## Solving sudoku puzzles by Pair-Plus schemes

Post the puzzle or solving technique that's causing you trouble and someone will help

### Solving sudoku puzzles by Pair-Plus schemes

Hi,
In this post I describe a technique wich is based on a simple idea. I consider a group of three candidates in two cells as (a, b, f) (a, b) lying in the region of a row, a column or a box. The group is called here Pair-Plus. In fact there exist two pairs of candidates a and b in two cells incorporating in one cell a single fellow f. Now, I consider that the first cell ( a, b, f) of the group consists of two parts: the subgroup [a, b] and the fellow f. It is true that only one of the two entities [a, b] and f being in the same cell must be true.

If the subgroup [a, b] of the first cell ([a, b], f) is true, by ignoring f, this subgroup cell together with the other cell (a, b) form the naked pair (a, b) (a, b) that acts in the region. If the fellow f is true, then the Pair-Plus scheme reduces to (f) (a, b), where f constitute a single candidate acting in the region. In case that a candidate is rejected by the action of [a, b] and also by f, it becomes obvious that there exists a contradiction meaning that both [a, b] and f being in the same cell are true. Therefore we come to the conclusion that the candidates which are rejected by both parts [a, b] and f must be removed.

From above, we ascertain that there are two processes:
1) One is due to action of the naked pair (a, b)(a, b)
2) The other is due to action of the single fellow f
If the two processes lead to common rejections of candidates, these candidates must be removed from the puzzle.

Based on the above explained technique for a Pair-Plus group the same procedure can also be applied to a Triple-Plus or a Quad-Plus schemes. Let us consider for example that a Triple-Plus group appears as (a, b, c) ([a, b, c], f ) (b, c ). We can proceed in the same way as before assuming that the fellow f is missing in the second cell. This allows the first cell (a, b, c) to joint with (a, b, c) (b, c) to form a naked triple acting in the region. Also, assuming that the subgroup [a, b, c] is missing in the second cell, the remaining group is (a, b, c) (f) (b, c), where the fellow f acts as a single in the region. The Triple-Plus and also Quad-Plus schemes are hard to spot and rarely give common rejections.

+---- ----------------------+---------------------------+-------------------------------------+
| 129 7 4 | 2^89* 19 1*2^8 | 1*2*^368* 5 1*2*^369* |
| 3 6 1289 | 4 1579 1258 | 1278 179 129 |
| 5 1289 1289 | 2789 6 3 | 1278 1479 1249 |
+--------------------------+---------------------------+--------------------------------------+
| 8 129 5 | 6 149 124 | 127 3 129 |
| 1279 4 1279 | 2359 8 125 | 1257 6 129 |
| 1269 3 1269 | 259 159 7 | 4 19 8 |
+ --------------------------+---------------------------+--------------------------------------+
| 4679 89 36789 | 1 2 468 | 36 4 5 |
| 146 18 1368 | 358 345 9 | 136 2 7 |
| 12467 5 12367 | 37 347 46 | 9 8 1346 |
+ --------------------------+---------------------------+--------------------------------------+

In the puzzle above there exists a Pair-Plus scheme (2, 1, 9) (1, 9) in cells A1 and A5.
1) The action of naked pair: Ignoring the fellow 2, the remaining naked pair
(1, 9) (1, 9) rejects 1 (note the symbol * ) from A6, A7 and A9, and also 9 from
A4, A9. Now, it “appeared” a naked pair (2, 8)(2, 8) in cells A4 and A6 which
rejects 2 in A7, A9 and 8 in A7.
2) The action of the fellow: The single fellow 2 in A1 rejects (note the symbol ^ )
the 2 in A4, A6, A 7, and A9.
3) There are common rejections of the candidate 2 in A7 and A9. These candidates
can be removed from the puzzle.

The above described technique is effective and easy to apply. The Pair-Plus schemes are very easy to spot in a puzzle and easy to check if they give common rejections. Although these schemes appear very often in a pazlle , few of them can give common rejections.
Any critisism and help on the subject would be welcome.
karybakas

Posts: 3
Joined: 06 May 2020

### Re: Solving sudoku puzzles by Pair-Plus schemes

Hello karybakas,

karybakas wrote:Any critisism and help on the subject would be welcome.

Good news: your logic is valid. Its generalized versions are commonly called Almost Locked Set (ALS) moves.

Bad news: your example is terrible. While the logic works, the same eliminations (2r1c79) are achieved directly with a hidden pair (36)r1c79 or a naked quad (1289)r1c1456. Besides, you have tons of unsolved singles and other basics left anyway. Here's the grid after all basics (singles, locked candidates, locked sets) are solved:

Code: Select all
`.---------------.------------.-------------.| 29   7   4    | 8   19  12 | 3   5   6   || 3    6   28   | 4   7   5  | 28  19  19  || 5    12  1289 | 29  6   3  | 28  7   4   |:---------------+------------+-------------:| 8    12  5    | 6   19  4  | 7   3   129 || 129  4   7    | 3   8   12 | 5   6   129 || 6    3   129  | 29  5   7  | 4   19  8   |:---------------+------------+-------------:| 7    9   3    | 1   2   8  | 6   4   5   || 4    8   6    | 5   3   9  | 1   2   7   || 12   5   12   | 7   4   6  | 9   8   3   |'---------------'------------'-------------'`

That is the point when you should start thinking in more complex logic, such as ALS moves. Can you use the "pair-plus" technique on that grid? It's actually possible to solve with that, though not quite as trivially as your own example. See the hidden text for how.

ALS solution: Show
Code: Select all
`.-----------------.--------------.--------------.| 29    7    4    | 8     19  12 | 3    5   6   || 3     6    28   | 4     7   5  | 28   19  19  || 5     12   1289 | 29    6   3  | 28   7   4   |:-----------------+--------------+--------------:| 8    b12   5    | 6    c19  4  | 7    3   129 || 129   4    7    | 3     8   12 | 5    6   129 || 6     3   a19'2 | 2-9   5   7  | 4   a19  8   |:-----------------+--------------+--------------:| 7     9    3    | 1     2   8  | 6    4   5   || 4     8    6    | 5     3   9  | 1    2   7   || 12    5    12   | 7     4   6  | 9    8   3   |'-----------------'--------------'--------------'`

Notice the almost-naked pair (your "pair-plus") 19+2 on row 6. If the pair 19 is true, it kills 9r6c4. If it's not true, then the 2 (your "fellow") must be. That doesn't help directly, but you could consider a short chain: 2r6c3 -> 1r4c2 -> 9r4c5. In other words, either the pair (19)r6c38 is true, or (2)r6c3 is true which implies (9)r4c5. Either way 9r6c4 can't be true so it can be eliminated. The rest is singles.

SpAce

Posts: 2514
Joined: 22 May 2017

### Re: Solving sudoku puzzles by Pair-Plus schemes

Hello SpAce,

Thanks for your comments and the interesting suggestions. I am new in this forum. From my own mistake in the loading of the puzzle, this shrank in the text. I don’t know if that can be recovered.
To explain the technique, I used a virgin puzzle and it was not my intention to solve it. Besides, I could use only one row from the puzzle for that purpose. The P-P (i.e. pair-plus) technique needs a row, a column or a box to be applied.
In the grid you have come up with, there are P-P schemes but no one can give a solution (few unsolved cells). The row 6 is saturated, apart from the P-P set (192)(19) it has only the unsolved cell (29), but it needs more cells. You provided a solution using the P-P (192)(19) by using the pair (19)(19), the single (2) and a chain. This is a nice solution a swell as any solution for which we use less basic techniques. On the other hand there are many basic techniques which we use and we need. These techniques provide a predetermined way for solving as do the P-P approach that is described here.
A set P-P is an ALS and it is a subset of the vast category of ALS, even more it is an almost-naked pair, but it’s role as a member of an ALS is inactive. The P-P set as it is used here is an autonomous entity. When it is spotted, the way of solving is given and this is it‘s advantage. Perhaps, it could be classified as a basic technique.
karybakas

Posts: 3
Joined: 06 May 2020

### Re: Solving sudoku puzzles by Pair-Plus schemes

use the code tag to prevent the grid from dissembling
for me show caseing the logic you are suggesting might be easier from the point past basics, which is here:
Code: Select all
`.---------------.------------.-------------.| 29   7   4    | 8   19  12 | 3   5   6   || 3    6   28   | 4   7   5  | 28  19  19  || 5    12  1289 | 29  6   3  | 28  7   4   |:---------------+------------+-------------:| 8    12  5    | 6   19  4  | 7   3   129 || 129  4   7    | 3   8   12 | 5   6   129 || 6    3   129  | 29  5   7  | 4   19  8   |:---------------+------------+-------------:| 7    9   3    | 1   2   8  | 6   4   5   || 4    8   6    | 5   3   9  | 1   2   7   || 12   5   12   | 7   4   6  | 9   8   3   |'---------------'------------'-------------'`

we also use the designation of Rows and Cols by numbers 1-9 to indicate which cells for easier reading and in some cases note the cells with () * or letters for the chain section it comes from} on the actual grid to show where we are working

Skyscraper: 2 in r4c2,r6c4 (connected by r3c24) => r6c3<>2
as one of many 2 -fish options for the grid to reduce it to singles at this point:

as for the discussion:

if what you are talking about is akin to this it is an als- xz, als-xy family of eliminations: simplistically known as an XY-wing.
Code: Select all
`+---------------+----------------+----------------+| 29   7   4    | 8     19  12   | 3   5     6    || 3    6   28   | 4     7   5    | 28  19    19   || 5    12  1289 | 29    6   3    | 28  7     4    |+---------------+----------------+----------------+| 8    12  5    | 6     19  4    | 7   3     129  || 129  4   7    | 3     8   (12) | 5   6     29-1 || 6    3   129  | (29)  5   7    | 4   (19)  8    |+---------------+----------------+----------------+| 7    9   3    | 1     2   8    | 6   4     5    || 4    8   6    | 5     3   9    | 1   2     7    || 12   5   12   | 7     4   6    | 9   8     3    |+---------------+----------------+----------------+`

or this one.
Code: Select all
`+-----------------+----------------+-------------+| 29   7     4    | 8     19    12 | 3   5   6   || 3    6     28   | 4     7     5  | 28  19  19  || 5    12    1289 | 29    6     3  | 28  7   4   |+-----------------+----------------+-------------+| 8    (12)  5    | 6     (19)  4  | 7   3   129 || 129  4     7    | 3     8     12 | 5   6   129 || 6    3     19-2 | (29)  5     7  | 4   19  8   |+-----------------+----------------+-------------+| 7    9     3    | 1     2     8  | 6   4   5   || 4    8     6    | 5     3     9  | 1   2   7   || 12   5     12   | 7     4     6  | 9   8   3   |+-----------------+----------------+-------------+`

or moving up into als-xz territory
Code: Select all
`+------------------+---------------+---------------+| 29   7     4     | 8    19    12 | 3   5     6   || 3    6     28    | 4    7     5  | 28  19    19  || 5    12    1289  | 29   6     3  | 28  7     4   |+------------------+---------------+---------------+| 8    (12)  5     | 6    (19)  4  | 7   3     129 || 129  4     7     | 3    8     12 | 5   6     129 || 6    3     (129) | 2-9  5     7  | 4   (19)  8   |+------------------+---------------+---------------+| 7    9     3     | 1    2     8  | 6   4     5   || 4    8     6     | 5    3     9  | 1   2     7   || 12   5     12    | 7    4     6  | 9   8     3   |+------------------+---------------+---------------+`
Some do, some teach, the rest look it up.

StrmCkr

Posts: 1163
Joined: 05 September 2006