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PostPosted: Wed Aug 25, 2010 6:51 pm 
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Grand Master
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 694
Location: Saudi Arabia
PS 4: 13 Cager

I've been working on minimum cage killers for a while and thought 14 cages was the smallest possible. However in trying to get one at Assassin level I came up with a formulation that gave paper solvable ones. As I played around trying to improve the puzzle I accidently found this 13-cager.

The requirement is for a plain killer (no cage overlaps, no zeros, no disjoint cages and no X) with the minimum number of cages. Without the zero constraint I've done a 10-cager.

I seem to remember Udosuk posting a 12-cager somewhere but cannot remember if he allowed zeros - does anyone remember?

I also have a human solvable I will post in a while.

SS gives it 0.87 which is a bit high for a paper solvable but I think it overstates it slightly.
Edit: I just solved it again over three pints of Guinness and I think the solver does not fully see the negatives of large cages.
By the way for these and other complicated ones like H3lix's (note he is the only person that I admit does weirder puzzles than me) I find that as well as marking dots for positive possibilities putting dashes for negative ones reduces the marking required.
I call it the "Morse Code Method" - well its Friday night and my extremely large G&T (in a good cut glass tumbler - {when I was younger I promised that I would be "drunk as a lord" not "drunk in the gutter"}) allows some leeway.


Image

Sumocue PS code: 3x3::k:10752:10752:10752:10752:8452:8452:8452:8452:8452:10752:8970:8970:8970:8970:8970:8970:7184:8452:10752:8970:5396:4373:4373:4373:7184:4373:9242:10752:8988:4893:5396:7184:7184:4373:9250:9242:8988:8988:4893:7184:5396:9250:9250:9250:9242:8988:4893:7184:4893:6193:5396:9250:9250:9242:8988:7184:6193:6193:4893:6193:5396:9250:9242:8988:9792:9792:9792:9792:9792:6193:5396:9242:8988:9792:10570:10570:10570:10570:10570:10570:10570:


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PostPosted: Sat Aug 28, 2010 3:20 am 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Thanks HATMAN for another fun puzzle.

I did my best to use a paper solvable approach, I definitely didn't use elimination solving, but it's quite likely that HATMAN found a more direct solving path using technically simpler steps.

I don't think I'm giving anything away when I say that I found one of the hardest things was spotting the hidden singles. Normally I feel I find it hard to spot hidden singles on my worksheet; however if people using software solvers in editor mode are using a paper solvable approach, they won't have started with all the candidates so won't have an artificial aid with the "Show all the N" function.

After I'd finished I felt that this puzzle probably wouldn't have been any easier using elimination solving, although the explanations for some steps would have been shorter.

HATMAN wrote:
I call it the "Morse Code Method" - well its Friday night and my extremely large G&T (in a good cut glass tumbler - {when I was younger I promised that I would be "drunk as a lord" not "drunk in the gutter"}) allows some leeway.
I'm a single malt man, my favourite being Talisker; I hope to have one later this evening in an etched glass bought at a Scottish distillery.

Maybe HATMAN would like to show us his Morse Code Method in a few days?

Rating Comment:
HATMAN wrote:
SS gives it 0.87 which is a bit high for a paper solvable but I think it overstates it slightly.
I'll go the other way and rate my walkthrough at 1.0. I used several hidden killer pairs, which I felt were necessary for my solving path; if I'd only used one I may have rated my walkthrough at Easy 1.0 which is pretty close to the SS score of 0.87. It can be argued that my step 6b is actually a hidden killer quad but I won't increase my rating because of that because the step was so obvious.

Here is my walkthrough:
This was posted as a Paper Solvable puzzle so I won’t start with Prelims although some will get used during my solving path.

Candidates are only placed when stated, even though they may be implied by "only remaining combination" for cages.

1. 45 rule on R9 2 innies R9C12 = 4 = {13}, locked for R9 and N9

2. 38(6) cage at R8C2 cannot contain 1,2,4 -> R9C2 = 3, R9C1 = 1
2a. Rest of 38(6) cage contains 5,6,7,8,9, locked for R8
2b. R8C1 = {24}, 1,3 in R8 only in R8C789, locked for N9
[Alternatively 45 rule on R89 5(4+1) innies R8C1789 + R9C1 = 11 -> R9C1 = 1 (because min R8C1789 = 10) but that’s a higher rated step even though it’s a paper solvable one.]

3. 42(7) cage at R1C1 cannot contain 1,2 -> 33(6) cage at R1C5 must contain 1,2 in R1, 35(7) cage at R4C2 must contain 2 in C1 (1 is already placed)

4. 33(6) cage at R1C5 contain both of 1,2 = {126789} -> 42(7) cage at R1C1 must contain 3,4,5 in R1
4a. 1,2 of 33(6) must be in R1 -> R2C9 = {6789}

5. 35(7) cage at R4C2 must contain both of 1,2 so must contain one of 3,4
5a. Hidden killer pair 3,4 in R1C1 and 35(7) cage at R4C2 for C1 -> R1C1 = {34}, no 3,4 in R45C2

6. 42(7) cage at R1C1 = {3456789} with 3,4,5 in R1 -> R234C9 = {6789}
6a. 35(7) cage at R4C2 = {1235789/1245689}, CPE no 8,9 in R4C1 -> R4C1 = {67}
6b. Only one of 6,7,8,9 can be in C1 (because R234C1 contain three of 6,7,8,9) -> R45C2 = {6789}

7. 28(7) cage at R2C8 = {1234567}
7a. 17(5) cage at R3C4 must contain 1,2,3, CPE no 1,2,3 in R3C7 -> R3C67 = {4567}

8. 36(6) cage at R3C9 must contain 7,8,9, locked for C9 -> R2C9 = 6

9. 6 in R1 only in R1C234, locked for 42(7) cage at R1C1 -> R4C1 = 7
9a. R1C234 = {3456}
9b. Naked pair {89} in R23C1, locked for C1 and N1

10. 35(7) cage at R4C2 = {1245689} (only remaining combination) -> R45C2 = {89}, locked for C2 and N4

11. R1C1 = 3 (hidden single in C1), R1C234 = {456}
[I suppose for a paper solvable approach I ought to have said only available place in C1 but I’ll stick with hidden single even though I’m not using elimination solving.]

12. 45 rule on C9 2 remaining innies R19C9 = 3 = [12], R8C1 = 2 (hidden single in R8)

13. 35(7) at R2C2 = {1345679/2345678} (cannot contain both of 8,9 which would clash with R23C1), 3 locked for R2
13a. Hidden killer pair 1,2 in 35(7) cage at R2C2 and R3C3 for N1, 35(7) cage contains one of 1,2 -> R3C3 = {12}, 1 or 2 in 35(7) cage must be in N1
13b. 1 in R2 only in R2C23, locked for N1 -> R3C3 = 2
13c. R6C2 = 2 (hidden single in C2)
13d. R2C2 = 1 (hidden single in C2, because 1 in N1 must be in R2)

14. 35(7) at R2C2 (step 13) = {1345679} with 9 in R2 -> R23C1 = [89], R3C2 = 6
14a. R2C3 = 7 (hidden single in N1)
14b. R1C4 = 6 (hidden single in R1)

15. 17(5) cage at R3C4 must contain 1,2,3 -> R4C7 = 2, 1,3 locked for R3 with 1 in N2

16. R3C9 = 8 (hidden single in R3 because 17(5) cage at R3C4 cannot contain 8,9)

17. R2C8 = 2 (hidden single in R2 because 35(7) cage at R2C2 doesn’t contain 2)

18. R5C6 = 2 (hidden single in N5 because 2 already placed for 21(6) and 28(7) cages)
18a. R1C5 = 2 (hidden single in R1)
18b. R7C4 = 2 (hidden single in C4)

19. R6C45 = {89} (only remaining places for 8,9 in N5 because 21(6) and 28(7) cages cannot contain 8,9), locked for R6

20. R5C4 = 7 (hidden single in N5 because 21(6) cage cannot contain 7)

21. R1C6 = 8 (hidden single in N2 because 17(5) cage cannot contain 8)
21a. R1C78 = {79}, locked for N3

22. R8C2 = 7 (hidden single in C2)

23. 17(5) cage at R3C4 = {12347} (only remaining combination), 4 locked for R3 -> R3C7 = 5
[I ought to have noticed that at step 15 although it wouldn’t have placed R3C7 until step 20.]

24. 36(6) cage at R3C9 and 36(7) cage at R4C8 must both contain 7 and 9 (36(7) cage must contain 7 because it contains 2), only one pair of these can be in N6 -> the other pair must be in R7C89 -> R7C89 = {79}, locked for R7 and N9

25. Hidden killer pair 4,5 in R1C2 and R7C2 for C2 but R7C2 cannot be 5 -> R7C2 = 4, R1C23 = [54]
[Alternatively can use two hidden singles in C2.]

26. 4 in C1 must be in R56C1 -> 19(5) cage at R4C3 cannot contain 4 = {12358} (only remaining combination containing 2 and one of 8,9) -> R6C4 = 8, R6C5 = 9

27. 24(5) cage at R6C5 contains 2,9 = {12489/23469} -> R8C7 = 4

28. R8C8 = 1, R8C9 = 3 (hidden singles in R8)

29. R7C7 = 6 (only candidate of {123456} which can be placed in this cell), R4C4 + R5C5 + R6C6 = {345}
29a. R9C8 = 5, R9C7 = 8 (hidden singles in N9)

30. R4C56 = {16} (hidden pair in N4), locked for R4)
30a. 28(7) cage at R2C8 = {1234567} (step 7) -> R6C3 = 3

31. 19(5) cage at R4C3 (step 26) = {12358} -> R7C5 = 3, R45C3 = [51]

32. R7C1 = 5 (hidden single in C1), R56C1 = {46}

33. R7C36 = {18} (hidden pair in R7) -> R7C3 = 8, R7C6 = 1, R4C56 = [16]

34. R8C5 = 8, R8C3 = 6 (hidden singles in R8), R9C3 = 9 (hidden single in C3)

35. R6C7 = 1 (hidden single in C7)

36. Naked pair {79} in R17C8, locked for C8 -> R6C9 = 7 (hidden single in R6), R7C89 = [79], R1C78 = [79]

37. R6C6 = 5 (hidden single in R6), R8C46 = [59]

38. R9C5 = 6, R9C6 = 7, R9C4 = 4 (hidden singles in R9), R4C4 = 3, R5C5 = 4, R56C1 = [64]

39. R3C4 = 1, R2C4 = 9 (hidden singles in C4)

40. R2C5 = 5, R3C5 = 7 (hidden singles in C5)

41. R2C67 = {34} (hidden pair in R2) -> R2C7 = 3, R2C6 = 4, R3C6 = 3 (hidden single in C6), R3C8 = 4 (hidden single in R4)

42. R5C7 = 9 (hidden single in C7), R45C2 = [98]

43. R4C8 = 8, R4C9 = 4 (hidden singles in R4), R5C9 = 5 (hidden single in C9), R5C8 = 3, R6C8 = 6 (hidden singles in C8)

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


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