## New pattern: the Turbot Fish

Advanced methods and approaches for solving Sudoku puzzles

### New pattern: the Turbot Fish

While studying double forcing chains, I noticed a common pattern that can be used in thousands of problems.

It is something between X-Wings and Swordfish.

X-Wings requires the possibilities for a number to be in 4 cells, while swordfish requires them to be in 6 to 9 cells.

The new pattern instead uses 5 cells.

The 5 cells must be at the vertices of a 5-sided polygon; it's easy to see that for this to happen, two sides must be along a row, two sides along a column, and the fifth side must have the two vertices in the same box.
The possible layouts are therefore like these:

Code: Select all
` . . . | . . . | . . .    . . . | . . . | . . .    . . . | . . . | . . . . . . | *---------* .    . . *-------------* .    . . *-------------* . . . . | | . . | . | .    . . | | . . . | . | .    . . | | . . . | . | .---------|---------|---  -----|-------------|---  -----|-------------|--- . . . | *\. . | . | .    . . | | . . . | . | .    . . *-----------* | . . . . | . \ . | . | .    . . | | . . . | . | .    . . . | . . . | .\| . . . . | . .\*-----* .    . . | | . . . | . | .    . . . | . . . | . * .-----------------------  -----|-------------|---  ----------------------- . . . | . . . | . . .    *\--|-------------* .    . . . | . . . | . . . . . . | . . . | . . .    . \ | | . . . | . . .    . . . | . . . | . . . . . . | . . . | . . .    . .\* | . . . | . . .    . . . | . . . | . . .`

Due to the second layout, I've called this pattern "Turbot Fish".

We'll call a side "strong" if the two vertices it connects are the only two places where the number can be placed in the row/column/box defined by the side; "weak" if the number can be placed somewhere else.

To have a Turbot Fish, you need four or three strong sides (in any position), or two non consecutive ones.

Let's see what happens with four strong sides, for example let's say the diagonal side is the weak one (the reasoning is the same for any side).

Code: Select all
` . . . | . . . | . . . . . . | B---------C . . . . | | . . | . | .---------|---------|---- . . . | A . . | . | . . . . | . . . | . | . . . . | . . E-----D .------------------------ . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . `

If our number is in A, then it must also be in C and in E. But A and E are in the same box, so it cannot be in both of them at the same time; therefore we have a contradiction.
The number must therefore be in B and in D. So we can put it there and continue with the puzzle.

This also shows that a Turbot Fish cannot have five strong sides, because that would lead to an unavoidable contradiction.

If there are three strong sides, the two weak sides can be either consecutive, or separated by a strong side. Let's see both cases.

Code: Select all
` . . . | . . . | . . . . . . | B---------C . . . . | . . . | . | .-------------------|---- . . . | A\. . | . | . . . . | . \ . | . | . . . . | . .\E | . D .------------------------ . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . `

Here the two weak sides are separated.
We can say that our number must be either in A or in E.
If it is in A, then it's not in B, so it must be in C.
If it is in E, then it's not in D, so it must be in C.
In either case, our number is in C. So we can put it there and continue with the puzzle.

Code: Select all
` . . . | . . . | . . . . . . | B---------C . . . . | | . . | . | .---------|---------|---- . . . | A . . | . | . . . . | . . . | . | . . . . | . . E | . D .------------------------ . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . `

Here the two weak sides are consecutive.
The only thing we can say in this case is that out number is either in A and C or in B and D. In either case, it cannot be in E. Therefore we can remove the number from the possibilities for E, and continue with the puzzle.

Finally, let's see what happens with two non consecutive strong sides.

Code: Select all
` . . . | . . . | . . . . . . | B . . | . C . . . . | | . . | . . .---------|-------------- . . . | A . . | . . . . . . | . . . | . . . . . . | . . E-----D .------------------------ . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . `

Our number cannot be in both A and E at the same time. Therefore it must be in at least one of B and D. So it cannot be in C. Therefore we can remove the number from the possibilities for C, and continue with the puzzle.

Well, that's all. The description was a bit long, but I think this pattern is quite neat, and easy to understand.

Here are a few problems that can be solved using this pattern.

Code: Select all
`..4.9.8.6.316.....2.............5.....6.1.9.....4.............7.....742.8.9.6.3....5.9.8.6.316.....2.............5.....9.1.3.....4.............7.....742.8.6.3.9....6.2.7.5.396.....8.............5.....2.1.6.....4.............4.....732.7.8.9.1....6.2.1.5.396.....7.............5.....2.1.6.....4.............7.....732.8.1.6.9..`
Last edited by Nick70 on Sat Jul 16, 2005 6:08 pm, edited 2 times in total.
Nick70

Posts: 156
Joined: 16 June 2005

### Re: New pattern: the Turbot Fish

Nick70 wrote:While studying double forcing chains, I noticed a common pattern that can be used in thousands of problems.

I'm not sure that this different to simple coloring. I gather each cell in your pentagon has a conjugate relationship with at least other cell forming a chain. Applying simple coloring will reveal that one color will appear in the same group twice, indicating that that color is the false color.
Have I misunderstood something?

Edit: You have me stumped on your third example, so perhaps I am missing something???
angusj

Posts: 306
Joined: 12 June 2005

### Re: New pattern: the Turbot Fish

angusj wrote:I'm not sure that this different to simple coloring.

It is a special case of coloring, or of double forcing chains, just like x-wing is. It seemed so common that it deserved its own name: it is a lot more common than swordfish or even x-wing. In a library of 30,000 problems, I have less than 100 x-wing examples and almost 1500 turbots.

angusj wrote:I gather each cell in your pentagon has a conjugate relationship with at least other cell forming a chain.

No, as explained in the description you can have two consecutive weak sides, which leave a cell "dangling", with no conjugate relationship with the others.

angusj wrote:Applying simple coloring will reveal that one color will appear in the same group twice, indicating that that color is the false color.

A turbot with 1 weak side can be solved using simple coloring with one color pair by finding an immediate contradiction.

A turbot with 2 non consecutive weak sides can be solved using simple coloring with two color pairs by finding a double exclusion.

A turbot with 2 consecutive weak sides can be solved using simple coloring with one color pair by finding a double excluding pair on an uncolored cell.

So turbot is also nice because it uses all techniques that can be applied with simple coloring.
Nick70

Posts: 156
Joined: 16 June 2005

### Re: New pattern: the Turbot Fish

Nick70 wrote:No, as explained in the description you can have two consecutive weak sides, which leave a cell "dangling", with no conjugate relationship with the others.

Code: Select all
`[4]    [1]    [6]    [39]   [2]    [389]  [7]    [389]  [5]    [5]    [3]    [9]    [6]    [7]    [148]  [24]   [18]   [12]   [8]    [2]    [7]    [15]   [35]   [49]   [49]   [6]    [13]   [369]  [78]   [4]    [239]  [368]  [5]    [29]   [139]  [1237] [39]   [5]    [2]    [7]    [1]    [39]   [6]    [4]    [8]    [369]  [78]   [1]    [4]    [368]  [2369] [5]    [39]   [237]  [2]    [9]    [3]    [15]   [56]   [16]   [8]    [7]    [4]    [1]    [6]    [5]    [8]    [4]    [7]    [3]    [2]    [9]    [7]    [4]    [8]    [23]   [9]    [23]   [1]    [5]    [6]`

This is as far as I can get with your 3rd example (using a single pair of colors) but can't see where to apply your Turbot Fish.

However, I can use "double colorings" on the 9s to progress to the end (ie using what I think you called "double exclusion" here ).
angusj

Posts: 306
Joined: 12 June 2005

### Re: New pattern: the Turbot Fish

angusj wrote:However, I can use "double colorings" on the 9s to progress to the end (ie using what I think you called "double exclusion" here ).

And that's indeed where the Turbot is.

(3,6)-(3,7)-(4,7) are two consecutive strong sides. (1,4)-(4,4) is the third strong side that completes the Turbot. Therefore you can place 9 in (3,7).
Nick70

Posts: 156
Joined: 16 June 2005

### Re: New pattern: the Turbot Fish

Nick70 wrote:And that's indeed where the Turbot is.

OK. Then, isn't Turbot really just another way of looking at or applying colorings? (I do prefer your former definition (double exclusion) as it can be applied more widely and I believe is easier to see.)
angusj

Posts: 306
Joined: 12 June 2005

### Re: New pattern: the Turbot Fish

angusj wrote:OK. Then, isn't Turbot really just another way of looking at or applying colorings?

Yes, as I said above - just like X-Wing and even naked pairs and hidden pairs are are applications of colorings.

It's just that it's a common pattern and I thought it was good to have it identified. But it doesn't allow to solve any problem that simple coloring doesn't.

It shouldn't be harder to see than a swordfish - after all, you need (at least) one cell less.
Nick70

Posts: 156
Joined: 16 June 2005

### Re: New pattern: the Turbot Fish

Nick70 wrote:It's just that it's a common pattern and I thought it was good to have it identified.

OK, I've been having a fairly close look at puzzles requiring mutli-color (or double exclusion) solutions and I'm happy to concede that your Turbot Fish pattern is more common than I originally thought. It seems to account for very roughly half of the steps requiring the more general mutli-color technique. Sorry if I seemed less than enthusiastic before .
angusj

Posts: 306
Joined: 12 June 2005

### Re: New pattern: the Turbot Fish

angusj wrote:OK, I've been having a fairly close look at puzzles requiring mutli-color (or double exclusion) solutions and I'm happy to concede that your Turbot Fish pattern is more common than I originally thought.

Yes it's very common, even this is a Turbot

I don't have precise figures because I just discovered that my simple coloring implementation wasn't detecting double exclusions if the cell to exclude wasn't itself colored. I'll fix it and see what happens.

For now I can say that of about 10000 problems that cannot be solved with standard techniques up to quadruples/jellyfish, about 1500 can be solved with turbot.
Nick70

Posts: 156
Joined: 16 June 2005

I have just realized that the Turbot also works if there are only two strong sides, provided that they are not consecutive. I have edited the post at the top of the thread to reflect that.

It is probably also the case where the exclusion is less obvious when looked at with coloring.
Nick70

Posts: 156
Joined: 16 June 2005

### Re: New pattern: the Turbot Fish

Nick70 wrote:Here are a few problems that can be solved using this pattern.

Code: Select all
`..4.9.8.6.316.....2.............5.....6.1.9.....4.............7.....742.8.9.6.3....5.9.8.6.316.....2.............5.....9.1.3.....4.............7.....742.8.6.3.9....6.2.7.5.396.....8.............5.....2.1.6.....4.............4.....732.7.8.9.1....6.2.1.5.396.....7.............5.....2.1.6.....4.............7.....732.8.1.6.9..`

Could you please take these samples up to the point where the Turbot Fish pattern applies and tell which candidate will be solved by the pattern? I want to make sure I fully understand and use this for future puzzles.

Thanks,

Kerry
zvalue

Posts: 4
Joined: 13 September 2005