1)Definition

Between two rows(columns similar) in a band, we define their distinction to be (columns occupied)+(total kinds of digits)-(number of digits).

As in

- Code: Select all
`.1..23.4.`

4...156..

distinction between the rows is 4.

It could also be seen as "number of open chains",if cells in a same column or having a same digit are linked.

As in

- Code: Select all
`..12..34.`

..34..1..

it can be seen as 1-3-3-1(loop) and 2-4-4, so the distinction is 1.

2) Two rows with distinction 0(and not fully filled)

It is easily seen that the rows are "locked", no information can distinguish the rows. Digits in those empty cells can always be switched to make another solution.

..12..34. 571296348<---> 691285347

..34..21. 693485217<---> 573496218

Related Links:

forming-mugs-from-bug-lite-composites-t3210.html (About 2 lock rows)

post36383.html#p218026 (Example with 58 in r78)

3) Two rows with distinction 1

We could feel that there is little information to distinguish the rows. In some of the 17/18-given puzzles there are rows having such patterns:

- Code: Select all
`....x.... or ....x....`

......... ....x....

If the whole puzzle except two rows like this are filled without any basic contradiction(no repeat), then there is at least a solution.

So if we ignore the empty cells in these 2 rows and focus on the rest of the grid, there must be unique way to fill.

3.1)

The Pattern

- Code: Select all
`* . . . . . . . .`

* . . . . . . . .

* * * . . . . . .

* * * * * * * * *

* * * * * * * * *

* * * * * * * * *

* * * * * * * * *

* * * * * * * * *

* * * * * * * * *

is a magic pattern.

Proof of 3.1:

Distinction between r12 is 1. If there is a unique solution, there must be only one way to fill r3. For a row with pencilmarks, if it can be solved without extra conditions, there must be a hidden single and a naked single(this can be proved by induction.)And it is obvious that there cannot be a hidden single in r3.

Related Links:

investigation-of-one-band-free-patterns-t30199.html (mentioned this pattern)

3.2)

The Pattern

- Code: Select all
`abcd | abcd abcd`

abcd | abcd abcd

-----+----------

| abcd abcd

is a deadly pattern.

Proof of 3.2:

Distinction between r12 is 1. So there must be one way to fill r3. But given abcd are all in r12c23, r3c23 can be switched then produce another solution.

Related Links:

post57490.html?hilit=MUGs#p57940 (mentioned this pattern)

3.3)

For the pattern (letter=hidden *=given/irrelevant)

- Code: Select all
`a a...`

a a...

* * *...

b b...

both b's cannot be include in a's simultaneously if distinction between r12 is 1.

Proof of 3.3:Switch b's and there would be another solution.

Here is an example:

Original Puzzle(Source 奕数独)

- Code: Select all
`12.....5..731.5.......23..1...8..41...45.2..3...4..58.....17..9.913.8...74.....3.`

After SDC in c1 and n4

- Code: Select all
`+----------------------+----------------------+----------------------+`

| 1 2 689 | 679 6789 4 | 3 5 678 |

| 69 7 3 | 1 689 5 | 2689 269 4 |

| 4 58 5689 | 679 2 3 | 6789 679 1 |

+----------------------+----------------------+----------------------+

| 2359 356 279 | 8 3679 69 | 4 1 267 |*

| 8 1 4 | 5 679 2 | 679 679 3 |

| 239 36 279 | 4 3679 1 | 5 8 267 |*

+----------------------+----------------------+----------------------+

| 35* 358* 2568 | 26 1 7 | 268 4 9 |

| 26 9 1 | 3 4 8 | 267 267 5 |

| 7 4 268 | 269 5 69 | 1 3 68 |

+----------------------+----------------------+----------------------+

* rows have distinction 1.

35 in N4 are in r46c12, so r7c12 must have another digit, so r7c2=8.

The following puzzle is solvable by r37c23 UR and X-chains, but there is another copy of this pattern.

**Hidden Text:**Show

Related Links:

post57490.html?hilit=MUGs#p57940 (example)

**Hidden Text:**Show

3.4)

If two rows and two columns both having distinction 1, and their crossing (4 cells that they shares)has at most 2 givens, then it is impossible to have unique solution.

- Code: Select all
`. . . . . . . . .`

. . . . . . . . .

. .

. .

. .

. .

. .

. .

. .

We can make addition between pairs, e.g.12+23=13, as they have same effect.

After addition, r3~9c12 will become two pairs. But as distinction between c12 is 1, only one of them can be fixed,so the other can be switched, making another solution due to r12 distinction is 1.(way too fast, sorry..)

Related Links:

symmetric-18s-t2388-135.html (discussed one form of this pattern)

investigation-of-one-crossing-free-patterns-t30977.html (mostly patterns with 3 rows and columns)

qiuyanzhe, 2018/Sep/08