I agree with Myth Jellies that the number of times the candidate appears in a group is one *reason* that there are logical relationships -- they are not the relationships. Those logical relationships are:

ALL links: NOT (A and B)

Strong Links: A if and only if NOT B(By "ALL links" I mean weak, any, default, etc. Actually, ALL cells in the same group are weakly linked by the candidate(s) they share based on the underlying rules of the game. A blank sudoku grid has weak links between every candidate in every cell and every other cell in its group. Unless we're working on a sudoku variant that allows, say 2 or 3 copies of each digit in a group, the logical relationship is assumed.)

Strong links are "if and only if" or IFF. They could be called IFF links.

ALL links, including Strong links are NOT AND.

One possible drawback from this way of defining the links is that there is no simple way to include

Non-Strong Links without falling back on the number of candidates in the group.

One advantage of using IFF and "if/then" rather that candidates=2 and candidates>=2 is that there can be a logical relationship between two candidates that is IFF even though there are more than two candidates. For example:

- Code: Select all
`[12][23][13]|[12345]`

Because of the naked triple in r1c123, r1c1=2=r1c2=3=r1c3=1=r1c1.

Of course, you could take a first step of removing candidates 1, 2 and 3 from r1c4 based on the naked triple -- but you don't have to. The IFF relationship and the strong links already exists. If it were defined based in the number of candidates, the links would have to be considered a weak until after the simplification.

Then there more complex links:

Here is a xy-ring:

- Code: Select all
` 12 . . | . . 23 `

. . . | . . .

14 . . | . . 34

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

[12]-2-[23]-3-[34]-4-[14]-1-[12]

These are all weak links -- until this ring is recognized. At that point, they can all be considered strong links before making any exclusions it allows.

In fact, the exclusions are made BECAUSE these are strong links -- NOT the other way around! Same thing in an x-wing -- as soon as you see the pattern, you can mark all four edges as strong links -- and then make exclusions.

The situation is clearer if you allow links between groups of candidates:

- Code: Select all
` 12 . . | . . 234 `

. . . | 34 . .

15 . . | . . 345

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

. . . | . . 34

. . . | . . 346

[12]-2-[234]-34-[34]-34-[345]-5-[15]-1-[12]

All these weak links can be used as stong links once this pattern is noted. There is a strong link between [234]=34=[345], as exactly one of them can be reduced to [34] allowing the exclusion r5c6<>[34] -- yet there are 3 of each candidate 3 and 4 in the 6th column,

even after the exclusion, showing that in these situations, weak vs strong MUST be defined based on IFF rather than number of candidates=2.

Jeff:

The definition "Strong link - if A is false, then B is true" is not accurate, as it allows both A and B to be true at once.

Also

'Strong link' is currently accepted to imply

"if 'A is false, then B is true' or 'if A is true then B is false'." should be:

"if 'A is false, then B is true' AND 'if A is true then B is false'." or simply "A iff NOT B"