## 222(2) N-tuples and Forcing Chains

Everything about Sudoku that doesn't fit in one of the other sections

### 222(2) N-tuples and Forcing Chains

# original puzzle
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` *-----------* |..1|59.|..3| |6..|..2|..8| |...|...|14.| |---+---+---| |..6|..9|.51| |...|2.5|...| |53.|8..|6..| |---+---+---| |.15|...|...| |3..|9..|..4| |8..|.21|9..| *-----------*`

# after a number of reductions
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`*--------------------------------------------------------------------* | 4      28     1      | 5      9      678    | 27     67     3      | | 6      79     3      | 1      4      2      | 5      79     8      | | 27     5      89     | 36(7)  3(78)  36     | 1      4      269    | |----------------------+----------------------+----------------------| | 27     28     6      | 347    37     9      | 48     5      1      | | 1      49     89     | 2      6      5      | 34     38     7      | | 5      3      47     | 8      1      47     | 6      29     29     | |----------------------+----------------------+----------------------| | 9      1      5      | 3467   378    347    | 237    23678  26     | | 3      67     2      | 9      5      678    | 78     1      4      | | 8      467    47     | 367    2      1      | 9      36     5      | *--------------------------------------------------------------------*`

At this point, my solver makes the following assignments.

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`r1c7    =  2     Templates -- Pass Br3c1    =  2     Templates -- Pass Br4c2    =  2     Templates -- Pass B`

How is this possible? If you examine [b1], you find a Naked Quad in <2789> where [r3c1]=7 forces [r3c3]=8 and, together, they force minirow [r3b2] to have two candidates -- <3> and <6> -- for three cells. Thus, [r3c1]=2 and the puzzle cracks (with the other two assignments being made in the process).

Now, it occurs to me that class 222(2) n-tuple in a box might be a good place to start when forcing chains are considered.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

Maybe it is detecting the pure bilocation chain ...

r3c1-7-r2c2=7=r2c8=9=r3c9=2=r3c1, implying r3c1<>7

... which can be found with advanced coloring.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

### Re: 222(2) N-tuples and Forcing Chains

daj95376 wrote:
Code: Select all
`r1c7    =  2     Templates -- Pass Br3c1    =  2     Templates -- Pass Br4c2    =  2     Templates -- Pass B`

How is this possible?

3 x cycles on 7 ( - for weak link, strong otherwise, last cell linked to first to form cycle)
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`[76]-[31][22]-[63][41]- => [76]^7[86]-[31][22]-[63][41]- => [86]^7[17]-[87][82]-[22][31]- => [17]^7`

some of the weak links like [41]-[76] are (collapsed) hinges
gsf
2014 Supporter

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Joined: 21 September 2005
Location: NJ USA

Thanks ronk and gsf for the additional insight into solving this puzzle.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006