## 4-Cell Bivalue Loop

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### 4-Cell Bivalue Loop

...54..32....629.5.25.9346.5.138924..4.276.5.2..415.9..3.92.5....2.513.4.58.3..29
I thought this puzzle was interesting. Having done no good in skyscrapers or kites I looked at the forced moves caused by the two possible locations of 3 in Box6. This led to the interesting result that in both cases there would be a locked 67 pair in Boxes 4 & 6, and these cells were all in Rows 4 & 6, forming a 4-cell bivalue chain, and in both outcomes three of the four cells were the same. The resultant two versions of the middle band look like this:

4CellLoop.png (4.17 KiB) Viewed 752 times

In the upper version the 8 could only go in r6c9 as putting it in r6c7 would cause a Unique Rectangle. Can this 4-cell bivalue loop be used to solve the puzzle, with 3 of its points known?
Yogi
2017 Supporter

Posts: 190
Joined: 05 December 2015
Location: New Zealand

### Re: 4-Cell Bivalue Loop

N(>1)-"Fish" : only possible for the digits 6, 7, 8.
Among the possible "fishes", the following 2 are sufficient.
Code: Select all
`+-------------------+---------------+---------------------+| 6789   16789  679 | 5      4  78  | 18(7)  3      2     || 34     18(7)  34  | 18(7)  6  2   | 9      18(7)  5     || 8(7)   2      5   | 18(7)  9  3   | 4      6      18(7) |+-------------------+---------------+---------------------+| 5      6(7)   1   | 3      8  9   | 2      4      6(7)  || 389    4      39  | 2      7  6   | 18     5      138   || 2      678    367 | 4      1  5   | 68(7)  9      3678  |+-------------------+---------------+---------------------+| 1467   3      467 | 9      2  478 | 5      178    1678  || 69(7)  69(7)  2   | 68(7)  5  1   | 3      8(7)   4     || 1467   5      8   | 6-7    3  47  | 16(7)  2      9     |+-------------------+---------------+---------------------+"Starfish"{7R2348C7} -> [7r238c4==7r9c7] -> -{7r9c4}; 10 singlesProof :||7r2c2 -> r4c9=r3c4=7||7r2c4||7r2c8 -> ||7r6c7 -> r4c2=7, XWing(7r38c14)           ||7r9c7+----------------+-------------+----------------+| 9     178   6  | 5    4  78  | 178  3    2    || 34    178   34 | 178  6  2   | 9    178  5    || 7(8)  2     5  | 178  9  3   | 4    6    1-8  |+----------------+-------------+----------------+| 5     6     1  | 3    8  9   | 2    4    7    || 3(8)  4     9  | 2    7  6   | 18   5    138  || 2     7(8)  37 | 4    1  5   | 6    9    3(8) |+----------------+-------------+----------------+| 147   3     47 | 9    2  478 | 5    178  6    || 6     9     2  | 78   5  1   | 3    78   4    || 147   5     8  | 6    3  47  | 17   2    9    |+----------------+-------------+----------------+Kite{8R6C1} -> -{8r3c9}; stte`

Yogi wrote: Can this 4-cell bivalue loop be used to solve the puzzle, with 3 of its points known?
3r6c9 -> 0 solution via, for example, the "Starfish";
8r6c9 -> 1 solution
JC Van Hay

Posts: 719
Joined: 22 May 2010

### Re: 4-Cell Bivalue Loop

Yes this is an interesting puzzle. As i found below two steps are required.
step1: Because of r8c2<>6 and r9c4<>7, single-digit status of 8 is moved from A to B.
step2: 6-8s can be eliminated, via FISHes, then stte.
Code: Select all
`Marked [*] as be false.A:after del 6and7   B: then stte*---+---+---*      *---+---+---*|*8.|..8|8..|      |.*.|..*|8..||.8.|8..|.*.|      |.8.|8..|...||8..|8..|..8|      |8..|8..|..*|*---+---+---*      *---+---+---*|...|.8.|...|      |...|.8.|...||8..|...|8.8|      |8..|...|*.8||.8.|...|*.8|      {.8.|...|..8|*---+---+---*      *---+---+---*|...|..8|.8*|      |...|..8|.*.||...|8..|.8.|      |...|*..|.8.||..8|...|...|      |..8|...|...|*---+---+---*      *---+---+---*`

Regards kozo
Kozo Kataya

Posts: 33
Joined: 06 July 2012