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This cage pattern created some very hard Killers and I hade to adjust the cage sums several times to get the difficulty right, but I think it was worth it. Here you can find lots of I/O moves, so have fun spotting and using them!

Assassin 152


Note that this is an X-Killer and R2C8+R3C7+R5C5+R7C3+R8C2 is a 29(5) remote cage.

3x3:d:k:4608:4608:4608:4608:3844:3844:3846:3846:2568:3849:3849:3851:3596:4877:3844:3846:7440:2568:3849:3851:3596:3596:3606:4877:7440:3097:3097:3355:5148:5148:3596:3606:3606:4877:4386:3097:3355:5148:4134:6183:7440:3606:4877:4386:3116:4653:5148:4134:6183:6183:6450:4386:4386:3116:4653:4653:7440:4134:6183:6450:6450:2621:2622:2367:7440:5697:2114:4134:6450:2621:2622:2622:2367:5697:5697:2114:2114:6477:6477:6477:6477:

Solution:

SS Score: 1.26
Estimated rating: 1.25

Edit: No further version planned.

Thanks for this interesting Assassin Afmob. There is nothing too much difficult (my walkthrough uses some cage blockers and a min-max argument to open the puzzle), and the steps flow nicely. There are many opportunities which makes this Assassin very enjoyable ; I guess there should be others different solving paths.

Are you planning a V2 Afmob ?

Walkthrough A152



Edit : Ed has pointed out some steps that needed further explanations that have been added to my WT.

Perfect difficulty thanks Afmob. No need for a V2!Was difficult for me! But I missed manu's really neat step 3d. A very elegant trick which I'd give an Easy 1.5 rating since I think it is very hard to find. Here is a more conventional (though much longer) way to solve this puzzle which....I'll rate at Easy 1.50 [but see edit following] I took sooo many tangents before getting to this optimised, simple to follow WT. Also, I like to rate puzzles that have remote cages and diagonal cages a bit higher than standard patterns. [edit: I'll up that rating to 1.50 since step 13c is chainy then a bit extra for the non-standard pattern]. It's cracked by step 17.

Assassin 152 (39 steps)
This is an optimised solution so I have occasionally missed some obvious eliminations since they aren't essential. However, I try and do clean-up as I go. Let me know of any corrections or clarifications.

Prelims
i. 10(2)n3: no 5
ii. 15(2)n1 = {69/78}
iii. two 14(4)n2 cages: no 9
iv. 13(2)n4: no 1,2,3
v. 12(2)n6: no 1,2,6
vi. 10(2)n9: no 5
vii. 10(3)n9: no 8,9
viii. 9(2)n7: no 9
ix. 22(3)n7: no 1,2,3,4
x. 8(3)n8: no 6,7,8,9

1. 29(5)n357: max. any three cells = {789} = 24 -> min. any two cells = 5 (important for next step)

2. "45" n7: 1 outie r6c1 - 4 = 2 innies r7c3 + r8c2
2a. min. 2 innies = 5 (step 1) -> r6c1 = 9
2b. r7c3 + r8c2 = h5(2) = {14/23}
2c. no 3 in r5c9
2d. no 4 in 13(2)n4

3. split cage 24(3)r2c8 + r3c7 + r5c5 = 24 = {789}
3a. all locked for D/
3b. no 1,2 in r8c1
3c. no 1,2,3 in r2c9

4. "45" n4: 2 remaining innies r56c3 = 3 = {12}
4a. both locked for c3 & n4 & 16(4)n4
4b. no 3,4 in r8c2 (h5(2)n7)

5. "45" r9: 3 outies r8c134 = 19 (no 1)

6. 8(3)n8 = 1{25/34}
6a. 1 locked for r9 & n8
6b. no 8 in r8c1

7. h19(3)r8c134 (step 5) = {379/469/478/568}(no 2) ({289} blocked by no candidates in r8c1)
7a. = one of 3/4/5 which must go in r8c4 -> no 3,4,5 in r8c13
7b. and = one of 8/9 -> r8c3 = (89)
7c. r9c1 = (23)

8. [32] blocked from h5(2)n7 by r9c1 (missed this first two times through - originally got it from hidden single 4 in n7)
8a. -> r7c3 = 4 & r8c2 = 1 (both placed for D/)
8b. no 6,9 in r2c9
8c. no 6 in r8c7
8d. no 9 in r7c8

9. r7c12 = 9 (cage sum) = {27/36}(no 5,8)

10. 5 locked in n7 in 22(3) = {589}: 5 locked for r9

11. 13(2)n4 = {58} ({67} blocked by r8c1)
11a. both locked for c1 & n4

12. 9 in n5 only in r5: 9 locked for r5
12a. no 3 in r6c9

13. 12(2)n6 = {48/57}
13b. 10(2)n3 = [28/37/46]
13c. -> 7 locked in these cages (since if 12(2) is not 7 it is {48} -> 10(2) = [37])
13d. 7 locked for c9

14. "45" n3: 1 outie r4c9 + 8 = 2 innies r2c8 + r3c7
14a. min. 2 innies = {78} = 15 -> min. r4c9 = 7
14b. -> r4c9 = (89)
14c. 2 innies = 16/17 = {79/89}
14d. 9 locked for n3 & D/

15. r5c4 = 9 (hsingle n5)

16. "45" n1: 1 outie r1c4 - 3 = r3c3
16a. r1c4 = (68)
16b. r3c3 = (35)

17. 14(4)n2 must have 3/5 for r3c3 = {1238/1256/1346/2345}(no 7)

This cracks it. Now trying to get to singles ASAP so cleanup missing.
18. r7c4 = 7 (hsingle c4)

19. r8c5 = 6 (cage sum)

20. r89c1 = [72] (2 placed for D/)

21. 8(3)n8 = {134}: 3 & 4 locked for n8

22. "45" c1234: 3 innies r689c4 = 12 and must have 3/4 for r8c4 = {345} only.
22a. r6c4 = 5
22b. r89c4 = {34}: both locked for c4 & n8
22c. r9c5 = 1

23. r67c5 = 10 (cage sum) = {28}: both locked for c5

24. r5c5 = 7 (placed for both D\ & D/)

25. naked pair {89} in n3: 8 locked for n3

26. "45" n3: 1 remaining outie r4c9 = 9 (finally caught up with manu!)

27. r3c89 = 3 (cage sum) = {12}: both locked for r3 & n3

28. naked pair {68} in r13c4: both locked for c3 & n2

29. 14(4)r3c5 must have 1/2 to keep below the cage sum
29a. r5c6 = (12)

30. naked pair {12} in n5 and r5c36: both locked for n5 & r5

31. r67c5 = [82]

32. "45" n89: 1 remaining outie r6c6 = 3 (placed for D\)

33. r3c3 = 5 (placed for D\)

34. "45" n1: 1 remaining outie r1c4 = 8
34a. r3c4 = 6

35. r34c5 = [34]

36. "45" n6: 2 remaining outies r2c5 + r3c6 = 12 = [57]

37. r1c9 + r4c6 = [36]
37a. r2c9 = 7 (cage sum)

38. r1c5 = 9
38a. r12c6 = 6 (cage sum) = {24} both locked for n2 & c6

39. r3c1 = 4
39a. r2c12 = 11 (cage sum) = [38]

rest are naked singles
Cheers
Ed

Thanks Afmob for a challenging puzzle!

It took me quite a long time to find the key breakthrough in step 22, which is the same as manu's step 3d. With hindsight it's the same sort of move as my step 2 but there's something I can't explain which makes it a much harder move to spot.

I'll rate my walkthrough for A152 at Hard 1.25 because although step 22 isn't technically difficult IMHO it's hard to spot.

I've probably also factored that into my rating.

Here is my walkthrough for A152.

Prelims

a) R2C3 + R3C2 = {69/78}
b) R12C9 = {19/28/37/46}, no 5
c) R45C1 = {49/58/67}, no 1,2,3
d) R56C9 = {39/48/57}, no 1,2,6
e) R7C8 + R8C7 = {19/28/37/46}, no 5
f) R89C1 = {18/27/36/45}, no 9
g) 22(3) cage in N7 = {589/679}, 9 locked for N7
h) 8(3) cage in N8 = {125/134}, 1 locked for N8
i) 10(3) cage in N9 = {127/136/145/235}, no 8,9
j) 14(4) cage at R2C4 = {1238/1247/1256/1346/2345}, no 9
k) 14(4) cage at R3C5 = {1238/1247/1256/1346/2345}, no 9

1. 45 rule on R9 2 outies R8C34 = 1 innie R9C1 + 10
1a. Max R8C34 = 14 -> max R9C1 = 4, clean-up: min R8C1 = 5
1b. Min R8C34 = 11, no 5 in R8C3, no 1 in R8C4
1c. 1 in N8 locked in R9C45, locked for R9, clean-up: no 8 in R8C1
1d. Min R9C1 = 2 -> min R8C34 = 12, no 6 in R8C3, no 2 in R8C4
[I enjoy recursive steps!]

2. 45 rule on N7 1 outie R6C1 = 2 innies R7C3 + R8C2 + 4
2a. Min R7C3 + R8C2 = 5 (because max R2C8 + R3C7 + R5C5 = 24) -> R6C1 = 9, clean-up: no 4 in R45C1, no 3 in R5C9
2b. R7C3 + R8C2 = 5 = {14/23}, R2C8 + R3C7 + R5C5 = 24 = {789}, locked for D/, clean-up: no 1,2,3 in R2C9
2c. R7C12 = 9 = {18/27/36/45}
2d. 9 in N5 locked in R5C45, locked for R5, CPE no 9 in R7C5, clean-up: no 3 in R6C9
[With hindsight 45 rule on N7 4(3+1) outies R2C8 + R3C7 + R5C5 + R6C1 = 33 -> R6C1 = 9, R2C8 + R3C7 + R5C5 = 24 = {789}, locked for D/ … is more direct. I’m not sure how I’d rate that step; I’m tending to rate 2+1, 3+1 etc. outies higher than I used to do.]

3. 45 rule on N4 2 remaining innies R56C3 = 3 = {12}, locked for C3, N4 and 16(4) cage at R5C3, clean-up: no 3,4 in R8C2
3a. R56C3 = 3 -> R7C4 + R8C5 = 13 = {49/58/67}, no 3

4. 45 rule on R9 3 outies R8C134 = 19 = {379/469/478/568}
4a. 8,9 of {379/478} must be in R8C3 -> no 7 in R8C3
4b. 6 of {568} must be in R8C1 -> no 5 in R8C1, clean-up: no 4 in R9C1

5. R7C3 + R8C2 (step 2b) = [41] (cannot be {23} which clashes with R9C1), placed for D/, clean-up: no 6,9 in R2C9, no 5,8 in R7C12 (both step 2c), no 9 in R7C8, no 9 in R8C5 (step 3a), no 6 in R8C7
5a. 4 in N4 locked in R456C2, locked for C2
5b. 5 on D/ locked in R4C6 + R6C4, locked for N5

6. Naked quad {2367} in R7C12 + R89C1, locked for N7
6a. 5 in N7 locked in R9C23, locked for R9

7. R45C1 = {58} (cannot be {67} which clashes with R9C1), locked for C1 and N4

8. 45 rule on N3 2 innies R2C8 + R3C7 = 1 outie R4C9 + 8
8a. Min R2C8 + R3C7 = 15 -> min R4C9 = 7
8b. Min R4C9 = 7 -> max R3C89 = 5 -> R3C89 = {1234}

9. 45 rule on C1 2 outies R27C2 = 1 innie R1C1 + 10
9a. Max R27C2 = 16 -> max R1C1 = 6
9b. Min R27C2 = 11, no 2,3 in R2C2

10. Hidden killer pair 1,4 in R1C1 and R23C1 for C1, R23C1 cannot contain both of 1,4 -> R1C4 = {14}, R23C1 must contain one of 1,4
10a. 15(3) cage in N1 = {168/249/348/456} (cannot be {258/267/357} which don’t contain 1 or 4, cannot be {159} because 5,9 only in R2C2), no 7
10b. 5,8 of {168/456} must be in R2C2 -> no 6 in R2C2
10c. 7 in C1 locked in R78C1, locked for N7, clean-up: no 2 in R7C1 (step 2c)

11. 45 rule on N1 1 outie R1C4 = 1 innie R3C3 + 3, R1C4 = {689}, R3C3 = {356}

12. 45 rule on N9 1 innie R7C7 = 1 outie R9C6, no 1,5 in R7C7, no 4 in R9C6

13. 6,7 in R9 locked in R9C6789 = {3679/4678}, no 2, clean-up: no 2 in R7C7 (step 12)

14. 45 rule on N9 4 innies R7C7 + R9C789 = 25 = {3679/4678}, 6,7 locked for N9, clean-up: no 3,4 in R7C8 + R8C7

15. 45 rule on N1 4 innies R1C123 + R3C3 = 15 = {1239/1257/1356} (cannot be {1248} because R3C3 only contains 3,5,6, cannot be {1347} because 1,4 only in R1C1, cannot be {2346} which clashes with 15(3) cage in N1), no 4,8 -> R1C1 = 1, placed for D\
15a. 2 of {1239/1257} must be in R1C2 -> no 7,9 in R1C2

16. 45 rule on N6 3 innies R45C7 + R4C9 = 16 = {169/259/268/349/367} (cannot be {178/358/457} which clash with R56C9)
16a. R4C9 = {789} -> no 7,8,9 in R45C7

17. 14(4) cage at R2C4 = {1238/1256/1346/2345} (cannot be {1247} because R3C3 only contains 3,5,6), no 7

18. 45 rule on R1 3 outies R2C679 = 13 = {148/157/238/247/346} (cannot be {139/256} because R2C9 only contains 4,7,8), no 9

19. 45 rule on C1234 2 remaining innies R56C4 = 2 outies R89C5 + 7
19a. Max R56C4 = 15 -> max R89C5 = 8, no 8 in R8C5, no 4 in R9C5, clean-up: no 5 in R7C4 (step 3a)
19b. Min R89C5 = 5 -> min R56C4 = 12, no 1,2,3,4,6 in R5C4, no 2 in R6C4
19c. 1 in N5 locked in R4C5 + R5C6 + R6C5, CPE no 1 in R3C5
19d. 7 in C4 locked in R57C4, CPE no 7 in R7C5

20. 45 rule on N89 1 outie R6C6 = 1 remaining innie R7C5 + 1, no 2,8 in R6C6, no 8 in R7C5

21. 24(4) cage at R5C4 = {2589/2679/3579/3678/4569} (cannot be {1689/3489/4578} because 1,4,7,8,9 only in R5C4 + R6C5), no 1
21a. 4,7,8,9 only in R5C4 + R6C5 -> no 2,3,6 in R6C5

22. 45 rule on N3 2(1+1) remaining outies R4C9 + R5C5 = 16 = {79} (cannot be [88] which clashes with R45C1), no 8, CPE no 7 in R5C9, clean-up: no 5 in R6C9
22a. 8 on D/ locked in R2C8 + R3C7, locked for N3, clean-up: no 2 in R1C9
22b. Killer pair 4,7 in R2C9 and R56C9, locked for C9 -> R4C9 = 9, R5C5 = 7 (step 22), placed for D/ and D\, clean-up: no 6 in R7C4 (step 3a), no 6 in R7C5 (step 20), no 7 in R9C6 (step 12)
22c. 9 on D/ locked in R2C8 + R3C7, locked for N3

23. R4C9 = 9 -> R3C89 = 3 = {12}, locked for R3 and N3

24. R5C4 = 9 (hidden single in R5), clean-up: no 6 in R3C3 (step 11), no 4 in R8C5 (step 3a)
24a. 24(4) cage at R5C4 (step 21) = {2589/4569}, no 3, clean-up: no 4 in R6C6 (step 20)

25. 45 rule on N8 4 remaining innies R7C5 + R789C6 = 24
= {2589/2679} (cannot be {3489/3678} because R7C5 only contains 2,5, cannot be {3579/4569/4578} which clash with 8(3) cage), no 3,4, 2 locked for N8, clean-up: no 3 in R7C7 (step 12), no 5 in R8C4 (prelim h)

26. R9C1 = 2 (hidden single in R9), placed for D/, R8C1 = 7, clean-up: no 9 in R2C2 (step 10a)
26a. Naked pair {36} in R7C12, locked for R7, clean-up: no 6 in R9C6 (step 12)

27. R1C2 = 2 (hidden single in C2)
27a. R1C12 = [12] -> R1C123 + R3C3 (step 15) = {1239/1257}
27b. 7,9 only in R1C3 -> R1C3 = {79}

28. R7C7 = 9 (hidden single on D\), R3C7 = 8, R2C8 = 9, R8C7 = 2, R7C8 = 8, R7C4 = 7, R8C5 = 6 (step 3a), R9C6 = 9 (step 12), clean-up: no 7 in R2C3, no 6 in R3C2

29. Naked pair {25} in R7C56, locked for R7 and N8 -> R7C9 = 1, R3C89 = [12], R8C6 = 8, R8C3 = 9, R1C3 = 7, R3C2 = 9, R2C3 = 6, R4C3 = 3, R3C3 = 5, placed for D\, R2C2 = 8, R9C23 = [58]
29a. R7C9 = 1 -> R8C89 = 9 = [45], clean-up: no 7 in R6C9
29b. R8C4 = 3, R9C45 = [41], R3C4 = 6, R1C4 = 8, R24C4 = [12], R6C4 = 5, R4C6 = 6, placed for D/, R6C6 = 3, placed for D\, R7C56 = [25], R6C5 = 8 (step 25)

and the rest is naked singles

Good walkthroughs from all of you! I used a Killer quad (step 3f) and an unusual IOD using a nonet and a column (step 4e) to solve it .

A152 Walkthrough:

1. R789
a) Outies R9 = 19(3) <> 1; R8C13 <> 2,3,4,5 since R8C4 <= 5
b) 8(3) = 1{25/34} -> 1 locked for R9+N8
c) 9(2) = [63/72]
d) 22(3) = {589} locked for N7 since {679} blocked by R8C1 = (67); 5 also locked for R9
e) 18(3) <> 1 since R7C12 <> 8,9
f) 1 locked in 29(5) @ N7 for D/ -> 29(5) = 189{47/56} -> 8,9 locked for D/
g) Innies+Outies N7: 4 = R6C1 - (R7C3+R8C2) -> R6C1 = 9 because R7C3+R8C2 >= 5
-> R7C3+R8C2 = 5 = {14} locked for N7+D/
h) 4 locked in 29(5) @ N7 = {14789} -> 7 locked for D/

2. C123
a) Innies N4 = 3(2) = {12} locked for C4+N4+16(4)
b) R7C3 = 4, R8C2 = 1
c) 13(2) = {58} locked for C1+N4 because {67} blocked by R8C1 = (67)
d) 15(3) <> 1 because R23C1 <> 5,9 and {168} blocked by Killer pair (68) of 15(2)
e) Hidden Single: R1C1 = 1 @ C1
f) 4 locked in 15(3) @ C1 for N1 -> 15(3) = 4{29/38/56} <> 7; R2C2 = (589)
g) Innies+Outies N1: 3 = R1C4 - R3C3 -> R3C3 <> 7,8; R1C4 = (689)

3. N356 !
a) 9 locked in R5C45 @ N5 for R5
b) 12(2) <> 3
c) Innies+Outies N3: -8 = R4C9 - (R2C8+R3C7): R4C9 = (789) since R2C8+R3C7 >= 15
d) 12(3): R3C89 = (1234) since R4C9 >= 7
e) 10(2) = [28/37/64]
f) ! Killer quad (4789) locked in 15(3) + R2C89+R3C7 for N3
g) 10(2) <> {28} since it's a Killer pair of 12(3)
h) Killer pair (47) locked in R2C9 + 12(2) for C9
i) 12(3) = 1{29/38} -> 1 locked for R3+N3
j) 5 locked in 15(3) @ N3 <> 9
k) 9 locked in R2C8+R3C7 @ N3 for D/

4. C456 !
a) 14(4) @ N1 <> 7 because R3C3 = (356)
b) Hidden Single: R5C4 = 9 @ N5, R7C4 = 7 @ C4
c) 16(4) = {1267} -> R8C5 = 6
d) R8C1 = 7 -> R9C1 = 2
e) ! Innies+Outies N1+C4: -4 = R9C5 - R6C4 -> R9C5 = 1, R6C4 = 5
f) 8(3) = {134} -> 3,4 locked for C4+N8
g) 24(4) = {2589} -> 2,8 locked for C5
h) R5C5 = 7
i) Innies+Outies N9: R7C7 = R9C6 = (89)
j) 25(4) = 89{26/35} -> R6C6 = (36)
k) Naked pair (36) locked in R46C6 for C6+N5

5. N59
a) Naked pair (28) locked in R4C4+R6C5 for N5
b) 14(4) = {1346} @ R3C5 -> R5C6 = 1, R4C5 = 4, R3C5 = 3. R4C6 = 6
c) R1C9 = 3 -> R2C9 = 7
d) R6C6 = 3
e) 10(3) = {145} -> R8C9 = 5, R8C8 = 4, R7C9 = 1
f) 10(2) = {28} locked for N9
g) Innies+Outies N9: R7C7 = R9C6 = 9

6. Rest is singles.