BTW, are there any defined names for higher-order patterns (ie: 6x6, 7x7, 8x8, 9x9) or are they even possible?

If a 9x9 is possible, I reckon we should call it a Cthulu...

N=9 isn't possible unless your grid is completely empty with no known cells at all. And even then, once you find the pattern you remove the possibilities from every row (or column) not in the pattern. If the pattern has nine rows, then there aren't any rows not in it, and so you remove nothing.

Patterns with N>4 rows exist of course, but aren't necessary, as a corresponding pattern of size 9-N exists in columns.

For example, in this grid X, +, and * are places a certain digit can go, while . represents places it can't.

Code:

a b c d e f g h i
1 X . X . . . . . .
2 X . . . X . . . .
3 . . . + . . . + +
4 X . X . X . . . .
5 . . * . . . . . +
6 . + . . . . . + .
7 . . . . * + + . .
8 . + * + . + . . .
9 . . . . * . + . .

There is a N=3 locked set with rows 1, 2, and 4 (the X's). Row 1 can be in columns {ac}, row 2 in {ae} and row 4 in {ace}. The union of these sets is {ace} which has three columns in it for three rows. So we can remove as a possibility the columns {ace} from all the rows other than {124}, which are the cells with *'s.

There is also a locked set with size N=6 with the columns b, d, f, g, h, and i (all the +'s). These are the columns that weren't used in the previous locked set. Column b can has rows {68}, column d has {38}, and so on. The union is the rows {356789}, which are all the rows that weren't in the N=3 locked set. This means we can remove the rows {356789} from the columns that aren't {bdefgi}. Which is, again, all the *'s.

So there's really no point in naming N=5 or more, because there always exists a corresponding pattern with N<5 that does the exact same thing.

Posted: Tue Oct 11, 2005 9:42 am Post subject: Re: Why cant I use a sword 3-fish

DHallman wrote:

After many steps Simple Sudoku gives me the following filter on 4s

I certainly don't understand your b's an B's.
Anyhow this is what I get when filtering on 4's (and it certainly isn't the way forward so I don't see why you might be interested in them):

DHallman wrote:

BTW how can I copy your image.png into here? I see others doing it.

1. Save the image to file (File | Save Image As) in SS.
2. Upload the image here - http://www.imageshack.us/
3. Copy the "direct link" given by imageshack into your post enclosed in IMG tags.

nb: Don't paste images here unless you think it'll help illustrate your problem (or solution) as it's much easier to copy and paste the text representation of the puzzle into SS than manually copying each given from an image.

Posted: Tue Oct 11, 2005 7:09 pm Post subject: Re: Why cant I use a sword 3-fish

Angus,

I normally highlight (mark) in blue the rows or columns of fish so I can better see them. b indicated actuak 4s and B the extension in c

I had eliminated the 4 at r5c8 and somehow at r3c4 & r3c6. Thus arriving at a swordfish using c4,c6 & c8. Which should have allowed me to eliminate r2c5 & r7c5. What did I do wrong?

Consider the b in r3c8 ... it does not share a row
with other b's. So the group of b cells is not quite
a swordfish (... but it would be if that same b was
in r4c8 instead).

Posted: Wed Oct 12, 2005 6:25 pm Post subject: Re: Why cant I use a sword 3-fish

Nick67,

Thank you. I did not realise that a fish had to have all rows AND
columns occupied when present. Ie no stragglers except for extensions for shorter rows (or columns).

Look for N columns (2 for X-wing, 3 for the Swordfish, 4 for a Jellyfish, 5 for a Squirmbag) with 2 to N candidate cells for ONE given digit. If these fall on exactly N common rows, then all N rows can be cleared of that digit (except in the defining cells!). The test can also be done swapping rows for columns.

I like your revised definition.

Quote:

It seems to me that you could actually look for columns with 1-N candidate cells (ie: includes solved cells), but if you find, say, a swordfish with such a cell, it degenerates to an x-wing.

.

I agree with "1 to N" instead of "2 to N".
Then, your pattern-finder could recognize patterns
containing singles. I would guess that most solvers
recognize singles before looking for such patterns,
but that doesn't seem to be necessary.

Quote:

Note also that the requirement "and each of those rows has at least 2 candidate cells" seems to be extraneous.

I agree. There seems to be no reason to make this additional check.
(For example, the proof above does not depend on this condition).

Nick,

Your last statement above does not seem to agree with your last post re my problem.

I had eliminated the 4 at r5c8 and somehow at r3c4 & r3c6. Thus arriving at a swordfish using c4,c6 & c8. Which should have allowed me to eliminate r2c5 & r7c5. What did I do wrong?

Looking at your grid, it satisfied only half the conditions required - ie three columns each contain three 4s, but the candidates in these columns are in more than three rows (rows 2,3,4 & 7).

Consider the b in r3c8 ... it does not share a row
with other b's. So the group of b cells is not quite
a swordfish (... but it would be if that same b was
in r4c8 instead).

I think it is a Swordfish - but not one based on columns - instead it's one based on rows as rows 2, 4 and 7 all contain b's in a combination of 3 columns, meaning that the b in R3C8 can be eliminated. Of course in this case there is an easier x-wing on the b's in locations (R4C4, R4C6, R7C4, R7C6)!

I think it is a Swordfish - but not one based on columns - instead it's one based on rows as rows 2, 4 and 7 all contain b's in a combination of 3 columns, meaning that the b in R3C8 can be eliminated. Of course in this case there is an easier x-wing on the b's in locations (R4C4, R4C6, R7C4, R7C6)!

The "b" notation certainly makes it looks like that.
But DHallman was using "b" to represent key cells
with the candidate 4.

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