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 Post subject: Decidoku NC repeat 5
PostPosted: Mon Jan 16, 2017 7:15 pm 
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Grand Master
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 693
Location: Saudi Arabia
Decidoku NC repeat 5

Decidoku structure 0 to 9, remember the hidden groups. (Moderator Note): See here.
Standard non-consecutive

Red twin killer cages are maximum repeat.


Image

Solution:
3905826174
8270461359
6427139580
1693574802
5148092637
7362908415
2084753961
0859317246
9531640728
4716285093


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 Post subject: Re: Decidoku NC repeat 5
PostPosted: Tue Mar 27, 2018 10:52 pm 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
The last of my backlog of HATMAN puzzles from the first 3 months of last year. Max Repeat NC FNC LS 4 and Human Solvable 26 (in the Killer forum) were fairly easy; this one seemed a lot harder. Lots of nibbling away, which got tedious at times.

Here is my walkthrough for Decidoku NC repeat 5:
Decidoku, candidates 0-9. Non-consecutive (NC). Cages are maximum repeat; the 18(4) cage must therefore have two repeated pairs, since two of its cells are in each of the decidoku houses and hidden houses. The other cages each have two numbers repeated which will be diagonally opposite each other so that these numbers may be in different decidoku houses, the other two numbers being different because they’re in the same house as one of the repeated numbers.
I’ve numbered the rows and columns 0 to 9, with the decidoku houses DR0C0, etc. Hidden decidoku houses, as in wellbeback’s diagram will be HR0C4, etc., but I may be able to use Law of Leftovers (LoL) instead or possibly neither.
The puzzle has symmetry so steps apply equally to the upper and lower halves; the cages interact in the same way for both rows and columns so, after doing eliminations for possible cage combinations, I’ve just given steps for the upper half of the puzzle. Then I’ll do the lower half later, by which time there may have been enough eliminations not to need to apply symmetry.

Possible cage combinations.
1a. 18(4) cage with two repeated pairs = {0099/1188/2277/3366} (cannot be {4455} NC), no 4,5
The other cages have single repeat
1b. 10(4) cages = {0028/0037/0046/1135/2206/4402} (other combinations blocked by NC) -> R2C9 + R3C8 and R6C1 + R7C0 = {0124}, R2C8 + R3C9 and R6C0 + R7C1 = {02345678}
1c. 11(4) cages = {0029/0038/0047/0056/1136/1145/2207/3305/4412} (other combinations blocked by NC) -> R0C7 + R1C6 and R8C3 + R9C2 = {01234}, R0C6 + R1C7 and R8C2 + R9C3 = {0123456789}
1d. 12(4) cages = {0039/0048/0057/1137/1146/2208/3306/3315} (other combinations blocked by NC) -> R0C2 + R1C3 and R8C6 + R9C7 = {0123}, R0C3 + R1C2 and R8C7 + R9C6 = {013456789}
1e. 15(4) cages = {0069/0078/1149/1158/1167/2247/2256/3309/3318/4407/4416/5523/6603/6612} (other combinations blocked by NC) -> R1C1 + R2C2, R0C6 + R1C5, R8C4 + R9C3 and R7C7 + R8C8 = {0123456}, R1C2 + R2C1, R0C5 + R1C6, R8C3 + R9C4 and R7C8 + R8C7 = {0123456789}
1f. 17(4) cages = {0089/1169/1178/2249/2258/2267/3356/4409/4418/4427/5507/6614/6623} (other combinations blocked by NC) -> R2C0 + R3C1 and R6C8 + R7C9 = {0123456}, R2C1 + R3C0 and R6C9 + R7C8 = {0123456789}
1g. 21(4) cages = {2289/3369/3378/4467/5529/5538/6609/6618/7725/7734/8805/8814/8823/9903/9912} (other combinations blocked by NC) -> R1C8 + R2C7 and R7C2 + R8C1 = {23456789}, R1C7 + R2C8 and R7C1 + R8C2 = {0123456789}

[There are a lot of interactions between the four cages which are partly in DR0C6, due to three of them each having three cells in it and overlapping by one cell each, also the 11(4) cage and 15(4) cage at R0C5 have two cells overlapping. To avoid being unnecessarily complicated, I’ll gradually build up on these interactions.]
2a. The 10(4) and 11(4) cages each have three cells in DR0C6 so cannot share any digits
-> 10(4) cage = {0028/0037/1135/2206/4402} (cannot be {0046} because 11(4) cage must contain at least one of 0,4,6), no 4 in R2C8 + R3C9
and 11(4) cage = {0029/0047/1136/1145/2207/4412} (cannot be {0038/0056/3305} because 10(4) cage must contain 0 or 3), no 3 in R0C7 + R1C6, no 0,8 in R1C7
2b. 11(4) cage and 15(4) cage at R0C5 overlap at R01C6 -> 11(4) cage = {0047/1136/1145/4412} (cannot be {0029/2207} because no valid combination for 15(4) cage containing 002/022}, no 0 in R0C6, no 2 in R0C7 + R1C6, no 9 in R1C7, also R0C6 + R1C5 are the repeated pair in 15(4) cage at R0C5 -> no 0 in R1C5
2c. 11(4) cage = {0047/1136/4412}/[4115] (cannot be [5114] because 15(4) cage cannot be [4551] NC), no 5 in R0C6, no 4 in R1C7, also R0C6 + R1C5 are the repeated pair in 15(4) cage at R0C5 -> no 5 in R1C5
2d. 11(4) cage = {0047/1136/4412}/[4115] -> 15(4) cage at R0C5 = {1149/2247/3318/4407/4416/6612} (other combinations don’t match with 11(4) cage) -> R0C5 = {26789}
2e. The 10(4) and 11(4) cages must each share one non-repeated digit with the non-repeated digits of the 21(4) cage -> one of R1C7 and R2C8 must be odd and the other even
2f. 11(4) cage = {1136/4412}/[4115] (cannot be {0047} = [4007] because 10(4) cage = {1135} would place two odd numbers in R1C7 and R2C8 and all other combinations for 10(4) cage would share a digit in DR0C6), 1 locked for DR0C6, no 0 in R0C7 + R1C6, no 7 in R1C7, also R2C9 + R3C8 are the repeated pair in 10(4) cage -> no 1 in R3C8
2g. 10(4) cage = {0028/0037/2206/4402}, no 5 in R2C8 + R3C9, 0 locked for DR0C6
2h. 11(4) cage = {1136} can only go with 10(4) cage = {0028/4402} which only contain even numbers -> R1C7 must be odd -> {1136} = [6113], no 3 in R0C6, no 6 in R1C7, also R0C6 + R1C5 are the repeated pair in 15(4) cage at R0C5 -> no 3 in R1C5
2i. 11(4) cage = {4412} can only go with 10(4) cage = {0037} which has one of 3,7 in R2C8 -> R1C7 must be even -> {4412} = [1442]
2j. From 2f, 2h and 2i, 11(4) cage = [1442/4115/6113], no 2 in R0C6, no 1 in R1C7, also R0C6 + R1C5 are the repeated pair in 15(4) cage at R0C5 -> no 2 in R1C5
2k. 1 in DR0C6 only in R0C67, locked for R0, also R0C2 + R1C3 are the repeated pair of 12(4) cage -> no 1 in R1C3
2l. 11(4) cage = [1442/4115/6113] -> 1 in R01C6, locked for C6
2m. 11(4) cage = [1442/4115/6113] -> either 1 repeated in R0C7 + R1C6 for 11(4) cage or 1 repeated in R0C6 + R1C5 for 15(4) cage at R0C5 -> 1 in R1C56, locked for R1 and DR0C5, also R1C1 + R2C2 are the repeated pair of 15(4) cage at R1C1 -> no 1 in R2C2
2n. 11(4) cage = [1442/4115/6113] -> 15(4) cage at R0C5 = [2661/6441/9114], no 7,8 in R0C5
2o. Possible combinations of 11(4) and 10(4) cages are [1442]+[3007]/[4115]+[2008]/[4115]+[0226]/[6113]+[8002]/[6113]+{4402) (cannot be [1442]+[7003] because 21(4) cage would be [2667] NC, cannot be [4115]+[8002] because 21(4) cage = [5448] would be NC and place a second 4 in DR0C6, cannot be [4115]+{0037} which would place two odd digits in R1C7 and R2C8, cannot be [4115]+[6220] which would make 21(4) cage = [5556], cannot be [6113]+[2008] because 21(4) cage = [3882] would place a second 8 in DR0C6) -> 10(4) cage = {0028/4402}/[0226/3007], no 6,7 in R2C8, no 3 in R3C9
2p. From the previous step -> R1C7 + R2C8 = [23/30/32/38/50/52] -> 21(4) cage = [2883/3990/3882/3558/5880/5772], no 2,3,4,6 in R1C8 + R2C7

[Now to look at the cages at the upper left-hand side of the grid.]
3a. 1 in DR0C0 only in 17(4) cage (step 1f) = {1169/1178/4418/6614}, no 0,2,3,5
3b. 12(4) cage (step 1d) = {0039/0048/0057/2208/3306}, 0 locked for DR0C0, also R1C1 + R2C2 are the repeated pair of 15(4) cage at R1C1 -> no 0 in R2C2
3c. 15(4) cage at R1C1 (step 1e) = {2247/2256/3309/3318/4407/4416} (cannot be {5523/6603} because no 0,2,3 in R2C1, cannot be {6612} because no 1,2 in R1C2), no 5,6 in R1C1 + R2C2, no 3 in R1C2
3d. 0 of {3309} must be in R1C2 -> no 9 in R1C2
3e. 1 of {3318} must be in R2C1 -> no 8 in R2C1
3f. Following these eliminations 15(4) cage at R1C1 = [2472/2562/2742/3093/3813/4074/4614]
3g. Now looking at interactions between these three cages, which are linked by overlapping cells R1C2 and R2C1, 15(4) cage at R1C1 = [2562/2742/3093/3813/4074] (cannot be [2472] because 12(4) cage = [0840] and 17(4) cage = [1781] put two 8s in DR0C0, cannot be [4614] which clashes with R2C0), no 4,6 in R1C2
3h. After those eliminations there are some simplifications to the other cages in DR0C0
12(4) cage = [0480/0570/0750/2082/2802/3603] (cannot be {0039} because no 3,9 in R1C2), no 3,9 in R0C3
17(4) cage = [1691/1781/1961/4184/6146/6416], no 7 in R3C0

[And now to look at the interactions between the upper left-hand side and the upper right-hand side. It looks like there are no direct one to one interactions along a pair of adjacent rows which can cause eliminations, so this step will also be heavy with multiple-overlapping cages involved on at least one side.]
4. First a summary of the remaining possibilities for each cage, to avoid having to repeatedly look back at steps 2 and 3.
12(4) cage (step 3h) = [0480/0570/0750/2082/2802/3603]
15(4) cage at R1C1 (step 3g) = [2562/2742/3093/3813/4074]
17(4) cage (step 3h) = [1691/1781/1961/4184/6146/6416]
15(4) cage at R0C5 (step 2n) = [2661/6441/9114]
11(4) cage (step 2j) = [1442/4115/6113]
21(4) cage (step 2p) = [2883/3990/3882/3558/5880/5772]
10(4) cage (step 2o) = {0028/4402}/[0226/3007]
4a. 15(4) cage at R1C1 = [2562/2742/3093/3813] (cannot be [4074] because combining it with 12(4) cage = [2802/3603], R0C23 + R1C123 = [28402/36403] clash with 15(4) cage at R0C5), no 4 in R1C1 + R2C2, no 7 in R2C1
4b. 17(4) cage = [1691/1961/6146/6416] (cannot be [4184] which overlaps with 15(4) cage at R1C1 = [3813] putting two 8s in DR0C0), no 4 in R2C0 + R3C1, no 8 in R3C0, 6 locked for DR0C0
4c. 12(4) cage = [0480/0570/0750/2082/2802], no 3 in R0C2 + R1C3
4d. Combining 12(4) cage with 15(4) cage at R1C1, R0C23 + R1C123 = [04380/05270/07250/20382/28302] -> 11(4) cage = [4115/6113] (cannot be [1442] which clashes with R0C23 + R1C123) -> R0C7 = 1, R1C6 = 1, no 2 in R1C7, no 0,2 in R2C6 (NC)
4e. 15(4) cage at R0C5 = [2661/6441], no 9, 6 locked for R0, C5 and DR0C5
4f. 21(4) cage = [3990/3882/3558/5880/5772], no 3 in R2C8
4g. 10(4) cage = {0028/4402}/[0226], no 7 in R3C9
4h. R1C1 = {23} -> no 2,3 in R0C1 + R1C0 (NC)
4i. R2C2 = {23} -> no 2,3 in R2C3 + R3C2 (NC)
4j. R1C3 = {02} -> no 1 in R2C3 (NC)
4k. R1C5 = {46} -> no 5 in R1C4 + R2C5 (NC)
[Now bringing NC into these interactions.]
4l. 15(4) cage at R1C1 = [2742/3093/3813] (cannot be [2562] because then 12(4) cage = [0750], 17(4) cage = [1691], R0C0 = 3 (hidden single in DR0C0), no 4 in R0C1 + R1C0 (NC) so cannot place 4 in DR0C0), no 5 in R1C2, no 6 in R2C1
4m. 15(4) cage at R1C1 = [2742/3813] (cannot be [3093] because then 12(4) cage = [2802], 17(4) cage = [1961], no 4 in R02C1 + R1C0 (NC) -> R0C0 = 4 (hidden single in DR0C0) -> no 5 in R0C1 + R1C0 (NC) so cannot place 5 in DR0C0), no 0 in R1C2, no 9 in R2C1
4n. 17(4) cage = [6146/6416] -> R2C0 = 6, R3C1 = 6, R2C1 + R3C1 = {14}, 4 locked for DR0C0, also no 5,7 in R1C0 + R3C2 + R4C1 (NC)
4o. R1C0 = {89} -> no 8,9 in R0C0 (NC)
4p. 12(4) cage = [0570/2082], no 7,8 in R0C3, 0 locked for R0 and C3
4q. 10(4) cage = {0028} (cannot be {4402} because 17(4) cage has one 4 in R23) -> R2C9 = 0, R3C8 = 0, 2,8 locked for DR0C6, no 1 in R4C8 (NC)
4r. 21(4) cage = [3558/5772], no 9 in R1C8, no 8,9 in R2C7
4s. R1C8 = {57} -> no 6 in R1C9 (NC)
4t. R0C6 = 6 (hidden single in DR0C6), R0C5 = 2, R1C5 = 6, R1C7 = 3 (cage totals), no 3 in R0C4, no 7 in R1C4 + R2C5 (NC)
4u. 12(4) cage = [0570] -> 15(4) cage = [2742], R3C0 = 1, no 4 in R0C4, no 0,2 in R4C0 (NC)
4v. 21(4) cage = [3558], R3C9 = 2, no 4 in R0C8 + R1C9, no 1,3 in R4C9 (NC)
4w. R1C9 = 9, R0C89 = [74], R1C0 = 8, R0C01 = [39], R01C4 = [84], no 3 in R2C4 (NC)
4x. 4,8 in DR0C5 only in R3C567, locked for R3 -> R3C2 = 9, R23C3 = [73], no 8 in R4C2, no 4 in R4C3 (NC)
4y. R23C4 = [15] (hidden singles in R2 and R3), no 4 in R3C5, no 6 in R4C4 (NC)
4z. R3C5 = {78} -> no 7,8 in R3C6 + R4C5 (NC)
4aa. R3C6 = 4, no 3 in R2C6, no 3,5 in R4C6 (NC)
4ab. R2C56 = [39]
4ac. R3C7 = {78} -> no 7,8 in R4C7 (NC)
4ad. Naked pair {18} in R4C13, locked for R4
4ae. Naked pair {45} in R4C02, locked for R4
4af. R4C9 = {67} -> no 6 in R4C8, no 6,7 in R5C9 (NC)
4ag. R4C5 = {09} -> 18(4) cage with two repeated pairs = {0099}, 0,9 locked for R45 and C45
4ah. R4C8 = 3 (hidden single in R4), no 2 in R4C7, no 2,4 in R5C8 (NC)
4ai. R4C679 = [267], no 7 in R35C7, no 3 in R5C6, no 8 in R5C9 (NC)
4aj. R3C57 = [78]

[Now to look at the lower half of the grid.]
5a. R5C6789 = [8415] (hidden quad in R5), no 9 in R5C5, no 7 in R6C6, no 2 in R6C8, no 6 in R6C9 (NC)
5b. R79C6 = [75] (hidden pair in C6), no 8 in R7C5, no 4 in R9C5 (NC)
5c. R6C8 = 6, R67C7 = [92] (hidden pair in DR5C6)
5d. R9C7 = 0 -> R8C6 = 0, R6C6 = 3, no 4 in R6C5, no 1 in R8C5 (NC)
5e. R5C45 = [90], R4C45 = [09], no 1 in R4C3 + R6C5 (NC)
5f. R4C13 = [18]
5g. R6C1 = 0 -> R7C0 = 0, no 1 in R6C2 (NC)
5h. 4 in C5 only in R78C5 -> no 5 in R78C5 (NC)
5i. R6C5 = 5 (hidden single in C5) -> no 4 in R7C5 (NC)
5j. R789C5 = [148], no 3 in R8C4, no 7 in R9C4 (NC)
5k. R7C7 = 2, R8C78 = [72] -> R7C8 = 4 (cage sum), no 3 in R7C9, no 1,3 in R8C9 (NC)

and the rest is naked singles, without using non-consecutive.

HATMAN wrote "remember the hidden groups":
In some cases, but not all, Law of Leftovers can be used instead. First time through I used LoL. However while checking my walkthrough I found that I'd prematurely eliminated a combination, so had to rework from that stage and found that I didn't need to use LoL or hidden groups.


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