Domain: a row, a column, or a box.
D1: a domain in which all cells containing candidate “x” belong to only two other domains, D2 and D3.
Cells “a”: cells containing candidate “x” and lying in the interception of domains D1 and D2.
Cells “b”: cells containing candidate “x” and lying in the interception of domains D1 and D3.
Na, Nb: the number of cells “a” or “b”.
Usually, in a nice loop, we draw a strong link between the only two cells in a domain where a candidate “x” can be placed. For example, [r1c1]=8=[r1c7], and the logic of this link is: if 8 is in r1c1, then it cannot be in r1c7, and if 8 is not in r1c1, then it must be in r1c7 (and vice-versa for r1c7). But we can extend this concept by making use of a special property of boxes: they can have more than one cell in their interception with a column or a row. Consider Grid 1 as an example:
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. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .
-------+-------+------
. . . | 5 . . | . . .
. . . | 5 . 5 | . . .
. . . | . . 5 | . . .
-------+-------+------
. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .
Grid 1
Box 5 has more than two possible cells for candidate 5. However, we can consider that these four cells are in only two possible locations inside box 5: columns 4 and 6. So, we can say that, if 5 is not in one of the cells r4c4/r5c4, then it must be in one of the cells r5c6/r6c6; and if 5 is in one of the cells r4c4/r5c4, then it cannot be in cells r5c6/r6c6. With this idea in mind, we are able to make the following generalization of the rule for draw a strong link:
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If a candidate ‘x’ is distributed among the cells of a domain D1 in such a way that these cells belong to only two other domains, D2 (cells ‘a’) and D3 (cells ‘b’), then we can draw a strong link between cells ‘a’ and cells ‘b’, with label ‘x’.
We can call this type of strong link a generalized strong link. It is easy to see that Na + Nb can have the maximum value of six. Grid 2 provides some examples, where “a” and “b” are cells containing candidate “x”:
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a . . | . a a | . b .
a . b | . . . | a . a
. . b | b b b | . b .
-------+-------+------
. . . | a . b | . . .
. . . | a . b | . . .
. . . | a . b | . . .
-------+-------+------
. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .
Grid 2
For this type of strong link, I propose a notation that is similar to the already known notation: the only difference is in putting the “a” or “b” cells between square brackets and separated by a vertical line. For example, for box 2 we can write: [r1c5|r1c6]=x=[r3c4|r3c5|r3c6].
Now, for this type of strong link to be inserted in a nice loop, it must be propagated but in a restricted way: we can only propagate from the set of cells “a” (or “b”) to another cell with a weak link having the same label “x”, and that cell can be of only one of two types: a bivalue cell containing candidate “x”, or a bi- or polivalue cell containing candidate “x” and having a strong link with same label “x”. Furthermore, that cell must be in the same domain common to all cells “a” (or “b”).
We can see that, besides its probable scarcity, the generalized strong link as I have defined so far cannot be used to make eliminations in cells “a” or “b”. However, this can be possible if Na (or Nb) = 1. Please consider the examples in grid 3 (“ab” denotes a cell that can function as an “a” cell or as a “b” cell):
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ab. b | a . . | . a .
. . ab| . . b | . . .
. . . | . . b | b . b
-------+-------+------
a . b | . . . | . a .
. . b | . . . | . . .
. . b | . . . | . . .
-------+-------+------
. . . | . . . | . b .
. a . | . b b | . b .
. . . | . . . | . b .
Grid 3
Boxes 1, 2, 3, and 4 show some examples with D1 = box and D2, D3 = rows and/or columns. Boxes 6 and 9, and 7 and 8 exemplify the case for D1 = column/row, and D2, D3 = boxes. In each case, we can draw a generalized strong link between cell “a” and the set of cells “b”.
Now, please note that this type of generalized strong link can be more useful because, although the set of cells “b” can only be propagated as described above, cell “a” can be propagated as in classic nice loops: with a strong or weak link, with the same or a different label, which can be used to remove a candidate from this cell.
Now, allow me to make the following definition:
Strong Nice Loop: a nice loop that includes one or more generalized strong links.
To exemplify all this, let’s return to the puzzle I have mentioned in the beginning of this thread:
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579 5789 12 | 3 4 12 | 589 59 6
459 6 249 | 7 8 29 | 1 3459 3459
34 89 1349 | 19 6 5 | 489 7 2
----------------+----------------+----------------
79 4 8 | 5 2 79 | 3 6 1
6 1 5 | 89 3 489 | 7 2 49
2 3 79 | 6 1 479 | 459 8 459
----------------+----------------+----------------
1 2 *39* | 4 7 *38 | 6 359 3589
3457 57 6 | 18 9 *138 | 2 *34 3478
8 *79 *3479 | 2 5 6 |*49 1 3479
With the definitions that I have made above, I have noted that Bennys’s observation can be described by the following strong nice loop:
[r7c3]=9=[r9c2|r9c3]-9-[r9c7]-4-[r8c8]-3-[r8c6]=3=[r7c6]-3-[r7c3] => r7c3<>3.
(in the grid, the cells marked with “*” are the ones used in the loop, and the cell marked “**” is where the discontinuity arises). Let’s confirm the logic inside this loop: if r7c3 is not 9 then it is 3; then one of the cells r9c2/r9c3 is 9; which means that r9c7 is 4, and r8c8 is 3, and r8c6 cannot be 3, so r7c6 is 3, and this is an impossible situation because we have already a 3 in row 7, in cell r7c3: so, r7c3 cannot be 3.
Another example can be seen in the following puzzle, which was posted in this forum in a thread entitled “Wondering if there is a solving technique for this?” (November 22).
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. . 1 | . 8 . | 6 . 4
. 3 7 | 6 . . | . . .
5 . . | . . . | . . .
-------+-------+------
. . . | . . 5 | . . .
. . 6 | . 1 . | 8 . .
. . . | 4 . . | . . .
-------+-------+------
. . . | . . . | . . 3
. . . | . . 7 | 5 2 .
8 . 2 | . 9 . | 7 . .
After some basic steps, we can remove a 4 from r7c2 with a Turbot Fish, a 7 from r5c2 with a nice loop, and a 2 from r4c5 and r6c5 with two forcing nets, and we get:
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29 29 1 | 57 8 3 | 6 57 4
4 3 7 | 6 25 129 | 129 1589 12589
5 6 8 | 1279 247 1249 | 1239 1379 1279
-------------------+------------------+--------------------
12379 124789 349 | 2789*367 5 | 12349 134679 12679
2379 249 6 | 279 1 29 | 8 34579 2579
12379 12789 5 | 4 *367 *2689 | 1239 13679 12679
-------------------+------------------+--------------------
1679 179 49 | 1258 2456 *12468*| 149 14689 3
1369 149 349 | 18 *46 7 | 5 2 1689
8 5 2 | 3 9 146 | 7 146 16
From here, we can make another elimination using the following strong nice loop:
[r7c6]=8=[r6c6]=6=[r4c5|r6c5]-6-[r8c5]-4-[r7c6]
which implies r7c6<>4.
Let’s look at a final example. The following puzzle was also posted by Bennys, in his recent thread entitled “Not almost locked sets”:
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12 7 126 | 5 126 9 | 3 4 8
3 5 9 | 8 126 4 | 126 26 7
4 8 126 | 3 1267 67 | 126 5 9
----------------+----------------+--------------
169 1349 7 | 12 46 368 | 5 *289 14
5 2 14 | 19 49 78 |*78 3 6
169 1349 8 |*127 4567 3567 |*27 *279 14
----------------+----------------+--------------
1289 149 124 | 6 *579 *57 | 478 *78 3
89 49 5 | *79* 3 1 | 4678 678 2
7 6 3 | 4 8 2 | 9 1 5
Bennys made a clever observation (which concludes that r6c4<>7) that allows the grid to be easily solved from here. However, I have noted that we can arrive at the same conclusion with the following strong nice loop:
[r8c4]=7=[r7c5|r7c6]-7-[r7c8]-8-[r4c8]=8=[r5c7]=7=[r6c7|r6c8]-7-[r6c4]=7=[r8c4]
(or [r8c4]=7=[r8c7|r8c8]-7-[r7c8]…), which implies r8c4 = 7.
Carcul
Excuse me for avoiding the term 'strong'.