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PostPosted: Sat Mar 07, 2020 12:28 pm 
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Grand Master
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 693
Location: Saudi Arabia
From Andrew's comments this set are just medium killers with an ending twist.

I'll have to look at creating ones where the constraint is embedded in the solution path, but these now go to the bottom of my stack.


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PostPosted: Tue Mar 24, 2020 7:03 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
Minimum Restraints 4 NC 7-cell is the exception to HATMAN's comment. It is a moderate Assassin level which required the use of some of the NC cells before the end.

I only solved it for the specified NC cells (I only used 6 of them so hope my solution is correct); solving using the specified set of values for the NC cells would make it a much harder puzzle.

Here is my walkthrough for Minimum Restraints 4 NC 7-cell:
The positions of seven NC cells are given (R1C5, R1C8, R2C8, R5C1, R5C5, R6C7 and R8C2; alternatively find them given that they are 2236789.

Prelims

a) R1C67 = {29/38/47/56}, no 1
b) R2C67 = {14/23}
c) R3C34 = {69/78}
d) R34C8 = {39/48/57}, no 1,2,6
e) R34C9 = {15/24}
f) R4C23 = {17/26/35}, no 4,8,9
g) R67C7 = {19/28/37/46}, no 5
h) R78C6 = {18/27/36/45}, no 9
i) 20(3) cage at R3C1 = {389/479/569/578}, no 1
j) 22(3) cage at R8C7 = {589/679}
k) 8(3) cage at R8C8 = {125/134}
l) 26(4) cage at R1C8 = {2789/3689/4589/4679/5678}, no 1

1a. 45 rule on N9 3 innies R789C7 = 23 = {689}, locked for C7 and N9, R9C6 = {57}, clean-up: no 2,3,5 in R1C6
1b. 22(3) cage at R8C7 = {589/679}, 9 locked for N9
1c. 8(3) cage at R8C8 = {125/134}, 1 locked for N9
1d. R7C7 = {68} -> R6C7 = {24}

2a. 45 rule on N478 2(1+1) innies R4C1 + R9C6 = 16 = [97] -> R89C7 = {69}, R7C7 = 8 -> R6C7 = 2, clean-up: no 9 in R1C6, no 4 in R1C7, no 3 in R2C6, no 3 in R3C8, no 4 in R3C9, no 2 in R7C6, no 1,2 in R8C6
2b. 45 rule on N1 1 outie R4C1 = 1 remaining innie R3C3, R4C1 = 9 -> R3C3 = 9 -> R3C4 = 6, clean-up: no 5 in R1C7, no 3 in R4C8
2c. R4C1 = 9 -> R3C12 = 11 = {38/47}, no 5

3a. 9 in C6 only in R56C6, locked for N5
3b. 16(3) cage at R5C6 = {169/259/349}, no 7,8
3c. 1,5 of {169/259} must be in R5C7 -> no 1,5 in R56C6

4a. 6,9 in N3 only in 26(4) cage at R1C8 = {3689/4679}, no 2,5
4b. Killer pair 3,7 in R1C7 and 26(4) cage, locked for N3, clean-up: no 2 in R2C6, no 5 in R4C8
4c. R3C9 = 2 (hidden single in N3) -> R4C9 = 4, clean-up: no 8 in R3C8
4d. Naked triple 1,4,5 in R2C7 + R3C78, 1 locked for C7, 5 locked for R3, 4 locked for N3
4e. 26(4) cage at R1C8 = {3689}, 3 locked for N3, R1C7 = 7 -> R1C6 = 4, R2C67 = [14], R3C8 = 5 -> R4C8 = 7, R3C7 = 1, clean-up: no 1 in R4C23, no 5 in R7C6, no 5,8 in R8C6
4f. 16(3) cage at R5C6 (step 3b) = {259} (only remaining combination) -> R5C67 = [25], R6C6 = 9, R4C7 = 3, R3C6 = 8, R4C6 = 5 (cage sum)
4g. Naked pair {26} in R4C23, 6 locked for R4 and N4
4h. Naked pair {18} in R4C45, locked for N5, R3C5 = 3 (cage sum)
4i. Naked pair {47} in R3C12, 7 locked for N1
4j. Naked pair {36} in R78C6, locked for N8
4k. 12(3) cage at R1C2 = {138/156}, no 2, 1 locked for R1
4l. 7 in N9 only in 14(3) cage at R8C8 = {257/347}
4m. 2,4 only in R8C8 -> R8C8 = {24}

5a. 45 rule on N8 2 remaining innies R78C4 = 1 outie R9C3 + 8
5b. R78C4 cannot total 16 -> no 8 in R9C3

6a. 21(5) cage at R7C5 = {12459/12468/13458} (cannot be {12369} because 3,6 only in R9C3)
6b. Consider combinations for 21(5) cage
21(5) cage = {12459} => R8C4 = 8 (hidden single in N8) => R4C4 = 1
or 21(5) cage = {12468} => R78C4 = {59} (hidden pair in N8)
or 21(5) cage = {13458} => R78C4 = {29} (hidden pair in N8)
-> no 1 in R7C4, no 1,4 in R8C4
6c. 1 in N8 only in R78C5 + R9C45, locked for 21(5) cage, no 1 in R9C3

[Seems like I need to start using the NC cells.]
7a. R2C7 = 4, R3C8 = 5 -> R2C8 (a specified NC cell) = {89}, no 8,9 in R1C8 + R2C9
7b. R1C7 = 7 -> R1C8 (a specified NC cell) = 3, R2C9 = 6
7c. Naked triple {124} in R789C8, 1 locked for C8 and N9
7d. 12(3) cage at R1C2 (step 4k) = {138/156} = {18}3/{16}5, no 5 in R1C23, no 8 in R2C3
7e. 12(3) cage = {18}3/{16}5 -> 13(3) cage at R1C1 = 6{25}/{28}{238}, no 5 in R1C1
7f. R1C6 = 4 -> R1C5 (a specified NC cell) = {29}
7g. R1C4 = 5 (hidden single in R1)
7h. 5 in N8 only in R789C5, locked for 21(5) cage at R7C5, no 5 in R9C3
7i. 21(5) cage at R7C5 (step 6a) contains 1,5 in N8 = {12459/13458}, no 6

8a. R5C5 (a specified NC cell) = {467} -> R56C6 = {46/47} (cannot be {67}), 4 locked for C5 and N5
8b. Naked pair {37} in R56C4, 7 locked for C4 and N5
8c. R2C5 = 7 (hidden single in N2)
8d. 21(5) cage at R7C5 (step 7i) = {12459} (only remaining combination, cannot be {13458} because then R78C4 = {29} (hidden pair in N8) clashes with R2C4), no 3,8, 4 locked for R9, 9 locked for N8
8e. R8C4 = 8 (hidden single in N8), R4C45 = [18]
8f. Naked pair {24} in R7C48, locked for R7
8g. 45 rule on N7 R7C13 + R9C3 = 12 = {37}2/{17}4/{35}4, no 6
8h. 25(5) cage at R6C2 = {14578/23578}
8i. R7C4 = {24} -> no 4 in R6C23
8j. 45 rule on N4 2 innies R6C23 = 1 outie R7C1 + 5, R6C23 cannot total 10 = {37} which clashes with R6C4 -> no 5 in R7C1
8k. 23(5) cage at R5C1 = {13478} (only possible combination), no 5
8l. 5 in N4 only in R6C12, locked for 25(5) cage
8m. R7C13 + R9C3 = 12 = {37}2/{17}4, 7 locked for R7 and N7
8n. R8C9 = 7 (hidden single in N9)
8o. 25(5) cage = {14578/23578}, CPE no 7 in R5C3
8p. 7 in C4 only in R67C4, locked for 25(5) cage, no 7 in R6C2

9a. R5C5 (a specified NC cell) = {46} -> R5C45 = [36/74] (cannot be [34/76])
9b. 23(5) cage at R5C1 = {13478}
9c. R67C1 cannot be [47] which clashes with R3C1 -> R5C123 must contain at least of 4,7
9d. R5C45 = [36] (cannot be [74] which clashes with R5C123), R6C45 = [74]
9e. R6C8 = 6 (hidden single in N6)
9f. R7C3 = 7 (hidden single in C3)
9g. 23(5) cage at R5C1 = {13478}, 3 locked for C1

10a. R7C13 + R9C3 (step 8m) = [174/372]
10b. 45 rule on N8 1 remaining innie R7C4 = 1 remaining outie R9C3, R7C1 + R9C3 = [14/32] -> R7C14 = [14/32]
10c. R8C9 = 7 -> R7C89 = 7 = [25] (cannot be [43] which clashes with R7C14) -> R7C14 = [14], R7C5 = 9, R9C34 = [42], R1C5 = 2, R9C58 = [51], R5C3 = 8, R25C8 = [89]
10d. R6C1 = 3 -> R5C1 (a specified NC cell) = 7

11. Consider placements for R7C2 = {36}
R7C2 = 3 => R2C12 = {25}, R1C1 = 6 (cage sum)
or R7C2 = 6, R9C1 = 8 => R1C1 = 6
-> R19C1 = [68], R1C23 = [81], R6C23 = [15], R2C3 = 3

12. R8C2 (a specified NC cell) = 9 (cannot be 2,3,5,6 because of NC with R7C2 + R8C13)

and the rest is naked singles, without using NC.

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


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