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HATMAN
 Post subject: Cage Pairs
PostPosted: Wed Dec 30, 2009 2:20 pm 
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Cage Pairs 2 X

A variant on cage placement. #1 which is easier is posted at:
http://www.sudoku.com/forums.html
under Sudoku variants - Cage Pairs

The dashed areas are not cages they are pairs of cages. The two numbers in each area are the sums for each of the cages.

Split the area horizontally or vertically to create the actual cages and put the sums in the appropriate ones. So each area must be split into two cages by a single vertical or horizontal line.

Cages can normally be singletons. In this one however singleton cages are not allowed (it probably not solveable without this constraint).

Note the areas may have repeats but not the true cages.

The solution is unique - however it is not necessary for the cage structure to be unique.

This is quite hard it took me a few attempts before I found an acceptable solution.

Remember the X.

I enjoyed solving it and came across one or two interesting techniques/implications .

Image

If you use JSudoku create the areas as "may repeat" killer cages then as you identify each cage put it in as a twin killer cage.
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Børge
PostPosted: Wed Dec 30, 2009 9:04 pm 
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Cage Pairs 1 X images with coloured udosuk Style Killer Cages: 
Image     Image
Cage Pairs 2 X images with coloured udosuk Style Killer Cages: 
Image     Image
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Ed
 Post subject: Re: Cage Pairs
PostPosted: Thu Jan 07, 2010 6:25 am 
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A couple of placements to get started but then things slowed up. Really slow. A wonderful, wonderful challenging puzzle right to the very end. Thanks HATMAN for putting in a tremendous amount of work to bring us this classic puzzle.

Walkthrough for Cage Pairs 2 X
48 steps:
Please point out any shortcuts missed! Also, all corrections welcome. Thanks HATMAN for quite a few.
1. "45" on c89: 2 outies r1c67 = 3 innies r367c8. Min. 3 innies = 6 -> min. r1c67 = 6
1a. 30,3(6)cages at r1c6 must have a 3(2) = {12}
1b. but the 3(2) can't be at r1c67 (step 1), must be at r12c9 or r2c89
1c. -> {12} locked for n3
1d. r2c9 = (12)
1e. 30(4) at r1c6 = {6789} -> r1c678 = (6789); r1c9 and r2c8 no 4 or 5

2. The 29,4(6) at r6c9 must have a 4(2) = {13} cage.
2a. Generalized X-wing on 1's in c89 with 3(2)n3: 1s locked for c89.

3. The 15,3(5) at r6c7 must have a 3(2) = {12} cage. Can't be at r67c8 (no 1) -> must be at r6c78 -> = [12]
3a. r7c78+r8c7 = 15(3) cage (note: can still have 2 in r78c7).

[edit after HATMAN's 2nd following post: need this bit to make step 4 work:

3b. "45" on n9: 1 remaining outie r6c9 - 3 = 1 innie r9c7. Min. r6c9 = 5, max. r9c7 = 6]

4. The 29,4(6) at r6c9 must have a 4(2) = {13} cage which only be at r89c8 or r9c89 [edit: can't be at r67c9 because no 1,3 in r6c9]: 1 & 3 both Locked for n9
4a. r9c8 = (13)
4b. 29(4) = {5789} -> r678c9 = (5789); r8c8, r9c9, no 2,4,6

5. 25,12(6) cages at r5c1 must be 25(4) and 12(2) since max. any 3 cells is 24 -> r5c12 = 12(2) (no 1,2,6)

6. "45" on r6789: 2 remaining outies r5c45 - 2 = 2 innies r6c69 -> min. r5c45 = 10
6a. 21,4(6) at r5c4 must have a 4(2) = {13} cage which cannot be at r5c45 (step 6) but must be at r45c4 (=[13]) or r7c56
6b. -> no 1,3 in common peers in r789c4 nor 3 in r7c7 (from diagonal /)

7. 14,5(5) at r7c2 must have a 5(2) cage (since min. any 3 cells is 6) = {14/23} = [1/3..]
7a. 5(2) can only be at r8c34 since no 1 nor 3 available in r78c4 -> r8c34 = [14/32]
7b. r7c234 = 14(3) cage

8. "45" on c12: 3 outies r6c3 + r9c34 - 13 = 3 innies r347c2 -> min. 3 outies = 19 and min. r9c34 = 10
8a. -> 28,5(6) at r8c1, which must have a 5(2) cage that can only be at r89c1 or r8c12 = {14/23} = [1/3..]
8b. r8c1 = (1234)
8c. 28(4) cage = {4789/5689} -> r9c234 = (4..9)

9. Killer pair 1,3 in implied 5(2)n7 (step 8a) and with r8c3: 1 & 3 both locked for n7

10. hidden pair 1,3 in r7 in r7c56: both locked for n8 (but not for 21,4(6) at r5c4!)

11. r7c234 = 14(3) (step 7b) = {248/257}(no 6,9)
11. 2 locked for r7

This next one is probably not needed but it is really fun so have put it in anyway.
12. one of the 5(2) cages in n7 (implied 5(2) and at r8c34) must have 2 (Locking cages); and the 14(3) at r7c234 must have 2 -> 2 locked at r78c4 for n8 and c4

13. "45" on n78: 1 remaining innie r7c1 - 5 = 1 outie r9c7 = [72/94]

13. "45" n9: 1 remaining outie r6c9 - 3 = 1 innie r9c7
13a. r6c9 = 5,7

14. There must be a 29(4) cage at r6c9 = {5789} with 8 & 9 only in n9: both locked for n9

15. 6 in r7 only in r7c78 in 15(3) cage
15a. 6 locked for n9
15b. 15(3) = 6{27/45}: no 7 in r8c7 since 2 must go there.

16. "45" on r89: 3 remaining innies r8c79 + one of r8c8/r9c9 = 19 = [2]{89}/[4]{78}(no 5 in any of those four possible cells)
16a. 8 locked for n9
16b. 5 in 29(4) at r6c9 only in c9: 5 locked for c9

17. naked pair {24} at r89c7: 4 locked for n9 & c7
17a. naked pair {24} at r8c47: both locked for r8

18. implied 5(2) at r8c1 = {14/23}, must have 2 or 4 which are only available at r9c1 -> r9c1 = (24)
18a. r89c1 = 5(2) -> no 1,3 in r8c2

19. 4 & 8 in r7 only in h14(3) at r7c2 = {248} only
19a. 8 blocked from r7c4 since {24} in n7 would clash with r9c1

20. 8 in r7 only in n7: locked for n7

21. naked pair {24} in r9c17: 4 locked for r9

22. 28(4) at r8c2 = {5689}(no 7)
22a. -> r9c4 = 8

23. hidden single 7 in n7 at r7c1
23a. no 7 in r6c23 (part of 28(4) cage

24. naked pair {56} = 11 at r7c78: 5 locked for r7
24a. and r8c7 = 4 (h15(3) cage sum)
24b. r9c7 = 2

25. "45" on n9: 1 remaining outie r6c9 = 5

26. r78c4 = [42]

27. r89c1 = [14] (h5(2) cage sum); note, 4 placed for D/

28. r8c3 = 3, r7c9 = 9

29. naked pair {78} at r8c89: 7 locked for r8 & n9

30. 7 in r6 only in n5: 7 locked for n5

As far as I can see, it's still not cracked! This next move was available near the beginning so it's needed after all.
31. The 17,7(5) at r3c9, min. any three cells is greater than 7 -> must have a 17(3) & 7(2) cage.
31a. The 7(2) can only be {34} at r45c8 or r5c89 -> r5c8 = (34)
31b. 3 and 4 locked for n6

32. 3 in c7 only in r23c7: locked for n3

33. "45" on n6: 1 outie r3c9 + 13 = 2 remaining innies r45c7
33a. max. r45c7 = 17 -> r3c9 = 4
33b. r45c7 = 17 = {89}: both locked for n6 & c7

34. "45" on n568: 2 remaining innies r4c45 = h12(2) = {39} only: both locked for n5 and r4

35. r45c7 = [89]

36. r3c7 = 3 (hidden single D/)

37. 4 on D\ only in r2c2 or r6c6 -> no 4 in r2c6 nor r6c2 (CPE)
37a. 4 in c6 only in r56c6: locked for n5
37b. and 4 locked in h12(3) (cages sum) at r456c6 = 4{17/26}(no 5,8)

38. we now know the 21,4(6) at r5c4 cannot have a 4(2) at r56c4 -> r7c56 is the 4(2) and r56c45 = 21(4)
38a. the 21(4) must have 8 for n5 = 8{157/256} = 6/7, not both -> no 6 or 7 in r5c45 or r6c5
38b. r6c5 = 8

39. 5 in n5 only in r5: 5 locked for r5

40. 12(2) at r5c12 = [84]

41. r4c8 = 4 (hidden single r4)
41a. r6c6 = 4 (hidden single n5) (placed for D\)
41b. r5c8 = 3
41c. r9c89 = [13] (3 placed for D\)

Just trying to get it to singles.
42. The 28,5(6) at r1c1 must have a 5(2). It can only be at r1c12 or r12c2 since min. r34c1 = 7.
42a. no 4 is available -> must be {32} -> R1C2 = 3
43b. 28(4) at r1c1 = {5689} -> r2c2 = 8 (placed for D\)
43c. r1c1 = 2 (5(2) cage sum) (placed for D\)
43d. r234c1 = {569}: all locked for c1 and 9 locked for n1

44. r6c4 = 7 (hsingle r6) (placed for D/)

45. hidden pair 2,6 in r45c6: both locked for c6

46. "45" on c6789: 2 outies r89c5 - 12 = r7c6 -> min. r89c5 = 13
46a. 18,11(5) at r8c5 must have an 11(2) cage which can't be at r89c5 (step 46) -> must be at r8c56 -> r8c56 = [65]

47. r8c2 = 9 (placed for D/)
47a. r2c8 = 6 (placed for D/)

48. 17,9(5) at r1c6 cannot have a 17(2) = {89}at r23c6 (blocked by ALS block at r19c6 = (789), and cannot have a 17(2) nor 9(2) at r2c67 (permutations) -> r23c6 must be 9(2) -> r23c6 = [18]

rest is naked singles.
solution: 
.-------------------------------.-------------------------------.-------------------------------.
| 2         3         5         | 6         4         9         | 7         8         1         |
| 9         8         4         | 3         7         1         | 5         6         2         |
| 6         7         1         | 5         2         8         | 3         9         4         |
:-------------------------------+-------------------------------+-------------------------------:
| 5         1         7         | 9         3         2         | 8         4         6         |
| 8         4         2         | 1         5         6         | 9         3         7         |
| 3         6         9         | 7         8         4         | 1         2         5         |
:-------------------------------+-------------------------------+-------------------------------:
| 7         2         8         | 4         1         3         | 6         5         9         |
| 1         9         3         | 2         6         5         | 4         7         8         |
| 4         5         6         | 8         9         7         | 2         1         3         |
'-------------------------------.-------------------------------.-------------------------------'
Cheers
Ed


Last edited by Ed on Sun Jan 10, 2010 7:43 pm, edited 2 times in total.

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HATMAN
 Post subject: Re: Cage Pairs
PostPosted: Thu Jan 07, 2010 11:12 am 
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Ed

Thank you - I'll go through it at the weekend and compare it with mine.

Maurice
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HATMAN
 Post subject: Re: Cage Pairs
PostPosted: Fri Jan 08, 2010 8:42 pm 
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Ed nice solution

A couple of logic jumps that need little patches and 31 needs to go before 4 else can still have [31] in r67c9.

Essentially you approached it as a classic killer using the cage total splits as clues to help in placement.

My approach as much as possible was to place cages or eliminate their placement. I admit some of my eliminations were a bit too T&E to be valid for posting a walkthrough.

Re-ordering your solution (46 starts to work after 12) and putting a couple of my bits in will give a shorter (but not simpler) solution. I'll do this over the week end.

I liked your extra bit on N7 and I note you got the double 13 pointer on N8 - which was the bit I really liked.

Cheers

Maurice
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HATMAN
 Post subject: Re: Cage Pairs
PostPosted: Sun Jan 10, 2010 1:46 pm 
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This is Ed's solution with some additions from mine, shorter (needs to be properly numbered to see how much shorter) and very slightly simpler.
I've left Ed's numbering as is and added my stuff as X etc. then deleted the unused bits of Ed's.

Critical to this is the mix of Ed's r89 innies and my c6789 innies.

If you use JSudoku I have found it helpful to put the areas in as may repeat cages and when I identify the actual cages put them in as twin cages.

Ed’s walkthrough reworked

1. "45" on c89: 2 outies r1c67 = 3 innies r367c8. Min. 3 innies = 6 -> min. r1c67 = 6
1a. 30,3(6)cages at r1c6 must have a 3(2) = {12}
1b. but the 3(2) can't be at r1c67 (step 1), must be at r12c9 or r2c89
1c. -> {12} locked for n3
1d. r2c9 = (12)
1e. 30(4) at r1c6 = {6789} -> r1c678 = (6789); r1c9 and r2c8 no 4 or 5

2. The 29,4(6) at r6c9 must have a 4(2) = {13} cage. Generalized X-wing on 1's in c89 with 3(2)n3: 1s locked for c89.

3. The 15,3(5) at r6c7 must have a 3(2) = {12} cage. Can't be at r67c8 (no 1) -> must be at r6c78 -> = [12]
3a. r7c78+r8c7 = 15(3) cage (can still be a 2 in r78c7).
3b. 17&7@r3c9 <>7(3) -> 7(2)&17(3), 7(2) = {34} locked N6, r5c8 = 3/4

4. The 29,4(6) at r6c9 must have a 4(2) = {13} cage which only be at r89c8 or r9c89: 1 & 3 both Locked for n9
4a. r9c8 = (13)
4b. 29(4) = {5789} -> r678c9 = (5789); r8c8, r9c9, no 2,4,6

5. 25,12(6) cages at r5c1 must be 25(4) and 12(2) since max. any 3 cells is 24 -> r5c12 = 12(2) (no 1,2,6)

6. "45" on r6789: 2 remaining outies r5c45 = 1 innies r6c6 +2 -> min. r5c45 = 5
6a. 21,4(6) at r5c4 must have a 4(2) = {13} cage which cannot be at r5c45 (step 6) but must be at r45c4 (=[13]) or r7c56
6b. -> no 1,3 in common peers in r789c4 nor 3 in r7c7 (from diagonal /)

7. 14,5(5) at r7c2 must have a 5(2) cage (since min. any 3 cells is 6) = {14/23} = [1/3..]
7a. 5(2) can only be at r8c34 since no 1 nor 3 available in r78c4 -> r8c34 = [14/32]
7b. r7c234 = 14(3) cage

7X. Innies C6789 r78c6r9c67 = 17(4) -> r8c6r9c67 = 18(3) -> r9c567 = 18(3), r8c56 = 11(2)
-> r9c5 = r89c6 + 1, r9c5>3, r89c6<9
7Y. O&I C6789 r89c5 = r7c6 +12 -> r89c5 min 13 <>123, r7c6 max 5 <> 6789
7Z tidies C7: 2 locked r78 <>C7, 3 locked r23 <>N3

8. "45" on c12: 3 outies r6c3 + r9c34 - 13 = 3 innies r347c2 -> min. 3 outies = 19 and min. r9c34 = 10
8a. -> 28,5(6) at r8c1, which must have a 5(2) cage that can only be at r89c1 or r8c12 = {14/23} = [1/3..]
8b. r8c1 = (1234)
8c. 28(4) cage = {4789/5689} -> r9c234 = (4..9)

9. Killer pair 1,3 in implied 5(2)n7 (step 8a) and with r8c3: 1 & 3 both locked for n7

10. Hidden pair 1,3 in r7 in r7c56: both locked for n8 (but not for 21,4(6) at r5c4!)
10X. form 7XY r8c5 <>8, r89c5 = 13/15

11. r7c234 = 14(3) (step 7b) = {248/257}(no 6,9)
11a. 2 locked for r7

This next one is probably not needed but it is really fun so have put it in anyway.
12. one of the 5(2) cages in n7 (implied 5(2) and at r8c34) must have 2 (Locking cages); and the 14(3) at r7c234 must have 2 -> 2 locked at r78c4 for n8 and c4
12X. r8c56<>{29} 9 locked N8 r456, r9c5<>45 as r89c5 =13/15 r9c6<>4 combos on 18(3)

13. "45" on n78: 1 remaining innie r7c1 - 5 = 1 outie r9c7 = [72/94]
13X. I&O N14 -> r12c3 = r7c1 +2 = 9/11(2) <> 7(2)/15(2) -> r2c34 = 7/15(12)

13a. "45" n9: 1 remaining outie r6c9 - 3 = 1 innie r9c7
13b. r6c9 = 5,7

14. There must be a 29(4) cage at r6c9 = {5789} with 8 & 9 only in n9: both locked for n9

15. 6 in r7 only in r7c78 in 15(3) cage
15a. 6 locked for n9
15b. 15(3) = 6{27/45}: no 7 in r8c7 since 2 must go there.

16. "45" on r89: r8c79 plus one of r8c8/r9c9 = 19 so 245 &5789 & 5789 = 2{89}/4{78}
(no 5 in any of those four possible cells)
16a. 8 locked for n9
16b. 5 in 29(4) at r67c9 only in c9: 5 locked for c9

17. Naked pair {24} at r89c7: 4 locked for n9 & c7
17a. naked pair {24} at r8c47: both locked for r8
17U. 11(2)@r8c56 = {56} locked R8 N8
17V. 18(3) @r9c567 = {79}2 -> r8c7=4, r6c9 =5, r8c34 =[32]
17W. -> r89c1 =[14] -> r8c2r9c234 = 9{56}8 <>47
17X. NS N7 r7c19 = [79] -> R7: 14(3) = {28}4 r5c2<>5 as 12(2) also r12c3 = 9(2)
17Y. 49 locked D/ 4 locked r3c89, 9 locked r1c678 also tidy 12(2) @r5c12
17Z. from 10X r89c5 = 67/69 -> r8c56 = [65]

23a. no 7 in r6c23 (part of 25(4) cage)

25. "45" on n9: 1 remaining outie r6c9 = 5

29. naked pair {78} at r8c89: 7 locked for r8 & n9 (now just an observation)

30. 7 in r6 only in n5: 7 locked for n5

31. Done at step 3
31X. 17(3)min in N6 ={67} -> n3c9 at least 4 17(30 =4{67}
31Y. -> r45c7 = {89} locked c7 -> 7 locked r123c7 for N3



34. "45" on n568: 2 remaining innies r4c45 = h12(2) = {39} only: both locked for n5 and r4

35. r45c7 = [89]

36. r3c7 = 3 (hidden single D/)
36X. r5c5 = 5 (hidden single D/) -> r5c12 = [84], r7c78 = [65] -> r12c7 = [75]
36Y. r5c2=4 -> r45c8 =[43] -> r89c89 = [7813]
36Z. D\ NS r4c45 =[93], r7c7=4 -> r1c1 =2 r2c2r3c3 ={18}, r7c56 =[13]

Almost singles from here
37X. HS r6 r6c5 =8 -> NS c5 r1c5 =4 -> r1c3 =5

38X. from 13X 7&15 @ r1c3 are horizontal and from 17X r12c3 =9(2)
38Y. -> r1c345r2c34 = [56443]

39X 9&17@r2c6 r23c7 =[53] and r2c3=4, r2c67 <>9/17(2), r2c7r3c78 <>7(3)
39Y. -> r2c7r3c78= 17(3) = [539]
Naked singles from here
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Ed
 Post subject: Re: Cage Pairs
PostPosted: Tue Jan 12, 2010 8:03 am 
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Finally ready to move on from this really satisfying puzzle. This is a reworked Walkthrough based on HATMAN'S combined one which gets it down to 39 steps. I took the liberty of stating some of HATMANS steps differently. Any mistakes are mine. Please let me know.

Cage Pairs 2 X

1. "45" on c89: 2 outies r1c67 = 3 innies r367c8. Min. 3 innies = 6 -> min. r1c67 = 6
1a. 30,3(6)cages at r1c6 must have a 3(2) = {12}
1b. but the 3(2) can't be at r1c67 (step 1), must be at r12c9 or r2c89
1c. -> {12} locked for n3
1d. r2c9 = (12)
1e. 30(4) at r1c6 = {6789} -> r1c678 = (6789); r1c9 and r2c8 no 4 or 5

2. The 29,4(6) at r6c9 must have a 4(2) = {13} cage. Generalized X-wing on 1's in c89 with 3(2)n3: 1s locked for c89.

3. The 15,3(5) at r6c7 must have a 3(2) = {12} cage. Can't be at r67c8 (no 1) -> must be at r6c78 -> = [12]
3a. r7c78+r8c7 = 15(3) cage (can still be a 2 in r78c7).

4. 17&7@r3c9 <>7(3) -> 7(2)&17(3), 7(2) = {34} locked N6, r5c8 = 3/4

5. The 29,4(6) at r6c9 must have a 4(2) = {13} cage which only be at r89c8 or r9c89: 1 & 3 both Locked for n9
5a. r9c8 = (13)
5b. 29(4) = {5789} -> r678c9 = (5789); r8c8, r9c9, no 2,4,6

6. 25,12(6) cages at r5c1 must be 25(4) and 12(2) since max. any 3 cells is 24 -> r5c12 = 12(2) (no 1,2,6)

7. "45" on r6789: 2 remaining outies r5c45 = 1 innie r6c6 + 2 -> min. r5c45 = 5
7a. 21,4(6) at r5c4 must have a 4(2) = {13} cage which cannot be at r5c45 (step 6) but must be at r45c4 (=[13]) or r7c56
7b. -> no 1,3 in common peers in r789c4 nor 3 in r7c7 (from diagonal /)

8. 14,5(5) at r7c2 must have a 5(2) cage (since min. any 3 cells is 6) = {14/23} = [1/3..]
8a. 5(2) can only be at r8c34 since no 1 nor 3 available in r78c4 -> r8c34 = [14/32]
8b. r7c234 = 14(3) cage

9. Innies C6789; r78c6+r9c67 = 17(4) -> r8c6r9c67 <> 18(3) -> r9c567 = 18(3), r8c56 = 11(2)
9a. -> r9c5 = r78c6 + 1,
9b. r9c5>3, r78c6<9 (no 8,9)

10. O&I C6789 r89c5 = r7c6 + 12 -> r89c5 min 13 <>123, r7c6 max 5 <> 6789

11. 3 locked r23c7 <> N3

12. "45" on c12: 3 outies r6c3 + r9c34 - 13 = 3 innies r347c2 -> min. 3 outies = 19 and min. r9c34 = 10
12a. -> 28,5(6) at r8c1, which must have a 5(2) cage that can only be at r89c1 or r8c12 = {14/23} = [1/3..]
12b. r8c1 = (1234)
12c. 28(4) cage = {4789/5689} -> r9c234 = (4..9)

13. Killer pair 1,3 in implied 5(2)n7 (step 8a) and with r8c3: 1 & 3 both locked for n7

14. Hidden pair 1,3 in r7 in r7c56: both locked for n8 (but not for 21,4(6) at r5c4!)
14a. r8c5 <> 8 (h11(2)), r89c5 = 13/15 (step 10)

15. r7c234 = 14(3) (step 8b) = {248/257}(no 6,9)
15a. 2 locked for r7

16. one of the 5(2) cages in n7 (implied 5(2) and at r8c34) must have 2 (Locking cages); and the 14(3) at r7c234 must have 2 16a. -> 2 locked at r78c4 for n8 and c4
16b. r8c56<>{29}(no 9)
16c. 9 locked N8 r456 for r9
16d. r9c5<>45 as r89c5 =13/15

17. (18(3) at r9c567 = {r9c6<>4 combos on 18(3)) This doesn't look correct.

18. "45" on n78: 1 remaining innie r7c1 - 5 = 1 outie r9c7 = [72/94]

19. I&O N14 -> r12c3 = r7c1 +2
19a. r12c3 = 9/11(2) <> 7(2)/15(2)
19b. -> r2c34 = 7(2)/15(2)

20. "45" n9: 1 remaining outie r6c9 - 3 = 1 innie r9c7
20a. r6c9 = 5,7

21. There must be a 29(4) cage at r6c9 = {5789} with 8 & 9 only in n9: both locked for n9

22. 6 in r7 only in r7c78 in 15(3) cage
22a. 6 locked for n9
22b. 15(3) = 6{27/45}: no 7 in r8c7 since 2 must go there.

23. "45" on r89: r8c79 plus one of r8c8/r9c9 = 19 so from (245) & (5789) = 2{89}/4{78}
(no 5 in any of those four possible cells)
23a. 8 locked for n9
23b. 5 in 29(4) at r67c9 only in c9: 5 locked for c9

24. Naked pair {24} at r89c7: 4 locked for n9 & c7
24a. naked pair {24} at r8c47: both locked for r8

25. 11(2)@r8c56 = {56} locked R8, N8

26. 18(3) @r9c567 = {79}[2]; 7 and 9 locked for r9 & n8
26a. -> r8c7=4, r6c9 =5 (step 20), r8c34 =[32](5(2))
26b. -> r89c1 =[14](5(2)) (4 placed for D/)
26c. -> r8c2r9c234 = 28(4) = [9]{56}[8] (9 placed for D/)
26d. NS N7 r7c19 = [79]
26e. -> R7: 14(3) = {28}[4]
26f. r5c2<>5 as 12(2)
26g. also r12c3 = 9(2) (step 19)
26h. no 7 in r6c23 (part of 25(4) cage)

27. 4 locked in r3c89 for r3

28. 9 locked in 30(4) in r1c678 for r1
28a. 12(2) @r5c12; no 3,7 in r5c1; no 8 in r5c2

29. from step 10, r89c5 = [67/69] -> r8c56 = [65]

30. 7 in r6 only in n5: 7 locked for n5

31. 17(3) at r3c9: min. in N6 = {67} = 13 -> r3c9 max. 4
31a. 17(3) = [4]{67}
31b. 6 & 7 locked for n6
31c. -> r45c7 = {89} locked c7

32. 7 locked r123c7 for N3

33. "45" on n568: 2 remaining innies r4c45 = h12(2) = {39} only: both locked for n5 and r4

34. r45c7 = [89]

35. r3c7 = 3 (hidden single D/, placed for D/)
35a. r5c5 = 5 (hidden single D/, place for both Ds) -> r5c12 = [84], r7c78 = [65] (6 placed for D\)
35b. -> r12c7 = [75]

36. r45c8 =[43](7(2) cage sum)

37. naked pair {67} at r45c9: both locked for c9
37a. -> r89c89 = [7813] (3 & 7 placed for D\)
37b. D\ NS r4c45 =[93](9 placed for D\)
37c. r1c1 =2
37d. r2c2r3c3 = {18}: both locked for D\ and n1
37e. r7c7=4, r7c56 =[13]

Almost singles from here
38. HS r6 r6c5 =8 -> NS c5 r1c5 =4
38a. from 26g, r12c3 = 9 -> = [54]
38b. 7&15 @ r1c3 are horizontal -> r1c345r2c34 = [56443]

39. 9&17@r2c6; r2c67 <>9/17(2), r2c7r3c78 <>7(3)
39a. -> r2c7r3c78= 17(3) = [539]
Naked singles from here

Cheers
Ed
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HATMAN
 Post subject: Re: Cage Pairs
PostPosted: Wed Feb 24, 2010 11:55 am 
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 694
Location: Saudi Arabia
KiCo 3 (cage pairs 4) posted at:

http://www.djape.net/sudoku/forum/viewt ... 0230#10230

This one has cage colours - not quite as Pyrrhon suggested.
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