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HATMAN
 Post subject: Minimum Cages
PostPosted: Fri Jul 23, 2010 7:13 pm 
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Joined: Wed Apr 30, 2008 9:45 pm
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I noted that Nikoli_freak's hands allowed 16 cages if twins and singles are allowed (less than minimum Vanilla). So I started wondering about the minimum number of cages.

This one is a 10+ cage zero that is theoretically solvable (about 20 fishy things). So the questions are for a plain killer (no X, no zeros, no twins):

What is the minimum number of cages?

What is the minimum number of cages at Assassin level?

Image

If there have been previous discussions on this please point them out.

Code at Andrew's request - corrected - thanks Ed
3x3::k:11:5642:5377:5378:9987:9987:9987:9987:9987:12:13:5642:5377:5378:9987:5378:8708:8708:14:15:8712:5642:5377:5378:5378:5637:8708:16:17:18:8712:5642:5377:5378:8708:5637:9991:19:6409:6409:8712:20:5377:5637:8708:21:9991:22:6409:6150:8712:6150:5377:5637:23:24:9991:6409:8712:6150:5637:5637:6150:25:26:27:9991:6409:8712:6150:6150:28:29:30:31:6409:9991:9991:32:33:34:


Note this counts as human solver as both SudokuSolver and JSudoku fail completely on it.


Last edited by HATMAN on Sat Aug 20, 2011 1:12 pm, edited 2 times in total.

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Andrew
 Post subject: Re: Minimum Cages
PostPosted: Tue Jul 27, 2010 8:31 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
HATMAN wrote:
... So I started wondering about the minimum number of cages.

This one is a 10+ cage zero that is theoretically solvable (about 20 fishy things).
Thanks for a fun puzzle. :D Quite an achievement to create a solvable 10 cage zero killer! Your introduction gave the impression that it would be very difficult but once I realised how to get into it I didn't find it too hard although I wasn't really happy with my step 23; I hope someone can find a better way at that stage.

HATMAN wrote:
What is the minimum number of cages at Assassin level?
That's a good question. I interpret it to mean one which can be solved without using chains, "clones", fishy moves (apart from X-Wings), etc.

Rating Comment:
I'll rate my walkthrough for Minimum Cages Killer at Hard 1.5. I used several closely related forcing chains, a "clone", a "big" caged X-Wing and a contradiction move using a "clone".

I've given an alternative to the forcing chains in steps 4 to 7. It's probably a caged fish; I look at it as a caged Super X-Wing for 2 numbers. Once spotted, this step is simple at a human level although technically probably in the 1.5 range.

Here is my walkthrough for Minimum Cages Killer:
Prelims

a) 21(6) cage at R1C3 = {123456}
b) 21(6) cage at R1C4 = {123456}
c) 39(6) cage at R1C5 = {456789}
d) 34(5) cage at R2C8 = {46789}
e) 22(6) cage at R3C8 = {123457}
f) 39(6) cage at R5C1 = {456789}

1. 39(6) cage at R1C5 = {456789}, CPE no 4,5,6 in R1C4

2. R2C7 + R3C78 = {123} (hidden triple in N3)
2a. Naked triple {123} in R1C4 + R23C7, locked for 21(6) cage at R1C4, no 1,2,3 in R2C5 + R3C6 + R4C7

3. 5 in N3 only in R1C789, locked for R1 and 39(6) cage at R1C5, no 5 in R2C6

[I assume the next steps are some of the 20 fishy things which HATMAN referred to, even though I don’t know how to express them as fishes.]

4. Consider placements for 4 in 21(6) cage at R1C4
4 in R4C7 => 4 in 34(5) cage at R2C8 must be in R2C89 + R3C9 => 4 in 39(6) cage at R1C5 must be in R1C56 + R2C6
or 4 in R2C5 + R3C6
-> 4 must be in R1C56 + R2C56 + R3C6, locked for N2

5. Consider placements for 6 in 21(6) cage at R1C4
Same logic as for 4 in step 4
-> 6 must be in R1C56 + R2C56 + R3C6, locked for N2

[Now to work those two steps the other way round.]

6. Consider the placements for 4 in 34(5) cage at R2C8
4 in R2C89 + R3C9 => 4 in 39(6) cage at R1C5 must be in R1C56 + R2C6 => 4 in 21(6) cage at R1C4 must be in R4C7
or 4 in R4C8 + R5C9
-> 4 must be in R4C78 + R5C9, locked for N6
6a. 4 in 22(6) cage at R3C8 only in R7C78, locked for R7 and N9
6b. 4 in 21(6) cage at R1C3 only in R1C3 + R4C6, CPE no 4 in R1C6 + R4C3

7. Consider the placements for 6 in 34(5) cage at R2C8
Same logic as for 6 in step 6
-> 6 must be in R4C78 + R5C9, locked for N6
7a. R1C3 + R4C6 = {46} (hidden pair in 21(6) cage at R1C3), CPE no 6 in R1C6 + R4C3

[With hindsight the main part of steps 4 to 7 can be written more directly as one step
21(6) cage at R1C4, 39(6) cage at R1C5 and 34(6) cage at R2C8 must each contain both of 4,6 -> 4,6 locked in 21(6) cage at R1C4, 39(6) cage at R1C5 and 34(6) cage at R2C8 for N236
I guess this is some variety of caged fish.]

8. R4C6 “sees” all 4,6 in 21(6) cage at R1C3 except for R2C5 -> R2C5 is a “clone” of R4C6 -> R2C5 = {46}
8a. R1C3 + R4C6 = {46}, R2C5 = R4C6 -> naked pair {46} in R1C3 + R2C5, CPE no 4,6 in R1C5 + R2C3

9. 5 in 21(6) cage at R1C3 only in N2 and N6, 5 in 21(6) cage at R1C4 only in N2 and N6 -> 5 locked in 21(6) cage at R1C3 and 21(6) cage at R1C4 for N26 (caged X-Wing)
9a. R7C78 = {45} (hidden pair in 22(6) cage at R3C8), locked for R7 and N9
9b. R1C9 = 5 (hidden single in C9)

10. 7 in 22(6) cage at R3C8 only in R4C9 + R5C8 + R6C9, locked for N6
10a. 7 in 34(6) cage at R2C8 only in R2C89 + R3C9, locked for N3
10b. 7 in 39(6) cage at R1C5 only in R1C56 + R2C6, locked for N2

11. R1C124 = {123} (hidden triple in R1)
11a. Max R1C2 = 3 -> min R2C3 + R3C4 + R4C5 = 19, no 1 in R2C3 + R3C4 + R4C5

12. 24(6) at R6C5 = {123468/123567} (cannot be {123459} because 4,5 only in R6C5), no 9
12a. 4,5 only in R6C5 -> R6C5 = {45}

13. Hidden killer pair 4,6 in R1C3 and R1C78 for R1, R1C3 = {46} -> R1C78 must contain one of 4,6 -> R2C6 = {46}

14. Naked pair {46} in R2C56, locked for R2 and N2 -> R3C6 = 5
14a. Naked pair {46} in R4C67, locked for R4
14b. Naked pair {46} in R24C6, locked for C6
14c. Naked triple {123} in R3C578, locked for R3

15. Naked triple {789} in R2C89 + R4C8, locked for 34(5) cage at R2C8, 7 also locked for R2
15a. Naked pair {46} in R35C9, locked for C9

16. R4C8 = 9 (hidden single in N6), R2C89 = {78}, locked for R2 and N3
16a. R1C7 = 9 (hidden single in N3)
16b. R3C4 = 9 (hidden single in N2)

17. R6C7 = 8 (hidden single in N6)
17a. 24(6) at R6C5 (step 12) = {123468} (only remaining combination) -> R6C5 = 4, R7C69 + R8C78 = {1236}, 6 locked for R8 and N9

18. R2C56 = [64], R4C67 = [64], R1C3 = 4, R1C8 = 6, R35C9 = [46], R7C78 = [54]

19. R6C8 = 5 (hidden single in N6)

20. R89C7 = [67] (hidden pair in C7)

21. At least one of 8,9 in N9 must be in R9C89 -> killer pair 8,9 in R9C6 and R9C89, locked for R9 -> R9C5 = 5
21a. Hidden killer pair 8,9 in R9C6 and R9C89 for R9, R9C6 = {89} -> R9C89 must contain one of 8,9 -> R8C9 = {89}

22. 22(4) cage at R1C2 = {1579/2389}
22a. 7,8 only in R4C5 -> R4C5 = {78}
22b. Naked pair {78} in R14C5, locked for C5

23. 34(6) cage at R3C3 = {136789/235789} cannot be {136789}, here’s how
6 of {136789} must be in R3C3 => one of 7,8 must be in R8C6 (both of 7,8 in N5 would clash with R4C5) => the other of 7,8 must be in N5 => R4C5 = R8C6
R4C5 = R8C6 = 7 => cannot place 7 in C4
R4C5 = R8C6 = 8 => cannot place 8 in C4
[Note. The “clone” R4C5 = R8C6 only applies for the {136789} combination.]
23a. -> 34(6) cage at R3C3 = {235789}, no 1,6 -> R4C4 = 5

24. Hidden killer triple 1,2,3 in R12C4 and R5679C4 for C4, R12C4 = {123} -> R5679C4 contains one of 1,2,3
24a. 6 in N8 only in 25(6) cage at R5C3 = {123469/123568/124567}
24b. One of 1,2,3 in {123469/123568} must be in R567C4 -> the other two of 1,2,3 must be in R5C3 + R8C5, no 8,9 in R5C3, no 9 in R8C5
24c. 25(6) cage = {124567} (only remaining combination, cannot be {123568} because R5C3 cannot contain 5 and one of 1,2,3), no 3,8 -> R5C3 = 5, R9C4 = 4, R7C4 = 6, 7 only in R56C4, locked for C4 and N5

[I had a look to see whether step 24 could be done before step 23, to avoid using the contradiction move using “clone”, but if that is done then step 24 only eliminates one combination from 25(6) cage at R5C3 when there is no placement in R4C4.]

25. R8C4 = 8, R9C6 = 9, R7C3 = 7, R5C1 = 4, R6C2 = 6

26. 34(6) cage at R3C3 (step 23a) = {235789} -> R8C6 = 7, R3C3 = 8, R3C12 = [67]

27. 22(4) cage at R1C2 (step 22) = {2389} (only remaining combination) -> R4C5 = 8, R1C2 + R2C3 = {23}, locked for N1 -> R1C1 = 1, R1C56 = [78], R8C9 = 9

28. Naked triple {123} in R149C2, locked for C2
28a. R5C2 = 8 (hidden single in N4), R7C2 = 9, R2C12 = [95], R8C2 = 4

29. Naked triple {123} in R8C3 + R9C12, locked for N7 -> R7C1 = 8, R8C1 = 5, R9C3 = 6
29a. R6C3 = 9 (hidden single in C3)
29b. R5C5 = 9 (hidden single in C5)

30. 34(6) cage at R3C3 (step 23a) = {235789} -> R6C6 + R7C6 {23}, CPE no 2,3 in R7C6 -> R7C6 = 1

and the rest is naked singles.

Solution:
1 3 4 2 7 8 9 6 5
9 5 2 3 6 4 1 7 8
6 7 8 9 1 5 3 2 4
2 1 3 5 8 6 4 9 7
4 8 5 7 9 3 2 1 6
7 6 9 1 4 2 8 5 3
8 9 7 6 3 1 5 4 2
5 4 1 8 2 7 6 3 9
3 2 6 4 5 9 7 8 1
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HATMAN
 Post subject: Re: Minimum Cages
PostPosted: Fri Jul 30, 2010 7:48 pm 
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 694
Location: Saudi Arabia
Well done Andrew
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