The New Sudoku Players' Forum

[ChrisT]

I happened to notice an extension of the "remote locked pairs" rule today. Unfortunately, although it was nearly there, it didn't quite fit in the puzzle that I was doing, so I can't give a real example, just an explanation.

Code: Select all

+-------------+-------------+-------------+
| 12  .   .   | 123 123 .   | .   .   .   |
| .   .   .   | .   .   .   | .   .   .   |
| .   .   .   | .   .   12  | .   .   .   |
+-------------+-------------+-------------+
| *   .   .   | .   .   12  | .   .   .   |
| .   .   .   | .   .   .   | .   .   .   |
| .   .   .   | .   .   .   | .   .   .   |
+-------------+-------------+-------------+


Because they both form a triplet with r1c4,5 the "12"s in r1c1 and r3c6 are linked - ie they can be both 1 or both 2, but not one of each. Therefore cells r1c1 and r4c6 are a remote locked pair, and the candidates 1 and 2 can be eliminated from r4c1.

It's not exactly ground-breaking, given that even normal remote locked pairs are rare to find, and in most cases the elimination will be possible using simple pairs or triplets, but it just struck me as rather neat. It could be extended to a couple more cases, such as a linking quadruplet:

Code: Select all

+-------------+----------------+-------------+
| 12  .   .   | 1234 1234 34   | .   .   .   |
| .   .   .   | .    .    .    | .   .   .   |
| .   .   .   | .    .    12   | .   .   .   |
+-------------+----------------+-------------+
| *   .   .   | .    .    12   | .   .   .   |
| .   .   .   | .    .    .    | .   .   .   |
| .   .   .   | .    .    .    | .   .   .   |
+-------------+----------------+-------------+

or a longer chain.

I just wondered
- if anyone has noticed this technique before
- if it falls under some other rule that I haven't considered
and
- if anyone has spotted an incidence of this in a real puzzle, preferably where simpler methods are not possible!

Your comments would be appreciated!

Chris

[ronk]

It could be extended to a couple more cases, such as a linking quadruplet:

Code: Select all

+-------------+----------------+-------------+
| 12  .   .   | 1234 1234 34   | .   .   .   |
| .   .   .   | .    .    .    | .   .   .   |
| .   .   .   | .    .    12   | .   .   .   |
+-------------+----------------+-------------+
| *   .   .   | .    .    12   | .   .   .   |
| .   .   .   | .    .    .    | .   .   .   |
| .   .   .   | .    .    .    | .   .   .   |
+-------------+----------------+-------------+

... with eliminations(*) r1c4<>1 and r1c4<>2 as you point out.

[edit: The following is accurate AFAIK, but it's quite unnecessary. If the simpler naked quads technique is first applied, the digit 1, 2, 3, and 4 candidates in locations tagged '#' will already have been eliminated.]

By inspection, it's obvious there are many more eliminations possible.

Code: Select all

+-------------+----------------+-------------+
| 12  #   #   | 1234 1234 34   | #   #   #   |
| .   .   .   | #    #    #    | .   .   .   |
| .   .   .   | #    #    12   | .   .   .   |
+-------------+----------------+-------------+
| *   .   .   | .    .    12   | .   .   .   |
| .   .   .   | .    .    .    | .   .   .   |
| .   .   .   | .    .    .    | .   .   .   |
+-------------+----------------+-------------+
... with elimination of digits 1 and 2 at the location flagged '*',
... and with elimination(#) of digits 1,2,3 and 4 at the 10 locations flagged '#'.

This configuration fits ...

1. the Two-Sector Disjoint Subsets thread authored by Sue De Coq, and
2. An Interesting Rule of Almost Locked Sets authored by Carcul

... where the almost almost locked set is A = {r1c4,r1c5} = {1234}. It is "almost almost" locked because there are N+2 possible candidates for N cells.

Combining set A with almost locked sets B = {r1c1} = {12} and C = {r1c6} = {34} takes out candidates 1,2,3, and 4 in row 1 (excepting the sets).

Combining set A with almost locked sets D = {r3c6} = {12} and C = {r1c6} = {34} takes out candidates 1,2,3, and 4 in box 2 (excepting the sets).

Set A in r1c45 acts like a grouped bivalue which, along with r1c1, r3c6, r4c6, and r4c1 is a turbot fish in both digits 1 and 2.

Sue's thread is titled two sector, but the technique applies equally well when the two sectors are one and the same sector, as noted in Carcul's thread.