Quick explanation of the vocabulary used:
(1-)truth – set of candidates at least one of which is true, such as all candidates in a cell or the ends of an AIC
(1-)link – set of candidates at most one of which is true, such as all candidates in a cell (or just some of them) or one digit in the targets of a JE
n-link – set of candidates at most n of which are true
(similarly we could create n-truths)
multi-link – n-link where n > 1
fish (multi-)link – a multi-link where all candidates are of one digit and which doesn't need other digits to be proven
- Code: Select all
+---------+---------+---------+
| . . . | . . . | . . . |
| . * . | . * . | . . . |
| . . . | . . . | . . . |
+---------+---------+---------+
| . . . | . * . | . . . |
| . * . | * * * | . . . |
| . . . | . * . | . . . |
+---------+---------+---------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+---------+---------+---------+
Some time ago, I posted a puzzle with a MSLS in the marked cells. The tricky part was to notate that no digit could appear in them more then twice.
For any digit X, consider the links Xr2, Xr5, Xc2, Xc5, Xb5. The following grid shows how many times is the candidate of the digit X covered in each cell.
- Code: Select all
+---------+---------+---------+
| . 1 . | . 1 . | . . . |
| 1 2 1 | 1 2 1 | 1 1 1 |
| . 1 . | . 1 . | . . . |
+---------+---------+---------+
| . 1 . | 1 2 1 | . . . |
| 1 2 1 | 2 3 2 | 1 1 1 |
| . 1 . | 1 2 1 | . . . |
+---------+---------+---------+
| . 1 . | . 1 . | . . . |
| . 1 . | . 1 . | . . . |
| . 1 . | . 1 . | . . . |
+---------+---------+---------+
This almost looks like the structure we're after, we just have to divide everything by two (and round down to the nearest integer).
- Code: Select all
+---------+---------+---------+
| . . . | . . . | . . . |
| . 1 . | . 1 . | . . . |
| . . . | . . . | . . . |
+---------+---------+---------+
| . . . | . 1 . | . . . |
| . 1 . | 1 1 1 | . . . |
| . . . | . 1 . | . . . |
+---------+---------+---------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+---------+---------+---------+
We've seen that more than double this pattern can be covered with just 5 links. Therefore only two of the candidates can be true. This pattern is therefore a valid fish 2-link (it can also be proven using a Finned X-Wing).
We could just write it down when describing a pattern:
Links: Xr25c25b5 / 2 etc.
But I think it makes more sense to substitute:
A = 2L(Xr2c25, Xr4c5, Xr5c2456, Xr6c5) = Xr25c25b5 / 2
The first part is a description of which candidates the 2-link covers, the second part is its proof.
When describing a pattern, one would simply write:
Links: (2)A etc.
(The '2' just explicitly states that it's a 2-link, I think it makes it easier to count the links and double-check the logic.)
Another use of fish links is that we can now find fish representations of Broken Wings.
The following is a general representation, see the post below for real-world examples.
Consider the oddagon Xr2c25, Xr4c5, Xr5c246, Xr6c5 with a set of guardians G.
- Code: Select all
+---------+---------+---------+ +---------+---------+---------+ +---------+---------+---------+
| . 1 . | . 1 . | . . . | | . . . | . . . | . . . | | . 1 . | . 1 . | . . . |
| 1 2 1 | 1 2 1 | 1 1 1 | | . 2 . | . 2 . | . . . | | 1 . 1 | 1 . 1 | 1 1 1 |
| . 1 . | . 1 . | . . . | | . . . | . . . | . . . | | . 1 . | . 1 . | . . . |
+---------+---------+---------+ +---------+---------+---------+ +---------+---------+---------+
| . 1 . | 1 2 1 | . . . | | . . . | . 2 . | . . . | | . 1 . | 1 . 1 | . . . |
| 1 2 1 | 2 3 2 | 1 1 1 | | . 2 . | 2 2 2 | . . . | | 1 . 1 | . 1 . | 1 1 1 |
| . 1 . | 1 2 1 | . . . | | . . . | . 2 . | . . . | | . 1 . | 1 . 1 | . . . |
+---------+---------+---------+ +---------+---------+---------+ +---------+---------+---------+
| . 1 . | . 1 . | . . . | | . . . | . . . | . . . | | . 1 . | . 1 . | . . . |
| . 1 . | . 1 . | . . . | | . . . | . . . | . . . | | . 1 . | . 1 . | . . . |
| . 1 . | . 1 . | . . . | | . . . | . . . | . . . | | . 1 . | . 1 . | . . . |
+---------+---------+---------+ +---------+---------+---------+ +---------+---------+---------+
the 5 truths double the 2-link A the guardian set G
We can now get a finned 5-fish:
5 truths: Xr2, Xr5, Xc2, Xc5, Xb5
4+n links: (2)A, (2)A, (n)G
The guardians are candidates covered by the truths and not covered by other links. The number of links required to cover them depends on the circumstances.
The rank of the pattern is n-1 (L - T), any n links then form a truth.
T(G)
Note that the candidate Xr5c5 is part of three truths, but only two links. Therefore it needs another link, which is part of the G set.
Edit: Renamed the thread, corrected typos.
