Regular Fish

In Sudoku, “fish” is a type of restriction that the given clues impose on candidates of a single value. This page defines and illustrates regular fish, the placement restriction that lines place on other lines. It also explains how the X-panel is used to find regular fish.

We start with illustrations from Antoine Alary’s More Extreme Sudoku 144, and. a restriction similar in principle to regular fish, the naked quad in c7. Four cells are found to contain only the four values 1, 4, 5 and 7. Therefore, no other c1 cells can contain these values. We say these values are locked. The  subset creates a 3-form placing 8 and the pair 69 in c1.

A fish is a similar lock on the placement of candidates of value X on a subset of lines. If n lines have candidates only in n crossing lines, other lines must give up candidates in these crossing lines.

In the line marked grid of More 144, examine the rows containing row 9-slinks. Right, 9 in the lower left corner. Do you find three rows with 9’s only in the same 3 column positions?

It’s a lock situation just like the naked quad. Among the 6 rows with 9- candidates, rows 1, 4 and 7 lock column positions 3, 6 and 9. These are lines of the swordfish, a regular fish of 3 lines.

So what happens to 9-candidates in these column positions in remaining rows 3, 4, and 9? They are fish food.

Yes, just verifying these facts on such a full grid is a chore. Imagine finding this swordfish on such a busy grid, when you don’t know it is there, or what else may be there.

How does it look on the X-panel? The left panel is freshly transcribed from the above grid.

 

 

Without the irrelevant clutter, notice how natural it is to match row patterns, especially if you are comparing lines of fewer X digits.

The right panel shows a swordfish identified for confirmation and analysis by Sysudoku’s blank line tally marking.  A blank column has minus (-) signs marking the rows of the fish. A blank row has plus (+) signs designating the locked columns.  Using the tally, candidates in locked columns, but not in fish rows are eliminated.

Back on the grid, we use template “fin” symbols to mark the candidates of the fish. The symbols point toward the eliminations, with row fish symbols pointing vertically and column fish symbols pointing horizontally.

As elsewhere in Sysudoku, diamonds mark victims.

 

 

Looking back at the naked quad example, four of the unplaced cells are locked in the quad, leaving a complementary pair 69.  With SE8,  there are 3 clues, and 6 unplaced cells. Four cells go to the quad, leaving two for the pair.

Likewise, a regular row (column) fish has one or more complementary column (row) fish. With 3 9-clues, the 3-row swordfish of More Extreme 144 has a complementary 3-column swordfish attacking the same victims. Find it on the 9-panel above.

I use vertical bars (on the keyboard above Enter) in rows to mark the fish columns, while the + signs mark locked rows. If I had found the column swordfish first, I would have marked it instead, with “fin” symbols pointing along rows.

By the math, if there are m clues, and n fish lines, then the complementary fish has, at the most, 9 – m – n lines. That means that if you are looking at both rows and columns, you never have to look for fish larger than (9 – m) / 2. This allows for an X-wing (2 lines), a swordfish (3 lines), or a jellyfish(4 lines).  When bragging about your fishing exploits, there’s no reason to go further than that.  If you do, your listener may point out the smaller complementary fish you probably  found first.

Here’s an interesting case, Royle 17-1007 encountered in a review of The Hidden Logic of Sudoku, by Denis Berthier. It was selected from a huge collection of 17 clue puzzles, but not to illustrate jellyfish, although it has one.  First, is there a complement? And what size is it? Now add the blank line tally of the jelly, and do the same for the complementary fish on a copy of the 9-panel.

To compare blank line tallies, you can find them in  the 12/06/16 post, Suset Jelly in nrc Space. At the end of the post, you’ll see an XY nice loop that destroys both fish.  That happens to fish all the time.

The simplest and most frequently encountered regular fish is the X-wing, in which two lines reserve two crossing line positions for themselves. The candidates of these positions form a slink in each line, therefore an X-wing present in the line marked grid can be spotted in line marking.

No searching required. As you shift candidate positions to mark a slink in the filled line, you notice a matching slink in a marked parallel line. Here, in line marking r6 of funster Extreme 74, the 6-slink in naked pair 46 matches the earlier slink marked in r2. No removal, but the fin markers now exclude 6-candidates from unmarked rows r3, r4, and r7 as line marking continues.

The X-wing slink match also comes into play  when a removal creates a new line slink. In the collapse of Rebecca Bean’s Extremely Hard VI-9, an XY nice loop removes a 7 which confirms 6r2c3, removing 1 to create a 1-wing. With slinks marked, X-wings are often found in advanced stage traces.

There are dead fish ( no victims) that we don’t bother to note or mark. After the removal of fish victims, the lines outside the fish form one or more subset fish, but that is because, like cells in a unit, subsets have to account for all unplaced lines. Also, hidden dublex marks in basic  generate dead X-wings we rightfully ignore.

Guide pages to follow will describe finned and kraken fish, and variously named fish that combine boxes and lines.

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