## Finding the possibility of single-candidate eliminations

Advanced methods and approaches for solving Sudoku puzzles

### Finding the possibility of single-candidate eliminations

Let me try to explain a technique I have used for years, and have not seen explained anywhere else. The question is, how do you see the possibility of a single-digit elimination? Most often, this means where could a skyscraper or kite (turbo fish) be lurking?

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
keith
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### Re: Finding the possibility of single-candidate eliminations

Kieth, as you have described it your method doesn't seem to be comprehensive. Would it catch this Turbot Fish with crossed pincers?
Code: Select all
`+-------+-------+-------+| . . 1'| . . . | . . . || . . / | . . . | . . 1 || 1"/ / | / / / | / / 1'|+-------+-------+-------+| . . / | . . . | . . . || . . / | . . . | . . . || . . / | . . . | . . . |+-------+-------+-------+| . . / | . . . | . . 1 || . . / | . . . | . . . || . 1 1"| . . . | . .-1 |+-------+-------+-------+`

Box 1 is the only one that must contain a 'C' pattern, but as shown, in boxes 3,7, & 9 the surviving candidates could be in the same row or column as the eliminated one. I appreciate that the chance of this happening is slight, but in Sudoku if it can happen then sooner or later it will happen.

DPB
.
David P Bird
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Location: Middle England

### Re: Finding the possibility of single-candidate eliminations

keith wrote: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
Those 2 propositions are too restrictive. So :

For N(>1)-Fish :
1. Replace strong link by constraint in proposition 1.
2. Replace four by four or six and rectangle by loop in proposition 2.
JC Van Hay

Posts: 719
Joined: 22 May 2010

### Re: Finding the possibility of single-candidate eliminations

David P Bird wrote:Kieth, as you have described it your method doesn't seem to be comprehensive. Would it catch this Turbot Fish with crossed pincers?
Code: Select all
`+-------+-------+-------+| . . 1'| . . . | . . . || . . / | . . . | . . 1 || 1"/ / | / / / | / / 1'|+-------+-------+-------+| . . / | . . . | . . . || . . / | . . . | . . . || . . / | . . . | . . . |+-------+-------+-------+| . . / | . . . | . . 1 || . . / | . . . | . . . || . 1 1"| . . . | . .-1 |+-------+-------+-------+`

Box 1 is the only one that must contain a 'C' pattern, but as shown, in boxes 3,7, & 9 the surviving candidates could be in the same row or column as the eliminated one. I appreciate that the chance of this happening is slight, but in Sudoku if it can happen then sooner or later it will happen.

DPB
.

David,

If I am reading your diagram correctly, you need to first clear the line - box interactions. The pattern in C9 is not valid unless there are other candidates in B3 and B9, making them Type C. (Similarly for R9 and B7.)

Keith
keith
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### Re: Finding the possibility of single-candidate eliminations

keith wrote:David,

If I am reading your diagram correctly, you need to first clear the line - box interactions. The pattern in C9 is not valid unless there are other candidates in B3 and B9, making them Type C. (Similarly for R9 and B7.)

Keith

Right, I understand where you're coming from now.

DPB
.
David P Bird
2010 Supporter

Posts: 1043
Joined: 16 September 2008
Location: Middle England

### Re: Finding the possibility of single-candidate eliminations

JC Van Hay wrote:
keith wrote: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
Those 2 propositions are too restrictive. So :

For N(>1)-Fish :
1. Replace strong link by constraint in proposition 1.
2. Replace four by four or six and rectangle by loop in proposition 2.

JC,

Thank you.

Do your two propositions cover all cases of empty rectangles and mutant or kraken fish?

I developed this technique as a way to find Turbot Fish when solving puzzles on paper, without overt pencil marks. I am interested in understanding if this rule applies generally to all one-digit eliminations.

Keith
keith
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### Re: Finding the possibility of single-candidate eliminations

Hi Keith, good to meet you again in another Sudoku forum!
Let's talk about puzzles and how to solve them ...

Your basic proposition gave me food for thought:
keith wrote:Proposition 1: Now, think about it: A useful strong link can only start or end in a Case C block.
Keith

Hm. In a perfect world, that could probably be true.

"perfect" and "true" in the sense that "useful" is narrowly understood: if we have sucessfully made all the lesser eliminations, then we've depleted some of the gold mines in the puzzle and new veins are hard to find. OK. I can understand that.

But being a pen&paper guy myself, it happens to me very often that while looking for x-wings or other single candidate eliminaions, I find that I missed a basic elimination. So, I seriously doubt the cost reduction of not looking for eliminations in certain places outweighs the potential benefit of discovering an oversight in my basics. On the other hand, you've used the method successfully for years.

Maybe there's something I don't understand in the exposition?

Example: I'm done with basics. No xy-wing to be found. Bummer. Look for x-wings and their kin, those wings and fish. Why NOT look at "type A" boxes? Two fives in a row, two fives in another row, same columns. Whoa! Great. What about this is not "useful" ?

Helmut
Nataraj

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Joined: 30 October 2015

### Re: Finding the possibility of single-candidate eliminations

Nataraj wrote:Hi Keith, good to meet you again in another Sudoku forum!
Let's talk about puzzles and how to solve them ...

Your basic proposition gave me food for thought:
keith wrote:Proposition 1: Now, think about it: A useful strong link can only start or end in a Case C block.
Keith

Hm. In a perfect world, that could probably be true.

"perfect" and "true" in the sense that "useful" is narrowly understood: if we have sucessfully made all the lesser eliminations, then we've depleted some of the gold mines in the puzzle and new veins are hard to find. OK. I can understand that.

But being a pen&paper guy myself, it happens to me very often that while looking for x-wings or other single candidate eliminaions, I find that I missed a basic elimination. So, I seriously doubt the cost reduction of not looking for eliminations in certain places outweighs the potential benefit of discovering an oversight in my basics. On the other hand, you've used the method successfully for years.

Maybe there's something I don't understand in the exposition?

Example: I'm done with basics. No xy-wing to be found. Bummer. Look for x-wings and their kin, those wings and fish. Why NOT look at "type A" boxes? Two fives in a row, two fives in another row, same columns. Whoa! Great. What about this is not "useful" ?

Helmut

Helmut,

I make (or consider) these diagrams quite early. Yes, they help with line / box interactions. My original post was though really about after the "basics" are done.

This is actually a pencil & paper method to find Turbot Fish (Kites and Skyscrapers), which don't seem to be in your diet! In my opinion, a kite or skyscraper is a mutant X-wing.

As JC van Hay posted, there is a generalization to n-fish.

Keith
keith
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### Re: Finding the possibility of single-candidate eliminations

Well I'm interested in Keith's idea as a quick-scan technique even if it only works most of the time. It seems to me that this principle can be summed up simply as:
'When scanning for possible Skyscrapers or Kites, ignore the boxes where the cells for the candidate you are looking at are all in one column or row, as they are unlikely to be useful.'
That's the way I read it, but I stand to be corrected. Now to Keith's example: In the very useful diagram highlighting the possible locations for candidate 4
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | . . * | . . * |
| . . . | . * . | * . . |
+-------+-------+-------+
| . . . | . . . | * * . |
| . . . | * * . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | * . * | . * . |
| . . . | * . . | . * * |
+-------+-------+-------+
I spotted the skyscraper (4c69) but it didn't do much that I could see except eliminate 4 as a candidate from r9c4 and r8c8.
However, in his corresponding diagram for candidate 3 the skyscraper (3R36) eliminates 3 from r4c7 = 4, which solves the puzzle.

Were there any Kites? I still have trouble spotting them.

Yogi
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### Re: Finding the possibility of single-candidate eliminations

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    4-3  5     |a34   6    8     ||-----------------+-----------------+-----------------|| 7    2    1     | 38   389  6     |b34   349  5     || 5    3    9     | 47   47   2     | 8    1    6     || 8    6    4     | 5   d39   1     | 2    7   c39    ||-----------------+-----------------+-----------------|| 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    |*-----------------------------------------------------*`

Here is a Kite that solves the above puzzle : (3) r3c7 = r4c7 - r6c9 = (3) r6c5 => - 3 r3c5; stte

When looking for kites look for a digit that has only one instance outside of the kite box in a row and a column.

Here the kite digit is 3, the box is Box 6, the row is Row 6 and the column is Column 7. The kite box can actually have up to 8 instances of the Kite digit and still work.

The one cell that can't have the Kite digit is at the intersection of the row and column in the kite box, r6c7 in this puzzle.

Here are some practice puzzles that all solve with basics plus a kite.

.56.7.9...4.62..8.28...........8....73....618..5....2...........7.5...49....487.1
.617....5842.95....5..6.4.........3..25........41...26..........8.....672.....349
32..479.6..6........5.9..3.....7.......9.1..5..28..7.........7.6.3..4....5..26143
.81.2............9..68...4...31..7....8.57916........45....6.9223...84...........

Leren
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### Re: Finding the possibility of single-candidate eliminations

Thanx. I will let you kow the results, probably by starting a new thread about Kites.

Yogi
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### Re: Finding the possibility of single-candidate eliminations

Nataraj wrote:Hi Keith, good to meet you again in another Sudoku forum!
Let's talk about puzzles and how to solve them ...

Your basic proposition gave me food for thought:
keith wrote:Proposition 1: Now, think about it: A useful strong link can only start or end in a Case C block.
Keith

Hm. In a perfect world, that could probably be true.

"perfect" and "true" in the sense that "useful" is narrowly understood: if we have sucessfully made all the lesser eliminations, then we've depleted some of the gold mines in the puzzle and new veins are hard to find. OK. I can understand that.

But being a pen&paper guy myself, it happens to me very often that while looking for x-wings or other single candidate eliminaions, I find that I missed a basic elimination. So, I seriously doubt the cost reduction of not looking for eliminations in certain places outweighs the potential benefit of discovering an oversight in my basics. On the other hand, you've used the method successfully for years.

Maybe there's something I don't understand in the exposition?

Example: I'm done with basics. No xy-wing to be found. Bummer. Look for x-wings and their kin, those wings and fish. Why NOT look at "type A" boxes? Two fives in a row, two fives in another row, same columns. Whoa! Great. What about this is not "useful" ?

Helmut

Nataraj,

Quite simply, Type A and Type B boxes cannot be bases or pincers in single-digit eliminations.

If the single-digit inference comes from a multi-digit pattern (like an XY-wing) my assertion does not apply.

Keith
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### Re: Finding the possibility of single-candidate eliminations

Here is a nice example of what JC Van Hay alluded to: There are 6 Type C blocks that lie in a swordfish pattern.

www.dailysudoku.com/sudoku/forums/viewtopic.php?p=41046#41046

Keith

JC Van Hay wrote:
keith wrote: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
Those 2 propositions are too restrictive. So :

For N(>1)-Fish :
1. Replace strong link by constraint in proposition 1.
2. Replace four by four or six and rectangle by loop in proposition 2.
keith
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Posts: 216
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### Re: Finding the possibility of single-candidate eliminations

I have now written these ideas into a thread

http://www.dailysudoku.com/sudoku/forum ... php?t=8180

which is actually more about the process of solving Sudoku rather than the tools and techniques.

Comments and criticisms will be gratefully received, in either this thread, or the one in the above link, or by pm on either site.

Keith
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