Sudoku Algorithms, Techniques & Strategies (Pointing Pairs, Pointing Triples, Box/Line Reduction, Intersection Removal Strategy, Killer Sudoku)

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This brief article is all about Sudoku Algorithms, Techniques & Strategies where you’ll learn Box/line Reduction Algorithm, Intersection Removal Strategy, Pointing Pairs, Pointing Triple’s, Killer Sudoku including types, variation’s and examples, I recommend you to give 10 minutes from your precious time to this article if you want to become a professional Sudoku Player.

Pointing pair and Box line reduction Sudoku

Intersection removal is a method for removing one candidate from a cell or a group of cells. Pointing pairs and box/line reduction are the two kinds of intersection elimination.

Box/Line Reduction

If a candidate only occurs two or three times in a row or column, and those candidates are all in the same box, you can utilize Box/Line Reduction. When this happens, you know the candidate must appear in that row or column since it can’t exist anywhere else, and it cannot appear anywhere else in that box.

Sudoku box line reduction

 

The red 4s in the illustration above is the only 4s in their row. Unfortunately, those four numbers also happen to be in the same box. Because they would prohibit a four from appearing in that row, the other 4s in the box is invalid. As a result, we may delete the other four numbers from the box. They’ve been highlighted in red.

Intersection Removal

We can delete a number from the intersection of another unit if it appears twice or three times in only one unit (any row, column, or box). Intersections may be divided into four categories:

  1. A pair or triple in a box can be detached from the remainder of the row if they are aligned on a row.
  2. If a pair or triple in a box is aligned on a column, the rest of the column can be eliminated.
  3. If all of the pairs or triples in a row are in the same box, then n can be detached from the remainder of the box.
  4. A Pair or Triple on a column can be detached from the remainder of the box.

Pointing Pairs, Pointing Triples

Sudoku pointing pair's and pointing triples

Pointing Pairs Example 1

There might be two or three occurrences of a specific number if you look at each box individually. If these numbers (as a pair or a triple) are aligned on a single row or column, we know that number MUST be on that line. As a result, it can be eliminated if the number appears elsewhere on the row or column than the box in which they are aligned: the pair or triple points to any number that can be deleted along the line.

On this difficult graded problem, there are two Pointing Pairs at the same time. For example, in inbox 3, the 3s in B7 and B9 are alone and aligned on the row. As a result, we may delete all of the 3s in Box 1 by glancing along the row. Similarly, the 2s in G4 and G5 refer to G2 down the row.

Sudoku poiting pairs and poiting triplets example 1

Pointing Pairs Example 2

This is a unique and challenging problem, but if we look at the entire board, we can see that I have highlighted a large cluster of Pointing Pairs. It is not required to spot everyone to advance the board, but there are so many notable instances worth considering. The eliminations are denoted with a yellow accent. You should be able to tell which Pointing Pair each elimination belongs to. A few are the result of prior removal.

Sudoku pointing pair's and pointing triplets example 2

From the Beginning or Pointing Triple: Load Example

A Pointing Triple was discovered in a difficult grade puzzle with several healthful and nutritious Pointing Pairs. Because all of the 3s can be found in G6, H6, and J6, but nowhere else in box 8, they indicate the column to another three that may be deleted.

Box Line Reduction

Sudoku Box Line Reduction

Load Example:

From the beginning, this method entails a thorough examination of rows and columns about the contents of boxes (3 x 3 squares). We can exclude numbers from the remainder of the box if we locate numbers in any row or column clustered together in only one box. Consider the following scenario:

This Sudoku has a lot of Pointing Pairs and Box/Line Reductions, so it’s worth starting from the beginning. Take, for example, row A. The only 2s left is in A4 and A5, indicating that the rest of the box should be checked. Thus, on row A, two must go someplace, and it will be in one of the two cells. 

Sudoku box line reduction example

Reduction of the box/line: Example 

There are two 4s alone in column 8 in the following stage of the same problem. This ensures that cell 4 is in either A8 or B8. We can acquire our next solution cell by removing the other 4s in box 3.

Sudoku box line reduction example

Triple BLR: Example

Combined two instances into this figure, and I hope it’s not too confusing. The one with the 6s comes right after the first one involving the 3s. So it’s a ‘triple’ version of Box/Line reduction in both cases.

The 3s occupy G1, H1, and J1 (shorthand = GHJ1) in column 1, all in box 7. Because the solution is pinned to column 1, the other three numbers in the box must be removed.

Column 2 has pinned the 6s as well. As you can see, there are additional 6s in column 3 (A3 and J3); therefore, removing the 6s in DEF3 is OK. You won’t run out of options.

If you enjoy Jigsaw Sudoku problems, you might be interested in reading the articles on Double Pointing Pairs and Double Line/Box Reduction, which expand on the concepts presented here but are only available in the Jigsaw version of the game.

Sudoku Algorithms – Box/Line Reduction In Row

sudoku algorithm box line reduction in row

 

The Hidden Single approach is comparable to the Box/Line Reduction technique. We’re seeking a number that shows as a viable candidate only in one cell for the whole box, row, or column in the Hidden Single approach. We’re seeking a number that occurs as a viable candidate in numerous cells but only in one box for the entire row or column in the Box/Line Reduction approach. Except for the cells that correspond to the selected row or column, we can eliminate this candidate from all cells in the box.

An example of Box/Line Reduction in a row can be seen on the left. Only inbox (1*1) does the number ‘1’ in cells C2 and C3 (marked yellow) appear in the list of probable candidates in row C. All cells inbox (1*1) except those in row C can have the number 1 deleted. Candidate ‘1’ is being removed from cells A2 and B3 (marked red).

Sudoku Algorithms – Box/Line Reduction In Column

An example of Box/Line Reduction in a column can be seen on the left. Only inbox (1*2) does the number ‘1’ in cells A4, and B4 (marked yellow) appear in the list of possible choices in column 4. All cells inbox (1*2) except those in column 4 can have the number 1 deleted. For example, candidate ‘1’ is being removed from cell B5, highlighted in red.

sudoku algorithm box line reduction in column

Basic-Intersection Removal

Removal of Intersections We can delete a number from the intersection of another unit if it appears twice or three times in only one unit (any row, column, or box). Intersections may be divided into four categories:

  1. A pair or triple in a box can be detached from the remainder of the row if they are aligned on a row.
  2. If a pair or triple in a box is aligned on a column, the rest of the column can be eliminated.
  3. If all of the pairs or triples in a row are in the same box, then n can be detached from the remainder of the box.
  4. A Pair or Triple on a column can be detached from the remainder of the box.

The first and second rules are also known as Pointing Pairs/Triples. Box/Line Reduction is another name for Rules 3 and 4.

Type 1 – Pairs/Triples Pointing Strategy (a.k.a. Intersection Removal) There might be two or three occurrences of a specific number if you look at each box individually. If these numbers (as a pair or a triple) are aligned on a single row or column, we know that number MUST be on that line. As a result, it can be eliminated if the number appears elsewhere on the row or column than the box in which they are aligned: the pair or triple points to any number that can be deleted along the line.

Sudoku Basic intersection removal

Take a look at the lower part of the puzzle board in the image above. In the middlebox, we’re looking at the number 7. It can only be found at A and B on the top row. In the left-hand box, the seven at Z can be eliminated. As a result of this revelation, the 7 in the last row must be in column 2 (where the current value is 5/7).

Type 2 – Reduction Strategy for Box Lines (a.k.a. Intersection Removal) This method entails a thorough examination of rows and columns about the contents of boxes (3 x 3 squares). We can exclude numbers from the remainder of the box if we locate numbers in any row or column clustered together in only one box.

Basic intersection removal in Sudoku

Line-Box Interaction

A method of problem-solving that makes use of the intersections of lines and boxes.

Intersection Removal and Line-Box Interaction are examples of aliases. To distinguish the two methods, the words pointing and claiming/box-line reduction are frequently employed.

This is a fundamental method of problem-solving. For example, we can eliminate the remaining candidates from the second home outside the junction when all candidates in a house are located inside the intersection with another house.

Type 1 (Pointing)

All of the contenders for the digit X are constrained to a single line (row or column) in a box. From the line segment that does not cross with this box, the excess candidates are deleted. This approach is also known as a Pointing Pair or Pointing Triple, depending on the number of potential candidates in the intersection.

Type 2 (Claiming or Box-Line Reduction)

A single box contains all of the choices for digit X on a line. The surplus candidates are removed from the section of the box where this line does not connect.

Type 2 (Claiming or Box-Line Reduction) example in sudoku

Example

The number 4 has a Locked Candidates Type 1 (Pointing) as highlighted in yellow. It is possible to remove the red candidates.

Killer Sudoku

Thanks to the cages, more variants of the locked candidate’s strategy may be found in Killer Sudoku. We’ll use two examples from Ruud’s Assassin 1 to demonstrate these concepts. First, consider the 17[3] cage in column 8 in the first example below.

Killer Sudoku

The 17[3] cage in column 8 has probable cage combinations of 2,7,8 or 3,6,8 or 4,5,8 or 4,6,7. In column 8, however, all cells with seven as a contender are cells in the 17[3] cage. As a result, the combinations 3,6,8 and 4,5,8 may be eliminated, leaving only the combinations 2,7,8, or 4,6,7. This means we can remove 3 and 5 from all of the cage’s cells. Consider the 9[3] cage in column 2 in the second example below.

Killer Sudoku

The 9[3] cage in column 2 has two potential cage combinations: 1,3,5 and 2,3,4. Because both combinations contain the number 3, one of the 9[3] cage cells must also contain the number 3. As a result, the number 3 may be removed from all other cells in column 2.

Pointing Pairs and Triples

When a candidate appears twice in a block and is aligned on the same row or column, this is a pointing pair. This means you know the candidate MUST appear in one of the two squares in the block, and you may rule out the candidate from any other cells on the row or column on which the candidate is aligned.

Sudoku Pointing Pairs and Triples

The red 2s appear twice in the block in the example above. This is because those two are also arranged in a row. Because we know the 2s must exist in that block, they can’t appear anywhere else in that row so that we may cross off any additional 2s.

Solving Techniques

Pointing Pairs


It’s not uncommon for the possibilities of boxes, rows, and columns to be exhausted, leaving only two cells empty (the technique also works with three remaining). There are other circumstances where the two remaining cells are in a row or column next to each other. In certain situations, this approach can be used.

Let’s take a look at the box in the upper center. Because [R8C4] contains a 4, it cannot be inserted in the fourth column.

Also, because the second row contains a 4, [R2C7], a four cannot be inserted there.

As a result, the cells in the top middle box where a four can be inserted are either [R1C6] or [R3C6].

As a result, a four may only be typed in column 5 in the center middlebox.

As you can see, you seek pairs or triple numbers in pointing pairs and rule out the chance that they belong in other cells.

Sudoku solving techniques pointing pair

Pointing Pairs-2

Only a six can be put in a row [2] in the upper center box. Similarly, a six may only be typed in a row [4] in the center middlebox.

Sudoku solving techniques pointing pairs 2

Because the upper-middle box now has a [6,6] pair, a six cannot be placed in the lower middle box’s columns 4 and 5.

As a result, a six can only be found in column 6 in the lower center box.

As you can see, this is a method of gradually reducing the number of candidates.              

Pointing Pair

The pointing pair solution approach is a strategy for solving problems at an intermediate level. This solution aims to limit the number of candidate lists of empty cells, which frequently reveal naked or concealed singles. There are two varieties of this problem-solving method that enable candidates to be eliminated from the row, column, or box.

Variations in Box Line Reduction

Variation 1:

Reducing row or column candidates

In the example below, the highlighted cells form a pointing pair. Therefore, the highlighted cells’ candidate “7” is not a candidate for any of the other cells in the highlighted cells’ box. This implies the “7” must be in one of the highlighted cells and can be removed from any of the other cells in the row’s candidate lists.

Sudoku Variations in Box Line Reduction(Sudoku Row or Column candidates)

Variation 2: 

Reducing box candidates

If a pair of empty cells share a candidate in the same row or column, that candidate can be deleted from the candidate list of all other cells in the box if it is not shared by any other cells in the row or column.

In the example below, the highlighted cells form a pointing pair. Thus, the highlighted cells’ candidate “7” is not a candidate of any other cell in the row to which the highlighted cells are “pointing.” This signifies that the “7” must be in one of the highlighted cells and that it can be removed from any of the other cells’ candidate lists.

Sudoku Variations in Box Line Reduction(Reducing box candidates)

Pointing Triple

The pointing triple solution approach is nothing more than a three-cell expansion of the pointing pair solving technique. Therefore, before looking at the example below, please read the description of the pointing pair approach above.

In the example below, the highlighted cells form a pointing triple. Thus, the highlighted cells’ candidate “6” is not a candidate for any of the other cells in the column to which they are “pointing.” This signifies that the “6” must be in one of the highlighted cells and that it can be removed from any of the other cells’ candidate lists.

Sudoku Pointing Triple

Sudoku Algorithms – Pointing Pairs In Column

Sudoku Algorithms - Pointing Pairs In Column

The Pointing Pairs approach is useful for removing candidates who are incompatible in the same row or column. If a number appears as a viable candidate in only one row or column in a box, it should not appear in any other box in that row or column. An example of the Pointing Pairs in column 1 is seen on the left. Only cells G1 and I1 (marked yellow) in the box (3*1) have the number ‘3’ in the list of eligible possibilities. Because the number ‘3’ is in column 1 outside the box (3*1), it may be deleted from cells B1 and C1 (marked red).

Sudoku Algorithms – Pointing Pairs In Row

Sudoku Algorithms - Pointing Pairs In the Row

An image on the left shows an example of the Row B Pointing Pairs. Only cells B1 and B3 (marked yellow) in the box (1*1) have the number ‘4’ in the list of eligible possibilities. Because the numbers ‘4’ are in row B outside the box (1*1), we can delete them from cells B4, B5, and B6 (marked red).

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