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Hi folks,

I'm not sure about the correctness of the quote: "A fin is a candidate or group of candidates in the defining set, which does not belong to the secondary set."

In this Killer Sudoku example (see candidates diagram below), the 23(3) cage at r3c67+r4c7 must contain the digits {689}. Unfortunately, this 23(3) cage and N3 cannot form a constraint set of size 2 on the digit 6, because the overlapping region (r3c7) contains the candidate 6, thus violating the above definition. If this candidate were not present at this position then the 23(3) cage and N3 (= defining set) would form a generalized X-Wing on 6 with R3 and C7 (= secondary set), thus eliminating the candidate 6 elsewhere in R3 and C7 (i.e., r3c1+r56789c7):

.-----------------------.-----------------------.-----------.-----------------------.-----------------------.
| 124         124       | 678         123       | 1234578   | 6789        6789      | 2345789     245789    |
|           .-----------'-----------------------:           :-----------------------'-----------.           |
| 124       | 6789        6789        5678      | 1245678   | 56789       12345       12345     | 245789    |
:-----------:           .-----------------------+-----------+-----------------------.           :-----------:
| 679       | 3         | 5           24        | 124       | 689         689       | 124       | 679       |
|           |           |           .-----------'           '-----------.           |           |           |
| 689       | 124       | 24        | 1357        123457      123457    | 69        | 57        | 689       |
:-----------'-----------+-----------'-----------------------------------'-----------+-----------'-----------:
| 456789      456789    | 12346789    12456789    2456789     1245678     12345678  | 124         245       |
:-----------.-----------+-----------.-----------------------------------.-----------+-----------.-----------:
| 123567    | 12456789  | 12346789  | 12456789    2456789     1245678   | 12345678  | 12345678  | 13        |
|           |           |           '-----------.           .-----------'           |           |           |
| 123567    | 12456789  | 12346789    123456789 | 123456789 | 12345678    123456789 | 123456789 | 13        |
:-----------:           '-----------------------+-----------+-----------------------'           :-----------:
| 123456789 | 12456789    12346789    123456789 | 12345678  | 12345678    123456789   123456789 | 2456789   |
|           '-----------.-----------------------:           :-----------------------.-----------'           |
| 123456789   12456789  | 46789       456789    | 12345678  | 123456      123456    | 123456789   2456789   |
'-----------------------'-----------------------'-----------'-----------------------'-----------------------'

However, it is possible to regard r3c7 as a fin, even though it does not obey the above rule "A fin ... does not belong to the secondary set.".

The logic is:

1. Either r3c7 does not contain a 6 -> no 6 in r3c1+r56789c7 due to generalized X-Wing mentioned above,
2. Or r3c7 does contain a 6 -> no 6 in r3c1+r56789c7
3. Either way, no 6 possible in r3c1+r56789c7

I would be interested to hear other views on this. - Regards, Mhparker 16:45, 30 January 2008 (CET)

To answer my own question, r3c7 is apparently a particular type of fin, namely a so-called endo-fin, since it belongs to more than 1 group within the defining (base) constraint set. Mhparker 22:13, 30 January 2008 (CET)