I think this also applies to other single-digit eliminations like simple coloring, hinges and empty rectangles.

This is not a computer algorithm. It is an aid for pencil & paper solvers to figure out where to look. It is also a good trick for weirdos like me who try to solve difficult puzzles without pencil marks.

The basic idea of a turbot fish is that you have two strong links that make pincers to eliminate a candidate. How can this be possible?

Well, in a Sudoku block you have nine cells, like this:

- Code: Select all
`+---+`

|...|

|...|

|...|

+---+

Case A: An unsolved digit (*) can occur in a row:

- Code: Select all
`+---+`

|*.*|

|...|

|...|

+---+

Case B: or in a column:

- Code: Select all
`+---+`

|..*|

|..*|

|..*|

+---+

Case C: or in a row and a column:

- Code: Select all
`+---+`

|...|

|**.|

|.*.|

+---+

Proposition 1: Now, think about it: A useful strong link can only start or end in a Case C block.

Proposition 2: Now, think some more: If a single digit elimination exists for a particular candidate, there must be four Type C blocks for that candidate in the puzzle, arranged in a rectangle.

As an example, let's take today's DS VH: (Tue Oct 20 2015)

- Code: Select all
`+-------+-------+-------+`

| . . 8 | 2 6 . | . . 7 |

| . . . | . . . | 5 . . |

| 9 . 2 | 1 . 5 | . . 8 |

+-------+-------+-------+

| . . . | . . 6 | . . 5 |

| . 3 9 | . . . | 8 1 . |

| 8 . . | 5 . . | . . . |

+-------+-------+-------+

| 3 . . | 6 . 8 | 9 . 2 |

| . . 6 | . . . | . . . |

| 1 . . | . 2 7 | 6 . . |

+-------+-------+-------+

Play this puzzle online at the Daily Sudoku site

After basics:

- Code: Select all
`+-------------+-------------+-------------+`

| 4 5 8 | 2 6 39 | 1 39 7 |

| 6 1 3 | 78 78 49 | 5 2 49 |

| 9 7 2 | 1 34 5 | 34 6 8 |

+-------------+-------------+-------------+

| 7 2 1 | 38 389 6 | 34 349 5 |

| 5 3 9 | 47 47 2 | 8 1 6 |

| 8 6 4 | 5 39 1 | 2 7 39 |

+-------------+-------------+-------------+

| 3 4 7 | 6 1 8 | 9 5 2 |

| 2 89 6 | 349 5 34 | 7 348 1 |

| 1 89 5 | 349 2 7 | 6 348 34 |

+-------------+-------------+-------------+

Let's look at the patterns of the candidates:

1 is solved.

2 is solved.

5 is solved.

6 is solved.

The pattern for the unsolved 7s is:

- Code: Select all
`+-------+-------+-------+`

| . . . | . . . | . . . |

| . . . | * * . | . . . |

| . . . | . . . | . . . |

+-------+-------+-------+

| . . . | . . . | . . . |

| . . . | * * . | . . . |

| . . . | . . . | . . . |

+-------+-------+-------+

| . . . | . . . | . . . |

| . . . | . . . | . . . |

| . . . | . . . | . . . |

+-------+-------+-------+

Not interesting at all.

Similarly, the pattern for the unsolved 8s is:

- Code: Select all
`+-------+-------+-------+`

| . . . | . . . | . . . |

| . . . | * * . | . . . |

| . . . | . . . | . . . |

+-------+-------+-------+

| . . . | * * . | . . . |

| . . . | . . . | . . . |

| . . . | . . . | . . . |

+-------+-------+-------+

| . . . | . . . | . . . |

| . * . | . . . | . * . |

| . * . | . . . | . * . |

+-------+-------+-------+

Also not interesting.

9 is a little more interesting:

- Code: Select all
`+-------+-------+-------+`

| . . . | . . * | . * . |

| . . . | . . * | . . * |

| . . . | . . . | . . . |

+-------+-------+-------+

| . . . | . * . | . * . |

| . . . | . . . | . . . |

| . . . | . * . | . . * |

+-------+-------+-------+

| . . . | . . . | . . . |

| . . * | * . . | . . . |

| . . * | * . . | . . . |

+-------+-------+-------+

but only B36 are of Type C. Not enough. That leaves candidates 3 and 4.

For candidate 4, the unsolved cells are:

- Code: Select all
`+-------+-------+-------+`

| . . . | . . . | . . . |

| . . . | . . * | . . * |

| . . . | . * . | * . . |

+-------+-------+-------+

| . . . | . . . | * * . |

| . . . | * * . | . . . |

| . . . | . . . | . . . |

+-------+-------+-------+

| . . . | . . . | . . . |

| . . . | * . * | . * . |

| . . . | * . . | . * * |

+-------+-------+-------+

Here is the point! Blocks 2389 are the only ones of Type C for candidate 4. The strong links for a skyscraper or kite must start or end in these blocks! (And, they are there!)

For the unsolved cells in candidate 3:

- Code: Select all
`+-------+-------+-------+`

| . . . | . . * | . * . |

| . . . | . . . | . . . |

| . . . | . * . | * . . |

+-------+-------+-------+

| . . . | * * . | * * . |

| . . . | . . . | . . . |

| . . . | . * . | . . * |

+-------+-------+-------+

| . . . | . . . | . . . |

| . . . | * . * | . * . |

| . . . | * . . | . * * |

+-------+-------+-------+

All of these blocks are type C and, sure enough, the skyscrapers are lurking!

I use this technique to solve moderately hard puzzles without pencil marks. For any candidate, look for the type C blocks. If there are four, arrayed in a rectangle, look for strong links. If not, move on.

With a little bit of practice, you can do this without pencil marks. If you use pencil marks, this observation on "Type C" blocks will still help you. If you

are a computer programmer, this is probably of no use at all.

Keith