Radial Killer

[mhparker]

Hi guys,

Here's my first puzzle for 2008, containing 8 radially-oriented 2-cell cages.


Radial Killer (Est. rating: 1.5)



3x3:d:k:2048:6145:614523076150:615058976145:61456150:61505393:5897:5897386030955393:539358972846:4383:43835393:489938772846:4383:43931322:48995678:4655:4383:4383:43933636:4899:5678:5678:4655:46553891363656785185:51854676:467636365185:518531474676:4676

Note: It's a Killer-X, where no repeated digits are allowed on either diagonal.

Hope you enjoy it!

[Afmob]

This was a fun killer which had a cool move you could apply right at the start (step 4a).

Radial Killer Walkthrough:

1. N456+D\
a) 7(3) = {124} locked for C1+N4
b) 17(2) = {89} locked for C6+N5
c) 11(2) = {47/56}
d) Killer pair (89) locked in 15(2) + R5C6 for R5
e) 19(3): R46C9 <> 2 because R5C9 <> 8,9
f) Killer pair (89) locked in 15(2) + R6C6 for D\
g) 8(2): R2C2 <> 6,7

2. C12345
a) Outies C123 = 28(4) = 89{47/56} -> 8,9 locked for C4
b) Innies C1234 = 6(3) = {123}
c) Killer pair (67) of 15(2) @ C5 blocks {67} of 13(2)
d) Killer pair (89) locked in 15(2) @ C5 + 13(2) for C5

3. R6789
a) Innies R89 = 14(3): R8C19 <> 8,9 because R8C5 >= 6
b) Innies R89 = 14(3): R8C9 <> 6,7 because R8C15 >= 9
c) Outies R789 = 25(4) <> 1,2 because 89{17/26} blocked by R6C6 = (89)
d) Outies R789 = 25(4) = 7{369/459/468} because {3589} blocked by R6C6 = (89)
-> 7 locked for R6
e) 11(2): R8C2 <> 7,9

4. C6789+D/ !
a) ! Innies D/ = R3C7+R7C3 = 15/16/17(2) because R6C4+R5C5+R4C6 @ 17(5) <= 14
-> R3C7+R7C3 = (6789) = {69/78/79/89}
b) 12(3) @ D/: R3C6+R4C7 <> 6,7,8,9 because R3C7 >= 6
c) 17(5): R46C5 = 3/4/5(2) (step 4a) -> R46C5 <> 5,6,7

5. R789
a) Innies+Outies N7: 8 = R6C2+R8C4 - R7C3; R7C3 = (6789)
-> R6C2+R8C4 <> 3,4 because they're at least 14

6. R1234
a) Innies R1234 = 26(5) must have 8 or 9 and it's only possible @ R4C9 -> R4C9 = (89)
b) Innies R1234 = 26(5) <> 89{126/135/234} (step 6a)
c) Innies R1234 = 26(5) must have 2 of (567) and they are only possible @ R4C46
-> R4C46 <> 1,2,3,4
d) 11(2): R5C4 <> 7
e) 17(5): R5C5 <> 7 because R4C6+R5C5 would be >= 12
f) 7 locked in R4C46 for R4

7. R6789 !
a) 7 locked in R789C5 for N8
b) Killer pair (89) locked in Outies R789 + R6C6 for R6
c) 19(3): R5C9 <> 2,3 because R46C9 <= 15
d) ! Innies R6789 = 20(5): R6C9 <> 3,4 because Innies would be <= 19
e) 19(3) <> 7 because R46C9 <> 4

8. R456+D\ !
a) 7 locked in 15(2) @ R5 = {78} locked for R5+N4
b) R5C6 = 9, R6C6 = 8
c) 19(3) = 6{49/58} -> 6 locked for C9+N6
d) 15(2) @ D\ = {69} -> R9C9 = 9, R8C8 = 6
e) 8(2) <> 2
f) 19(3) = {568} -> R4C9 = 8, 5 locked for C9+N6
g) 14(4) = 2{138/147/345} -> 2 locked for N9
h) Killer pair (57) locked in 8(2) + R4C4 for D\
i) ! 17(5) = {12347} because (56) only possible @ R4C6
-> R4C6 = 7; 4 locked for C5+N5
j) R4C4 = 5, R5C4 = 6
k) 8(2) = {17} -> R1C1 = 7, R2C2 = 1

9. C456
a) 13(2) = {58} locked for C5+N2
b) 15(2) = {69} -> R7C5 = 6, R8C5 = 9
c) Hidden Single: R9C5 = 7 @ C7
d) R8C4 = 8, R7C4 = 4
e) 12(3) @ N8 = {237} -> 2,3 locked for R9+N8
f) 15(3) @ N8 = {357} -> R7C7 = 3, R7C6 = 5, R6C7 = 7
g) 18(4) = {1458} -> R8C6 = 1; {458} locked for N9; 8 locked for R9

10. N13+D/
a) 11(2) = {56} -> R9C1 = 6, R8C2 = 5
b) 1 locked in R3C89 @ 21(4) for N3; R4C8 <> 1
c) 5(2) = {23} locked for N3+D/

11. Rest is singles without considering the diagonals.

Rating: Hard 1.25. Required combo analysis of Innies/Outies.

[Andrew]

Thanks Mike. Very enjoyable puzzle. As Afmob said, there's a neat move that's available right at the start and then I found the result of it useful again later.

I'll rate Radial Killer as a moderate 1.25.

Here is my walkthrough.

This is a Killer-X

Prelims

a) 8(2) cage in N1 = {17/26/35}, no 4,8,9
b) R23C5 = {49/58/67}, no 1,2,3
c) 5(2) cage in N3 = {14/23}
d) R5C23 = {69/78}
e) R45C4 = {29/38/47/56}, no 1
f) R56C6 = {89}, locked for C6 and N5, clean-up: no 2,3 in R45C4
g) R5C78 = {14/23}
h) 11(2) cage in N7 = {29/38/47/56}, no 1
i) R78C5 = {69/78}
j) 15(2) cage in N9 = {69/78}
k) R1C456 = {126/135/234}, no 7,8,9
l) R456C1 = {124}, locked for C1 and N4, clean-up: no 6,7 in R2C2, no 7,9 in R8C2
m) R456C9 = {289/379/469/478/568}, no 1
n) 14(4) cage at R6C8 = {1238/1247/1256/1346/2345}, no 9

1. Killer pair 8,9 in R5C23 and R5C6, locked for R5

2. Killer pair 8,9 in R6C6 and 15(2) in N9, locked for D\

3. R23C5 = {49/58} (cannot be {67} which clashes with R78C5)
3a. Killer pair 8,9 in R23C5 and R78C5, locked for C5

4. 45 rule on R12 3 innies R2C159 = 20 = {389/479/569/578}, no 1,2

5. 45 rule on C12 3 innies R159C2 = 19 = {289/379/469/478/568}, no 1

6. 45 rule on D/ 2 innies R3C7 + R7C3 = 2 outies R46C5 + 12
6a. Min R46C5 = 3 -> min R3C7 + R7C3 = 15, no 1,2,3,4,5
6b. Max R3C7 + R7C3 = 17 -> max R46C5 = 5, no 5,6,7
6c. 45 rule on D/ R3C7 + R7C3 = 15,16,17 -> R4C6 + R5C5 + R6C4 -> 12,13,14 = {147/156/237/256/356} (cannot be {157/167/246/247/257/345/346/347} which clash with R45C4 and/or with R1C9 + R2C8)

7. Min R3C7 = 6 -> max R3C6 + R4C7 = 6, no 6,7,8,9

8. 45 rule on N3 2 outies R2C6 + R4C8 = 1 innie R3C7 + 5, min R3C7 = 6 -> min R2C6 + R4C8 = 11, no 1, no 2,3 in R4C8

9. 45 rule on N7 2 outies R6C2 + R8C4 = 1 innie R7C3 + 8, min R7C3 = 6 -> min R6C2+ R8C4 = 14, no 1,2,3,4

10. 45 rule on N1 2 outies R2C4 + R4C2 = 1 innie R3C3 + 10, min R2C4 + R4C2 = 11, no 1

11. 45 rule on C89 3 innies R159C8 = 16, max R5C8 = 4 -> min R19C8 = 12, no 1,2

12. 45 rule on C1234 3 innies R169C4 = 6 = {123}, locked for C4

13. R9C456 = {147/156/237/246/345}
13a. 1 of {147/156} must be in R9C4 -> no 1 in R9C56

14. Hidden killer pair 8,9 in R78C4 and R78C5 for N8 -> R78C4 must contain 8/9
14a. 45 rule on N8 4 innies R78C46 = 18, min R78C4 = 12 -> max R78C6 = 6, no 6,7

15. 45 rule on N9 4 outies R6C78 + R78C6 = 17, max R78C6 = 6 -> min R6C78 = 11, no 1, no 2 in R6C7

16. 45 rule on C6789 3 innies R149C6 = 15 = {267/357/456}, no 1

17. 45 rule on R789 4 outies R6C2378 = 25 = {3679/4579/4678} (cannot be {2689/3589} which clash with R6C6), no 2, 7 locked for R6
17a. Killer quad 1,2,3,4 in R6C1, R6C2378, R6C4 and R6C5, locked for R6
17b. Killer pair 8,9 in R6C2378 and R6C6, locked for R6

18. R456C9 = {469/568} (cannot be {289/379/478} because R6C9 only contains 5,6), no 2,3,7, 6 locked for C9 and N6, clean-up: no 9 in R8C8
18a. 8,9 of {469/568} must be in R4C9 -> R4C9 = {89}

19. Hidden killer pair 1,2 in R4C7 and R5C78 -> R4C7 = {12}

20. Hidden killer pair 8,9 in R4C2378 and R4C9, R4C9 = {89} -> R4C2378 must contain 8/9
20a. 45 rule on R123 4 outies R4C2378 = 19 = {1369/1378/1459/1468/2359/2368/2458} (cannot be {1279} because 1,2 only in R4C7)
20b. Killer quad 1,2,3,4 in R4C1, R4C2378 and R4C5, locked for R4, clean-up: no 7 in R5C4

21. Killer triple 5,6,7 in R1C1 + R2C2, R4C4 and R8C8 + R9C9, locked for D\

22. 17(5) cage in N5 = {12347} (only remaining combination, cannot be {12356} because 5,6 only in R4C6) -> R4C6 = 7, locked for D/, clean-up: no 4 in R5C4, no 4 in R8C2
22a. R19C6 (step 16) = {26/35}, no 4
22b. 4 in N5 locked in R456C5, locked for C5, clean-up: no 9 in R23C5

23. Naked pair {58} in R23C5, locked for C5 and N2, clean-up: no 7 in R78C5
23a. Naked pair {69} in R78C5, locked for C5 and N8
23b. R9C5 = 7 (hidden single in C5), clean-up: no 8 in R8C8
23c. R9C46 = {23} (step 13), locked for R9 and N8, clean-up: no 8 in R8C2

24. Naked pair {56} in R45C4, locked for C4 -> R78C4 = [48]

25. R4C6 + R5C5 + R6C4 (step 6c) = {147/237}
25a. 1 of {147} must be in R6C4 -> no 1 in R5C5
25b. R4C6 + R5C5 + R6C4 = 12 -> R3C7 + R7C3 = 17 (step 6c) = {89}, locked for D/, clean-up: no 2,3 in R8C2
25c. Naked pair {56} in R8C2 + R9C1, locked for N7
[Alternatively 5 locked in R8C2 + R9C1 for D/ but I saw step 25 first.]

26. Naked pair {15} in R78C6, locked for C6

27. 12(3) cage at R3C6 = {129/138} (only remaining permutations with 8,9), no 4 -> R4C7 = 1, clean-up: no 4 in R5C78

28. Naked pair {23} in R5C78, locked for R5 and N6 -> R5C5 = 4, locked for both diagonals, R5C1 = 1, clean-up: no 1 in R1C9 + R2C8

29. Naked pair {23} in R1C9 + R2C8, locked for D/ and N3 -> R6C4 = 1

30. Naked pair {23} in R1C49, locked for R1 -> R1C56 = [16], R1C49 = [23], R23C6 = [43], R9C46 = [32], R2C8 = 2, R5C78 = [23], R7C7 = 3, locked for D\, clean-up: no 5 in R1C1 + R2C2 = [71], locked for D\, R3C3 = 2, R8C8 = 6, R9C9 = 9, both locked for D\, R45C4 = [56], R56C6 = [98], R456C9 = [856], R8C2 + R9C1 = [56], R9C23 = [41], R78C5 = [69], R78C6 = [51], R8C13 = [37], R8C79 = [42], R5C23 = [78], R7C3 = 9, locked for D/

and the rest is naked singles

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