THE ANTI-TRIDAGON PATTERN
While I use the expression "trivalue oddagon pattern" for (and only for) the precise contradictory pattern (a pattern that can appear in no well-formed puzzle), let me call anti-trivalue-oddagon pattern or in short anti-tridagon pattern the following very broad set of conditions (which, in and of itself, allows no assertion or elimination, but only a big OR conclusion). It is defined to cover all the cases of any number of additional candidates in any number of cells of the "trivalue oddagon pattern".
(As such, it covers the trivalue oddagon pattern as well as the pattern of the tridagon rule in the 2st post or the patterns of the Tridagon Forcing Whips. But it covers many more cases.)
The anti-tridagon pattern
Let there be four blocks forming a rectangle in two bands and two stacks:
b11 b12
b21 b22
Let there be three digits, say 1 2 3, such that:
in each of the four blocks, there are three cells in different rows and different columns such that:
-- the same additional conditions on the 4x3 cells as in the first post of this thread are satisfied;
-- each of these 4x3 cells contains the three digits
----------notice that this doesn't cover the case where digits might be missing in some cell - but I haven't seen any such case as yet;
----------notice also the only difference with the contradictory trivalue oddagon pattern: each cell may contain any number of additional candidates
First remark: in any such case, none of the three digits can be a given in the 2 bands and 2 stacks of the 4 blocks. As a result, at least two of the three digits must be given in the uniquely defined block not in the 2 stacks and not in the 2 bands. This justifies mith's additional criterion and may provide speed improvements in some implementations.
First result: each of the 63,137 min-expand puzzles in mith's database has at least one anti-tridagon pattern (and at most one possibility for the broader pattern of blocks and digits). (Thanks mith for pointing out an error in the first version of this statement.)
Second result: classification of the number of additional candidates when Subsets, Finned-Fish, whips[≤7], Tridagons and Tridagon-Forcing-Whips(≤22) are active:
- Code: Select all
1: 33,579 (including 29,356 solved in SFin+Trid+W7)
2: 19,016 (including 9089 solved in SFin+Trid+W7+TFW22 but not in SFin+Trid+W7)
3: 7,443
4: 2,201
5: 647
6: 191
7: 42
8: 7
9: 7
10: 2
11: 1
12: 1
Total: 63,137
[Edit:] I think the above results can best be appreciated if compared to the raw results after allowing only Subsets and Finned Fish:
- Code: Select all
1: 23,279
2: 16,830
3: 10,851
4: 6,165
5: 3,254
6: 1,718
7: 562
8: 233
9: 121
10: 59
11: 40
12: 20
13: 1
14: 0
15: 4
Total: 63137
