26/35}

6. 10(2) cage R78C2, no 5

7. 11(2) cage R34C3, no 1

8. 13(2) cages R45C5, R67C7, R89C1 and R89C5 = {49/58/67}

9. 14(2) cages R45C8 and R56C1 = {59/68}

10. 15(2) cages R34C7, R45C2, R56C6 and R78C8 = {69/78}

11. R9C234 = {126/135/234}, no 7,8,9

12. 10(3) cage in N3 = {127/136/235}, no 8,9

13. 45 rule on R89 2 innies R8C28 = 17 = {89}, locked for R8, clean-up: R7C2 = {12}, R7C8 = {67}, no 4,5 in R9C1 and R9C5

14. 45 rule on N1 3 innies R2C3 + R3C23 = 21 = {489/579/678}, no 1,2,3, clean-up: no 8,9 in R4C3

15. 45 rule on N4 5 innies R4C13 + R5C3 + R6C23 = 16 = {12346}, locked for N4 -> R56C1 = {59}, locked for C1 and N4, R45C2 = {78}, locked for C2 -> R8C2 = 9, R7C2 = 1 -> R8C8 = 8, R7C8 = 7, clean-up: no 4,6 in R3C3, no 6 in R45C8, no 2 in R6C3, no 5,6 in R6C7, no 2,4 in R7C3, no 4 in R8C1, no 6 in R8C4, no 1 in R8C6, no 8 in R9C1

16. R89C1 = {67}, locked for C1 and N7, clean-up: no 3 in R6C3, no 1 in R8C4

17. R45C8 = {59}, locked for C8 and N6, clean-up: no 6 in R3C7, no 4 in R56C9, no 4 in R7C7

18. R3C2 = {456} -> R23C3 = {789} (from combinations in step 14)

19. Only valid combination for R123C1 = {138}, locked for C1 and N1 -> R23C3 = {79}, locked for C3, R3C2 = 5, clean-up: no 4 in R2C6, no 3,6 in R4C3

20. R7C3 = 8 (hidden single), R6C3 = 1, clean-up: no 5 in R5C4, no 8 in R5C9

21. R4C13 = {24} (naked pair), locked for R4 and N4, clean-up: no 9 in R5C5

22. 45 rule on N5 3 innies R4C46 + R6C5 = 11, no 9

23. 45 rule on C5 3 innies R367C5 = 10 = {127/136/235}, no 8,9

24. 8/9 in C5 locked in the two 13(2) cages, no 6,7 in these cages
24a. Killer pair 4/5 in the two 13(2) cages, locked for C5

25. 45 rule on R9 3 innies R9C159 = 20, minmax R9C15 = 14..16 -> R9C5 = {456}, clean-up: R8C5 = {123}

26. 1 in R8 locked in R8C79, locked for N9

27. 7 in N8 locked in R89C6, locked for C6, clean-up: no 2 in R23C6, no 8 in R56C6

28. R56C6 = {69}, locked for C6 and N5, clean-up: no 3 in R23C6, no 4 in R5C5, no 2 in R8C7

29. R23C6 = [54] (naked singles), clean-up: no 3 in R8C7

30. R45C5 = {58}, locked for C5 and N5 -> R89C5 = [49], clean-up: no 1 in R5C4, no 3 in R8C34

31. R56C4 = {24}, locked for C4 and N5 -> R8C34 = [25], clean-up: no 9 in R3C3, no 3 in R8C6, no 6 in R8C7, no 5 in R9C9 -> R34C3 = [74], R2C3 = 9, R47C1 = [24], R8C67 = [71], R89C1 = [67], R89C9 = [34], R9C234 = [351], R9C6 = 8 (hidden single), R9C78 = {26}, locked for N9, R1C3 = 6 (naked single), clean-up: no 3 in R2C5

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

BTW When I first started solving it, I reached an impossible position and then found that I was using the wrong diagram. I'd put a 14(2) cage as a 15(2) cage. It would have been interesting, and embarassing, if that had come out and I'd posted a walkthrough for the wrong puzzle.

3x3::k:2048:2048:2306:5379:5379:5379:3846:3079:3079:2048:6922:2306:7948:7948:7948:3846:3088:3079:4114:6922:6922:7948:2326:7948:3088:3088:5146:4114:4114:6922:7948:2326:7948:3088:5146:5146:4388:4388:3622:3622:4136:3625:3625:3115:3115:4388:8750:8750:3622:4136:3625:9779:9779:3115:3894:8750:8750:2105:4136:2363:9779:9779:3902:3894:8750:8750:2105:3395:2363:9779:9779:3902:3894:3894:2634:2634:3395:2637:2637:3902:3902:

Very enjoyable, difficult Killer. Thanks a lot Ruud.

Took a couple of goes. Had to use hidden cages, lots of combination conflicts and a few nice (small?) chains to make some decent inroads. Maybe the chains weren't necessary - but got things moving nicely .

[edit 9th Feb: Nasenbaer has kindly pointed out a nice move after step 12, and some other hidden singles missed. Thanks Peter]

Assassin 36

0. 27(4)n14 = 9{378/468/567}(no 1,2)

1."45"n1 -> r4c3 + 1 = r3c1 -> r3c1 = {4..9}, r4c3 = {3..8}

2. -> 9 required in 27(4)n12 only in n1:9 locked n1 -> no 8 r4c3 (step 1)

3. 8(3)n1 = 1{25/34}: 1 locked n1 = [4/5..]
3a. no 8 9(2)n1

4. 9(2)n1 = {27/36} ({45} blocked by 8(3)n1 step 3)

5. {5679} blocked from 27(4) by 9(2)n1 (leaves no 6 or 7 for 9(2))
5a. 27(4) = 89{37/46} (no 5)
5b. 8 locked for n1 -> no 7 r4c3(step 1)
5c. no 6 r3c1 (step 1)

Now some chains make a big impact
6. from step 1 with r4c3 = {346}
6a.r4c3 = 3 -> r3c1 = 4 and 9(2) = {27}: Blocked: (no 4 or 2 for 8(3)n1)
6b.r4c3 = 6 -> r3c1 = 7 : Blocked: no 6 or 7 for 9(2)n1
6c. -> r4c3 = 4, r3c1 = 5

7. 8(3)n1 = {134}:locked for n1

8. 9(2)n1 = {27}: locked for n1, c3

9. 27(4)n14 now 23(3) = {689}

10. r4c12 = 11(2) = {29/38}(no 1,6,7) = [2/8,3/9,8/9..]

11. 17(3)n4 = {179/269/368}:no 5 ({278/359} blocked by r4c12 (step 10))

12. 17(3)n4 = [8/9], r4c12 = [8/9]:Killer pair [89] for n4

(NB:"45" on c1 -> r1459c2 = 11(4) = {1235} -> r9c2 = 5, naked triple on {123} for c2, 4 locked for c1 in r12c1 and this Killer pair [89] would also work for c1. Very handy! Sadly, not in this walk-through)

13. "45" n4 -> r5c3 + r6c23 = h13(3) and must have 5 = 5{17/26} (no 3)

14. h13(3)n4 must have 2 or 7(step 13): only available in r6c2 -> r6c2 = {27}
14a. ->5 for n4 only in c3 ->locked for c3

15. "45" n3 -> r4c7 + 6 = r3c9
15a. r4c7 = {123}
15b. r3c9 = {789}

16. "45" n3 -> r4c789 = 14

17. combining steps 15 and 16 and some more short chains
17a. r4c7 = 1 -> r3c9 = 7 -> r4c89 = 13 = {58}
17b. r4c7 = 2 -> r3c9 = 8 -> r4c89 = 12 = {57} ({39} blocked by r4c12 (step 10))
17c. r4c7 = 3 -> r3c9 = 9 -> r4c89 = 11 = {56} ({38} blocked by r4c7)

18. -> r4c89 = 5{6/7/8}(no 3,9):5 locked for r4,n6
18a. no 4 r3c5

19. 38(6)n69 = {356789}(no1,2,4)
19a. from step 18 -> 5 required in 38(6) in n9:5 locked for n9
19b. 5 only in r78c7 for c7 -> 5 locked for 38(6)

20. "45" n2 -> r4c456 = h16(3) = {169/178/367}(no 2) ({268} blocked by r4c12:step 10)
20a. no 7 r3c5

21. 21(3)n2 = {489/579/679}
21a. 31(7)n25 = 12347{59/68}
21b. 2 required in 31(7) in n2 only: 2 locked n2
21c. no 7 r4c5

22. 9(2)c5 = {18/36} = [1/6..]

23. "45" c5 -> r12c5 = 7 = [43/52] ([61] blocked by 9(2)c5 step 22)

24. 21(3) n2 must have 4 or 5 but not both = {489/579} (no 6) = 9{48/57}
24a. -> r1c46 = {789}
24b. 9 locked in r1c46 n2,r1
24c. no 6 r2c7

25. 6 in r1 now in n3:locked for n3

26. 12(4)n3 = {1245} (no 3)
26a. no 9 r3c9 (step 15)
[missed these next two but do them now if you like
26b. r2c8 = 5 (single 12(4)
26c. r1c5 = 5 (single n2)]

27. 20(3)n36 = {578}
27a. -> no 7 or 8 in r56c9
[if you did 26bc then 27c.r4c9 = 5 (single 20(3))]

28. 6 in r4 only in n5:locked for n5
28a. (from step 20) h16(3)n5 must have 6 = 6{19/37} (no 8)
28b. no 1 r3c5

29. 1 in n2 only in 31(7) -> no 1 r4c46

30. should have done this step earlier
31(7)n25 - the {1234579} combo can only have {45} in n2: blocked by r1c5
30a. -> 31(7) = {1234678} (no 5,9}
30b. r1c5 = 5 (hidden single n2)
30c. r2c5 = 2 (step 23)

Now things go crazy
31. r12c2 = [27], r1c46 = {79}:locked for r1,n2, r12c7 = [69]

32. from step 28a. r4c456 = {367} only:locked for r4,n5

33. r4c89 = {58}:locked for r4,n6
33a. r3c9 = 7, r4c7 1 (single r4)

34. r4c12 = {29}:locked for n4

35. r3c78 = {24}:locked for r3,n3
35a. r2c8 = 5

36. 9(2)c5 = {36}:locked for c5

37. 13(2)c5 = {49} locked for c5,n8

38. r56c5 = {18} locked for c5,n5
38a. r7c5 = 7

39. 17(3)n4 = {368}:locked for n4

40. r56c3 = {15}:locked for c3

41. 3 in c3 in n7:3 locked for n7

42. 14(3)n45 = {149}
42a. r5c3 = 1
42b. r56c4 = {49}:locked for c4,n5

the rest goes on.

Some neat moves there Ed! You solved the key area of N1 and R4C3 more directly than I did and also made use of some hidden cages that I never used. I saw the 16(3) one in R4C456 but didn't make any use of it as 16(3) isn't normally a particularly useful combination unless one knows that one cell has a small number. Bringing it into your walkthrough after 4,5 had already been eliminated from it certainly made it more useful. Don't think I even spotted the 14(3) one in R4C789.

A bit late but here's my walkthrough. I didn't get to solve the key area until a long way in because I was using more methodical moves first before I started looking for chains. However I did get that naked quad in C2 fairly early but it wasn't immediately as powerful as in Ed's comment because I hadn't yet solved the key area. There are more comments about Ed's moves after my steps 12 and 36.


Clean-up is used in various steps, using the combinations in steps 1 to 8 for further eliminations from these two cell cages; it is also used for the two cell split sub-cages that are produced by applying the 45 rule. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. R12C3 = {18/27/36/45}, no 9

2. R12C7 = {69/78}

3. R34C5 = {18/27/36/45}, no 9

4. R78C4 = {17/26/35}, no 4,8,9

5. R78C6 = {18/27/36/45}, no 9

6. R89C5 = {49/58/67}, no 1,2,3

7. R9C34 = 10(2), no 5

8. R9C67 = 10(2), no 5

9. 8(3) cage in N1 = 1{25/34}, 1 locked for N1, clean-up: no 8 in R12C3
9a. No 4,5 in R12C3 because {45} would clash with 8(3) cage
9b. Killer pair 2/3 in 8(3) cage and R12C3, locked for N1

10. R1C456 = {489/579/678}, no 1,2,3

11. 20(3) cage in N36, no 1,2

12. 27(4) cage in N14 = 9{378/468/567}, no 1,2
[I missed the clash with the 9(2) cage that Ed used to eliminate the {5679} combination and therefore 5. A neat one using the combination of the nonet and the column!]

13. 12(4) cage in N36 = 12{36/45}, no 7,8,9
[I should then have spotted that 12(3) cage in N3 cannot be {129} -> no 9]

14. 31(7) cage in N25 = 12347{59/68}

15. 34(6) cage in N47 = 9{13678/14578/23578/24568/34567}, contains two of 1,2,3,4 and three of 5,6,7,8

16. 38(6) cage in N69 = {356789}

17. 45 rule on N1 1 innie R3C1 – 1 = 1 outie R4C3 -> no 9 in R4C3
17a. 9 in N1 must be locked in 27(4) cage -> no 8 in R4C3, R3C1 = {45678}

18. 45 rule on N3 1 innie R3C9 – 6 = 1 outie R4C7 -> R3C9 = {789}, R4C7 = {123}

19. 45 rule on C1 4 outies R1459C2 = 11 = {1235}, locked for C2

20. 45 rule on C5 2 innies R12C5 = 7 -> R1C5 = {456}, R2C5 = {123}, R1C46 = {789} (step 10)

21. 16(3) cage in C5 cannot contain more than one of 1,2,3, R2C5 = {123} -> 9(2) cage in C5 must contain one of 1,2,3 -> no {45} in R34C5
21a. 16(3) cage in C5 must contain one of 1,2,3, valid combinations are {169/178/259/268/349/358/367}

22. 45 rule on C123 2 innies R59C3 = 9, no 9, clean-up: no 4 in R5C9, no 1 in R9C4

23. 45 rule on C789 2 innies R59C7 = 11, no 1, clean-up: no 6 in R5C7, no 9 in R9C6

24. 1 in C7 locked in R34C7, locked for 12(4) cage

25. 1 in N9 locked in 15(4) cage

26. 45 rule on C9 4 outies R1459C8 = 14, no 9

27. 9 in C8 locked in R678C8, locked for 38(6) cage

28. 16(3) cage in N14, valid combinations {169/178/259/268/349/358/367/457}, no 1,2 in R4C1

29. 17(3) cage in N4, valid combinations {179/269/278/359/368/458}, no 1,2 in R56C1

30. 15(4) cage in N7, valid combinations {1239/1248/1257/1347/1356/2346}, contains at least two of 1,2,3

31. 15(4) cage in N9 must contain 1 (step 25) and at least one of 2,4, valid combinations 1{239/248/257/347}, no 6, contains two of 2,3,4,5 and one of 7,8,9

32. Time to try contradiction moves. Looks like there may be interactions between R4C3 and R12C3 so try R4C4 = 3/6/7 (no 4/5 in R12C3).
If R4C3 = 3 => R2C2 + R3C23 = {789} => R12C3 = {36} clashes with R4C3 -> no 3 in R4C3
If R4C3 = 6 => R2C2 + R3C23 = {489} (cannot be {678}) => 8(3) cage = {125} => R12C3 = {36} clashes with R4C3 -> no 6 in R4C3
If R4C3 = 7 => R2C2 + R3C23 = {569} => 8(3) cage = {134} => R12C3 = {27} clashed with R4C3 -> no 7 in R4C3

33. R4C3 = {45} -> R2C2 + R3C23 = {6789} from combinations in step 12
33a. 8(3) cage = 1{25/34} [4/5] -> R3C1 = {45}
33b. Remaining valid combinations for 27(4) cage are {4689/5679} = 69{48/57}, 6 locked for N1, clean-up: no 3 in R12C3 = {27}, locked for C3 and N1, clean-up: no 3,8 in R9C4
33c. R2C2 + R3C23 = {689} -> R4C3 = 4, clean-up: no 5 in R5C3, no 6 in R9C4
33d. 8(3) cage in N1 = {134} with 4 locked for C1 -> R3C1 = 5
33e. R4C12 = 11 = [83/92], 2 in N4 locked in R45C2, locked for C2
[Alternatively after step 33a I could have used step 17 to give R3C1 = 5, R4C3 = 4 directly but I’d forgotten about that one. Poor short term memory as one gets older! Fortunately the remaining sub-steps above are almost as quick and make the other related eliminations.]

34. 2 in N7 locked in 15(4) cage, valid combinations {1239/1257} = 12{39/57}, no 6,8, 1 locked for N7, clean-up: no 8 in R5C3, no 9 in R9C4

35. 34(6) cage in N47 must contain 4 in R78C2, 7 in R678C2 and 5 in R678C3, remaining valid combinations 9{14578/34567} = 4579{18/36}

36. 1 in C3 locked in R56C3, locked for N4
36a. Remaining valid combinations for 17(3) cage in N4 are {269/278/359/368}
[I missed the killer pair 8/9 in R4C1 and the 17(3) cage. No real problem as I then got them in a naked triple in step 38b!]

37. 8 in C1 locked in R456C1, locked for N4

38. 6 in C1 locked in R56C1, locked for N4, clean-up: no 3 in R9C3, no 7 in R9C4
38a. 17(3) cage in N4 = 6{29/38}, no 3,7 in R56C1, no 5 in R5C2
38b. R456C1 = {689}, locked for C1 and N4 -> R6C2 = 7

39. R45C2 = {23}, locked for C2 and N4 -> R1C2 = 1, R12C1 = {34}, locked for C1, R9C2 = 5, R5C3 = 1, R6C3 = 5, clean-up: no 8 in R8C5
39a. R56C4 = 13 = {49}/[58]/[76]
39b. 34(6) cage in N47 = {345679}, no 8
39c. R9C3 = 8 (hidden single in N7), R9C4 = 2, clean-up: no 6 in R78C4, no 7 in R78C6, no 5 in R8C5

40. 9 in N8 locked in R789C5, locked for C5

41. R9C8 = 1 (hidden single in C8), clean-up: no 9 in R9C7

42. 2 in N9 locked in R78C9, locked for C9
42a. 15(4) cage in N9 = 12{39/48/57}, no 4,7 in R78C9

43. R9C1 = 7 (naked single), clean-up: no 3 in R9C67 = {46}, locked for R9, R9C5 = 9, R8C5 = 4, R9C9 = 3, R9C67 = [64], clean-up: no 3 in R2C5, no 3,5 in R78C6
[I should also have included R5C7 = 7 in the clean-up after fixing R9C7 = 4. Just spotted that while checking the walkthrough before posting it.]
43a. R78C9 = {29} (step 42a), locked for C9 and N9 -> R6C8 = 9 (hidden single in C8), clean-up: no 4 in R5C4
43b. R78C6 = {18}, locked for C6 and N8, clean-up: no 7 in R78C4 = {35}, locked for C4 and N8, clean-up: no 8 in R6C4
43c. R7C5 = 7 (naked single), R56C5 = 9 = {36}/[81], clean-up: no 2 in R34C5

44. R1C5 = 5, R2C5 = 2 (hidden singles in C5)

45. R7C2 = 4 (hidden single in N7)

46. 3 in 31(7) cage (step 14) locked in R234C6, locked for C6

47. R1C5 = 5 -> R1C46 = 16 = {79}, locked for R1 and N2 -> R12C3 = [27], clean-up: no 6,8 in R2C7 -> R2C7 = 9, R1C7 = 6
47a. R23C6 = {34}, locked for C6 and N2, clean-up: no 6 in R4C5

48. 8 in R1 locked in R1C89, locked for N3 -> R1C9 = 8, R1C8 = 3, R2C9 = 1 (only possible combination)
48a. R3C9 = 7, R4C89 = 13 = [85] (only possible combination)

49. R3C7 = 2, R3C8 = 4, R2C8 = 5 (naked singles) -> R4C7 = 1, clean-up: no 8 in R3C5

50. R3C6 = 3 (naked single), R2C6 = 4, R12C1 = [43]
50a. No 5 in 31(7) cage -> no 9 (step 14) -> {1234678} -> R4C6 = 7, R4C4 = 6, R2C4 = 8, R3C4 = 1

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

3x3::k:3328:6401:6401:6401:4100:4100:3334:5895:5895:3328:3328:3339:6401:4100:3334:3334:5895:3601:5906:3328:3339:3339:4118:4118:4118:5895:3601:5906:5906:4893:5918:5918:5918:3105:3105:3601:5906:4893:4893:1575:1575:3881:3105:3883:3883:3629:3886:3886:3888:1575:3881:3881:3883:5685:3629:3639:3886:3888:3898:3898:3132:3132:5685:3629:3639:3639:3888:3898:4420:3132:3398:5685:3400:3400:2378:2378:4420:4420:4420:3398:3398:

Was a fun puzzle.
Quick opening and steady progress till the end.
This walkthrough is a bit long but i just couldn't manage to get it to all singles till the last few cells. Must have missed a step somewhere along the way.
Any corrections or suggestions are appreciated.

Walkthrough Assasin 37

1) 13(4) in R1C1 = {1237/1246/1345} -->> no 8,9 ; 1 locked for N1
2) R4C456 = {689} -->> locked for R4 and N5
3) R5C45 + R6C5 = {123} -->> locked for N5
4) R678C9 = {589/679} -->> no 1,2,3,4; 9 locked for C9
5) R9C34 = {18/27/36/45} -->> no 9
6) R9C12 = {49/58/67} -->> no 1,2,3
7) 19(3) in R4C3 = {289/379/469/478/568} -->> no 1
8) 45 on R9 -->> 2 outies = 7 : R8C68 = {16/25/34} -->> no 7,8,9
9) 45 on C1234 -->> 2 innies = 7 : R45C4 = [61]
10) 45 on C89 -->> 2 innies = 3 : R47C8 = {12} -->> locked for C8
11) {23} locked in R56C5 for C5
12) 15(3) in R5C6 = {357/456} (has to contain 2 digits of 4,5,7) -->> 5 locked in C6 and N5
12a. R6C7 = {36}
13) 45 on R6789: 4 outies: R5C5689 = 14 = {2345} -->> locked for R5 -->> no 6,7,8,9
14) 45 on R6789: 4 innies: R6C5678 = 21
14a. Min R6C567 = 12; Max R6C567 = 14
14b. R6C8 = {789}
15) 7 locked in N5 for R6 -->> no 7 anywhere else in R6
16) 45 on N8: 3 outies R6C4+ R9C37 = 11(3) = {137/146/245/344} (has to contain either a 4 or 7 in R6C4)
16a. R9C37 = {123456}
17) Clean up
17a. R9C3 : no 3
17b. R9C4 : no 2
17c. R8C6 : no 6 (step 8)
17d. R5C9: no 5 (can’t form 15(3) that way)
18) 1 locked in N6 for R4 -->> nowhere else in R4
19) 13(3) in R2C3: no options left with 9: no 9
20) 45 on N9: 1 outie and 1 innie: R6C9 – R9C7 = 2
20a. R6C9: no 9; R9C7: no 1,2,5
21) 9 locked in R78C9 for N9
22) 12(3) in R7C7 = {138/147/156/237/246}
22a. R78C7 = {345678}
23) Hidden Pair {12} in R7C8 + R9C9 -->> R9C9 = {12}
24) 13(3) in R8C8 = {148/157/238/247/256} -->> no combinations possible with 3 or 4 in R9C8
25) Step removed. Really doing double work here.
26) Step removed, did it before. Don't know what i was thinking.
27) 45 on R123: R3C1 – R4C9 = 2 -->> no 2,8 in R3C1
28) Follow up on step 16
28a. 11(3) = {137/146/344}: {245} not possible
28b. R9C7: no 4; R9C3: no 2 or 5
28c. Clean up: R9C4: no 4 or 7; R6C9: no 6
29) Naked pair {36} in C7 in R69C7
30) 12(3) in R7C7 = {147}
30a. R7C8 = 1; R78C7 = {47} -->> locked for C7 and N9
31) Singles: R4C8 and R9C9 = 2
32) R45C7 = [19]
33) R6C8 = 8; R6C9 = 5
34) R5C89 = {34} -->> locked for R5 and N6
35) Singles: R4C9 = 7; R5C5 = 2; R5C6 = 5; R6C7 =6
36) More Singles: R6C6 =4; R6C5 = 3; R6C4 = 7; R9C7 = 3[/color]
37) R9C3 = 1 (step 16) -->> R9C4 = 8
38) R78C9 = {89} -->> locked for C9 and N9
39) R23C9 = {16} -->> locked for N3: {34} clashes with R5C9
40) {56} locked in C8 in R89C8
41) R78C4 = {35} -->> locked in C4 and N8
42) 17(4) in R8C6 = 3{149/167} -->> 1 locked in 17(4) for N8 -->> R8C6 = 1
42a. R9C56 = [49]/{67} -->> no 9 in R9C5
43) Hidden single 2 in R7C6
44) 9 locked in C4 for N2 -->> no 9 anywhere else in N2
44a. 9 locked in R12C4 for 25(4) -->> no 9 anywhere else in 25(4)
45) Hidden single 9 in R3C1
46) 13(2) in R9C1 = [49]/{67}-->> R9C1: no 5, R9C2: no 4,5
47) Hidden single 5 in R9C8 -->> R8C8 = 6
48) 23(4) = 9 + {347/356} -->> 3 locked in N4 in 23(4); R5C1: no 8
49) 8 locked in C1 for N7 -->> no 8 anywhere else in N7
49a. 8 locked in 14(3) in R6C1 in R78C1: 14(3) = {158/248}: R7C1: no 3 or 6; R8C1: no 2,3
50) 9 locked in N4 in 15(3) in R6C2: 15(3) = {159/249}: R7C3 = {45}
51) Naked pair {45} in C3
52) 2 and 3 locked in N7 in 14(3): 14(3) = {239)
53) 6 locked in 13(2) in N7: 13(2) = {67} -->> locked for R9
54) R9C56 = [49]
55) R4C56 = [98]; R78C5 = [67]; R78C7 = [74]
56) 25(4) in R1C2 has 2 digits of 2, 4 or 9 in R12C4: 25(4) = {2689/4579}
56a. R1C23 = [57]/{68}
57) Naked Pair {67} in R59C1 for C1
58) 1 and 4 locked in 13(5) in R1C1: 13(4) = {1246/1345}: no 7
59) 16(3) in R1C5 has to contain 2 digits of 1, 5 or 8: 15(3) = {178/358}
59a. 8 locked in 16(3) for N2
59b. no 6 in R1C6
60) 6 locked in R1 for N1 -->> no 6 anywhere else in N1
60a. 6 locked in R1C23 in 25(4) -->> 25(4) = {2689}
60b. R1C23 = {68} -->> locked for R1 and N1;
60c. R12C4 = {29} locked for N2
61) Singles:
61a. Naked single 4 in R3C4 for N2
61b. Hidden Single 8 in R2C5 for N2
61c. Hidden Single 8 in R3C7 for N3
62) R23C3 = {27} -->> locked for N1 and C3
63) Singles
63a. Naked single 9 in R6C3
63b. Naked single 3 in R8C3
63c. R78C2 = [92]; R78C4 = [35]; R78C9 = [89]
63d. R8C1 = 8; R6C2 = 1; R6C1 = 2
63e. R7C1 = 4; R7C3 = 5; R4C3 = 4
63f. R5C23 = [78]; R5C1 = 6; R9C12 = [76]
63g. R1C23 = [86]
63h. Hidden single 4 in R2C2
64. 13(3) in R1C7 = {256} -->> R2C6 = 6
65 Singles
64a. R23C9 = [16]
64b. Hidden single 1 in R1C1 and R3C5
64c. R3C6 = 7; R1C56 = [53]
64e. The rest is all naked singles; you can figure them out yourselves.

Edited for Ed and Andrew

Anyone else want to be mentioned here? Find some more mistakes!! Now hope there aren't.


greetings

Para

I think I may have got to the simple singles a little bit quicker:

Edited following some good comments from Andrew

1. 23(3)={689} locked for N5 and r4

2. 6(3)={123} locked for N5
2a.15(3) n56 must have {45},{47} or {57} - only {357} or {456} -> r6c7=3/6

3. 45 on n5 r6c4 minus r6c7 = 1 - no 5 in r6c4
3a. 15(3) n58=4{29/38/56} or 7{26/35} - no 1/4/7 in r78c4

4. 5 locked in n5 for c6

5. 45 on n9 r6c9 minus r9c7 equals 2. No 1/2/3/4 in 22(3) n69 so r9c7=3/4/5/6/7
5a. 9 locked in 22(3) n69

6. 45 on r9 r8c68 total 7 - only {16}/[25]/{34} possible

7. 45 on n47 r3c1 minus r9c3 equals 8 - only possibility [91]
7a 9(2) r9=[18]
7b. 23(4) n14 = 9{15}8 9{17}6 9{24}8 9{257} 9{347} 9{35}6 - no 1 in r5c1
7c. 13(2) n7 can only be {76}/[49]

8. 45 on n5689 - r4c9=7
8a. 22(3) n69={589} locked for c9
8b. 14(3) n36 now 7{16/43}
8c. 23(4) n14 now 9{15}8 9{24}8 9{25}7 9{34}7 9{35}6 - no 2/3/4/5 r5c1
8d. 19(3) n4. now 2{89} 3{79} 4{69} {478} 5{68} - no 2345 in r5c23
8e. 12(3) n6 no 1/2 r5c7

9. 45 on n3 r3c7 - r2c6 = 2 -> r2c6={1/2/3/4/6} r3c7={3/4/5/6/8}

10. 45 on n4 r7c3 - r6c1 = 3 -> min r7c3 is 4, max r6c1 is 6

11. 45 on n8 r6c4 r9c7 = 10 -> [46]/[73]
11a. 17(4) n89 = 1{367} 1{49}3 5{24}6 5{27}3 - no 3/6 in r8c6 - no 2 in r9c5
11b. (from step 5) no 9 in r6c9
11c. (from step 6) - no 1/4 in r8c8 -> no 7/9 in r9c8

12. naked pair {36} r69c7
12a. 12(3) n6 {12}9 [138] {4[3]5} - no 4/5 in r4c8
12b. 12(3) n9 {129} {1[3]8} {147} {1[6]5} {2[6]4} - no 5/8 in r7c8
12c. (from step 9) no 1/4 in r2c6

13. 45 on c89 r47c8 = 3 = {12} locked for c8.
13a. 12(3) n6 = {12}9 - r5c7=9 and {12} locked for r4 n6

14. 12(3) n9 =1{47} r7c8=1
14a. 13(3) n23 - no combo with 1
14b. 19(3) n4 - no combo with 3
14c. r4c78 now = [12]
14d. so 23(4) n14 = 9{34}7 9{35}6 - no 8 in r5c1
14e. 13(3) n9 now {56}2 - r9c9=2 - no 2 in r9c6, so from 11a no 4 in r8c6

15. hidden single 3 at r9c7 for n9
15a. r6c7=6
15b. 15(3) n56={45}6 - {45} locked for c6 and n5 -> r6c4=7
15c. 15(3) n6={348} locked for n6
15d. 15(3) n58 now 7{26/35} no 9
15e. 25(4) n12 r1c2 minimum 2

16. hidden single 5 r5c6 for r5 -> r6c6=4
16a (from 11a) no 9 in r9c5

17. hidden single 5 at r6c9 for n6
17a. 14(3) at n47 - no 2/3 in r78c1
17b. 15(3) at n47 [168] 195 {29}4 {38}4 {28}5 - no 7/8/9 in r7c3
17c. r78c9 = {89} locked for n9

18. 2 in n3 locked in 13(3) n23, locked for c7, no 4/7 in 13(3), no 2 in r2c6

19. {12} locked in r6 of n4 - r6c5=3
19a r6c8 = 8.
19b. 16(3) n23 no 8 in r3c6
19c. (from 17b) 15(3) n47 no 6 in r7c3

20. 4 locked in n8 for c5

21. 4 locked in n9 for c7

22. 5 and 6 locked in n9 for c8
22a 23(4) n3 no 1/6 in r1c9

23. naked pair {45} at r47c3
23a. 13(3) n12 - {238} {27}4 {26}5 {36}4 - no 1/6 at r3c4

24. naked pair {34} at r15c9

Killer move setting 6 values !!

25. 45 on c567 r45c4 total 7 = [61] -> r5c5=2
25a. 15(3) n58=7{35} {35} - {35} naked pair for c4 and n8
25b. 15(3) n8 2{49/67} - no 1 -> r8c6=1 and (from step 6) r8c8=6 -> r9c8=5
25c. (from 11a) 17(4) n89 = 1{67}3 1{49}3
25d. 25(4) n12 - only possible now {4[7]59} or {68}{29} - no 2/3/5/7 r1c23 (5 already gone from r1c3)

26. naked triple {249} at r123c4
26a. 16(3) n23 = [178]/{5[3]8} -no 6
26b. 16(3) n2= 7{18}/3{58} - locked 8 for n2

27. hidden single 2 at r7c6 for c6
27a 15(3) n8 =2{49}/2[67] - no 7 r7c5


28 13(4) n1, no 8, 1 locked for n1
28#. 8 locked in n7/14(3)n47 for n7
28a. 14(3)n47 = {158}/{248} - no 6/7
28b. 14(3)n7=[239}/{257}/{356} - no 4
28b. naked triple {458} at r78c1 r7c3 - so 28b -> 14(3)n7={239}
28c. 13(2)n7={67} - no 9

29. hidden single 6 at r7c5 for r7 -> r8c5=7
29a. r9c5=4 & r9c6=9 -> r4c6=8 -> r4c5=9
29b. r8c7=4 -> r7c7=7
29c. 16(3) n2 = {18}7 {58}3 - no 6 at r1c6

30. hidden single 6 at r2c6 for c6
30a. 13(3)n23=6{25} {25} locked for c7/n3 in 13(3)n23 -> r3c7=8
30b. r2c9=1 -> r3c9=6

31. naked pair {67} r59c1

32. hidden pair {15} at r3c25 for r3
32a. 13(4)n1 = {134}5/{345}1/{23}71

33. 2 locked in 13(3)n12 for r3 -> no 2 at r2c3 -> no 7 at r3c3

34. r3c3={2/3} 13(4)n1 cannot have both -> {1345} only - no 2/7
34a. must include 3 - so r3c3=2 -> 13(3)n12=[724]

all the rest are very simple singles


Cheers
Richard

And here is the quicker walkthrough, as if I'd seen my original steps 23 and 35 much earlier. Apart from moving these steps nearer the beginning, I've followed the same order of steps with appropriate simplification. I know that there are a few cases where I could have done things even quicker if I'd tried to solve it again knowing what I did from the first attempt.


Clean-up is used in various steps, using the combinations in steps 1 and 2 for further eliminations from these two cell cages; it is also used for the two cell split sub-cages that are produced by applying the 45 rule.

1. R9C12 = {49/58/67}, no 1,2,3

2. R9C34 = {18/27/36/45}, no 9

3. 19(3) cage in N4, no 1

4. 23(3) cage in N5 = {689}, locked for R4 and N5

5. 6(3) cage in N5 = {123}, locked for N5

6. 22(3) cage in N69 = 9{58/67}, 9 locked for C9

7. 13(4) cage in N1 = 1{237/246/345}, no 8,9, 1 locked for N1

8. 45 rule on R9 2 outies R8C68 = 7 = {16/25/34}, no 7,8,9

9. 45 rule on N9 1 outie R6C9 – 2 = 1 innie R9C7, R6C9 = {56789} -> R9C7 = {34567}

10. 45 rule on C9 3 innies R159C9 = 9 = {126/135/234}, no 7,8

11. 45 rule on C1234 2 innies R45C4 = 7 -> R4C4 = 6, R5C4 = 1, R56C5 = {23}, locked for C5, clean-up: no 3,8 in R9C3

12. 45 rule on N47 2 outies R3C1 + R9C4 = 17 = [98], R9C3 = 1, clean-up: no 4 in R9C2, no 5 in R9C12

13. 45 rule on R123 1 innie R3C1 – 2 = 1 outie R4C9, R3C1 = 9 -> R4C9 = 7 -> 22(3) cage in N69 = {589} (step 6), locked for C9

14. R4C9 = 7 -> R23C9 = 7 = {16/34}, no 2

15. 2 locked in R159C9 = 2{16/34}

16. 45 rule on R1234 4 outies R5C1237 = 30 = {6789}, locked for R5
16a. 7 in R5 locked in R5C123, locked for N4

17. R56C6 = {457} = 9, 11 or 12 -> R6C7 = {346} but cannot have {447} combination -> only valid combinations {357/456} = 5{37/46}, 5 locked for C6 and N5, R6C7 = {36}, clean-up: no 2 in R8C8

18. R6C4 = {47}, only valid combinations for R678C4 are {249/348/357}, no 4,7 in R78C4

19. 45 rule on C89 2 innies R47C8 = 3 = {12}, locked for C8, clean-up: no 6 in R8C6

20. 45 rule on R6789 4 innies R6C5678 = 21, max R6C567 = 14 (R6C67 cannot be {67}, step 17) -> R6C8 = {89} (7 already eliminated)

21. 1 in R6 locked in R6C12, locked for N4

22. 45 rule on N4, 1 remaining outie R7C3 – 3 = 1 innie R6C1 -> min R7C3 = 4, max R6C1 = 6

23. 1 in N6 locked in 12(3) cage = 1{29/38/56}, no 4

24. 13(4) cage in N1 = 1{237/246/345}, must contain two of 2,3,4 -> only one of 2,3,4 in remainder of N1 -> min R23C3 = {25} = 7 -> no 7 in R3C4

25. 45 rule on N1 2 remaining innies R1C23 – 10 = 1 outie R3C4, min R3C4 = 2 -> min R1C23 = 12, no 2,3

26. 45 rule on N8 2 remaining innies R78C4 – 5 = 1 outie R9C7, R78C4 = 8 or 11 (step 18) -> R9C7 = {36}

27. Naked pair {36} in R69C7, locked for C7

28. 12(3) cage in N6 = {129} (only remaining combination) -> R5C7 = 9, R4C78 = {12}, locked for R4 and N6
28a. R4C123 = {345}, locked for N4
28b. R5C123 = {678}, locked for N4

29. R6C8 = 8, R6C9 = 5 (naked singles), R78C9 = {89}, locked for N9

30. Naked pair {34} in R5C89, locked for R5 and N6 -> R5C56 = [25], R6C5 = 3, R6C7 = 6 -> R6C6 = 4 (cage sum), R6C4 = 7, R9C7 = 3

31. R6C4 = 7 -> R78C4 = 8 = {35}, locked for C4 and N8 -> R123C4 = {249}, locked for N4

32. 12(3) cage in N9 = {147} -> R7C8 = 1, R78C7 = {47}, locked for C7 and N9

33. R4C78 = [12]

34. R9C9 = 2 (hidden single in N9), R89C8 = {56}, locked for C8

35. R159C9 = {234} (cannot now be {126} because 1,6 in the same cell) -> R15C9 = {34}, locked for C9

36. R7C3 – 3 = 1 R6C1 (step 22), R6C1 = {12} -> R7C3 = {45}

37. 9 in N2 locked in R12C4, valid combinations for 25(4) cage are 9(268/457} -> no 4 in R1C23

38. R9C8 = 5 (hidden single in R9), R8C8 = 6

39. R8C6 = 1 (step 8), R9C56 = 13 = [49]/{67}

40. R7C6 = 2 (hidden single in N8), R78C5 = 13 = {49}/[67]

41. 23(4) cage in N14, valid combinations are 9{347/356} = 39{47/56}, no 8, 3 locked for N4

42. 14(3) cage in N47, valid combinations are {158/167/248/257}, no 3, no 2 in R8C1

43. 3 in N7 locked in 14(3) cage = 3{29/47/56}, no 8, no 5 in R7C2

44. 8 in N7 locked in R78C1 -> 14(3) cage in N47 = 8{15/24}, no 6,7

45. Naked triple {458} in R78C1 + R7C3, locked for N7, clean-up: no 9 in R9C2 -> R9C12 = {67}, locked for R9 and N7
45a. R9C56 = [49] -> R78C5 = [67], R78C7 = [74], R4C56 = [98]

46. Only valid combinations for 13(3) cage in N23 are {238/256} = 2{38/56}, no 7, 2 locked in R12C7 for N3

47. Only valid combinations for 16(3) cage in N2 are {178/358} (cannot be {367} because those digits are all in the same cell), no 6, 8 locked for N2

48. Naked pair {45} in R47C3, locked for C3

49. 4 in N1 locked in 13(4) cage = 14{26/35}, no 7

50. 6 in R1 locked in R1C123, locked for N1

51. Naked pair {67} in R59C1, locked for C1 -> no 6 in 13(4) cage in N1 = {1345}, locked for N1

52. R6C1 = 2 (hidden single in C1) -> R6C3 = 9, R6C2 = 1

53. 2 in N1 locked in R23C3, locked for C3 and 13(3) cage -> R8C3 = 3, R7C2 = 9, R8C2 = 2, R3C4 = 4 (naked singles), R23C3 = {27}, locked for C3 and N1

54. R3C7 = 8 (hidden single in R3), 16(3) cage = 8{17/35}, no 6

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

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:

Nice puzzle Peter.


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.

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:

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.

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

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.