How large nxn fishes of each type (shape, fins) do we need to check to find all possible fish eliminations?the largest fish with no smaller eliminations is a 7x7+1 { in nxn notation} [ found to date.]
1) 7x7+1 Base: R1458C9B27, Cover: R2C234589B8 Bi{Cells}: 5,73, Exclusions: 77,
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+---------------+-------------------+---------------------+
| 6 78 2 | 3 478 9 | 1 5 478 |
| 17 3 18 | 2 4578 578 | 789 489 6 |
| 9 5 4 | 78 1 6 | 278 3 278 |
+---------------+-------------------+---------------------+
| 2 6 789 | 5 789 4 | 3 789 1 |
| 4 1 789 | 6789 2 3 | 5 6789 789 |
| 5 789 3 | 1 6789 78 | 2789 246789 24789 |
+---------------+-------------------+---------------------+
| 3 79 679 | 4 6789 2 | 789 1 5 |
| 8 4 15 | 79 3 15 | 6 279 279 |
| 17 2 1569 | 6789 56789 1578 | 4 789 3 |
+---------------+-------------------+---------------------+
http://forum.enjoysudoku.com/post229153.html#p229153my nxn+ k fish that replicates nxn fish with no reuse age of sectors has found fish up to size 9x9+2 but all of them had a smaller fish for the same elimination.
nxn+k fish that reuse sectors can scale well beyond those size and has solved a few of the "no fish" grids.
most large fish past size 4 have an equivalent smaller fish.
{ just like subsets ie hidden set of size 4 has a complementary size 5 naked set}
- which is why we don't use quintuplets or larger subsets.
now if some one can prove the smaller size elimination that should in theory exists then fish searching wouldn't be so difficult.
as building base ((1 ->7 ) out of 27) sectors and cover (( 1 -> 7 ) out of 20 )+ ( (0 -> 2) out of 13 ) sectors = millions of fish to check for each digits 1-9.
{around 3 -1/2 hours for all of them on my solver that doesn't use template short cuts like hodoku and that time frame took me 5 years and many revisions to drop down from 16+ hours on 1 cycle. }
fact, should we make the connection explicit: LoadER Crane ? Or is that stupid?
nope, that was what i was hinting at with the ER separated.
For any of that to make sense, you'd have to define what you mean by "pattern" and how its internal logic is different from a chain (or coloring) in this case.
this one will be tricky to explain but ill attempt. patterns understood conjugated cells in a different light cells in a sector both could be true but not both. so, we built patterns around that idea.
that a sector is locked to holding a truth.
what happens if a is true and b is off, using line of sight we add another sector with the same concept
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on/off <-> on/off
|| ||
on/off <-> on/off
|| indicates 2 way arrows between the "Col" not the normal "strong link"
this example uses 4 cells with 4 way on/off { x-wing}
step 3 would be to evaluate the out come of each of the on/off sequences using line of sight and note the cells that see the "on" cell.
if any cell would see "on" for all the out comes of the cells selected then it would always be off.
this can be done with out needing or understanding "weak links" or even writing yes/no or using colors
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this one has 3 way on/off in 5 cells thus all cells aren't on/off directly.
the above check proves the eliminations thus the pattern sound {finned x-wing}
on/off <-> on/off
||
on/off <-> (on/off) ( on/off) (on/off)
loose rules to explain that system: {i don't technically have rules this is made up right here} as all i needed was
if any cell would see "on" for all the out comes of the cells selected then it would always be off. rule 1: any set of sectors with the same number yes/no relation ships then any cell that sees 2 yes's is always off
rule 2: any set of sectors with less then the same number yes/no relationships then any cell that sees all of the yes/no's that aren't line of sight connected must be false.
eventually i figured out how to use
(cell count = row count) as a truth. { 2 cells}
cell count of an intersections ( {Box * Row } + (Box * Row) = Row ) ie 3 cells in one box/row + 1 cell in another box/row = grouped link. { 3 + 3) or (3 + 2 ) is also possible.
easiest to show in base/cover
which is basically what I've been using since i got on this forum using mathematics limitation of placements. ie SETs {containers} to identify truth holders
{every thing in my solver is coded with that concept including all my "wing" patterns }
if you want you can add fancier words and explain the connections and call it a "chain" i can defiantly see how you would do that and can admit they would be similar to a chain/coloring comparitivly.
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ERI
was a label i created for advance chaining purposes to mark the direction / connection point of a node link that changes directions in the box row < = > col
it's easier and more compact to pre identify all the ERI Boxes and connect weak links off that 1 cell instead of 2 separate local groups.
you could list the ERI + Conjugated link but it was never really intended for anything out side of coding
but if you don't know what the ERI stands for it's overly complicates the simple ER which just needs the box + row/col listed that it works out of.
this example show cases how it can makes some stuff easier :
this grid has 4 linked ERi's we could reduce it to a box-style x-wing off the hubs instead of 8 grouped links.
eliminations are fun 2, all 4 hubs are removed, all boxes that see 2 boxes on row/col for that row/col <> digit. { row & Col is keyed to the hubs direction } making look up easy.
{and this is just from fish] the concept could be applied to all "chains" as an easier way to link box grouped nodes.
like this which could also fall under the turbot header as 4 grouped links using ERI's
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+----------------+------------+----------------+
| . (1) . | 1 1 1 | . (1) . |
| (1) (-1) (1) | -1 -1 -1 | (1) (-1) (1) |
| . (1) . | 1 1 1 | . (1) . |
+----------------+------------+----------------+
| 1 -1 1 | 1 1 1 | 1 -1 1 |
| 1 -1 1 | 1 1 1 | 1 -1 1 |
| 1 -1 1 | 1 1 1 | 1 -1 1 |
+----------------+------------+----------------+
| . (1) . | 1 1 1 | . (1) . |
| (1) (-1) (1) | -1 -1 -1 | (1) (-1) (1) |
| . (1) . | 1 1 1 | . (1) . |
+----------------+------------+----------------+
That's one reason why we have this predicament.
How about my second option: Turbot Crane? That has three syllables, doesn't it? (And like the 2-String Kite, its official name could be shortened in use.) I'm open to better suggestions, too.
