Sudoku Solving Techniques

Here is a list of some Sudoku solving techniques. The Simplified Lonely Loner and Simplified Swinging Single techniques only use knowledge of the digits already entered, whereas the rest take help from the indirect knowledge derived from the digits already entered. But depending on the difficulty level of the techniques it might still be possible to apply them without actually writing down the candidate lists.

If you apply the Simplified Lonely Loner and Simplified Swinging Single techniques repeatedly, you will be able to solve all puzzles generated by the Easy and Basic Techniques buttons.

For the puzzled puzzler it might be comforting to know that the somewhat curiously chosen names of these techniques are explained further down on this page.

Simplified Lonely Loner

For each row, column and box, see if there is only one empty cell in that row, column or box. If so, all but one digit can be excluded from that cell, so you can enter the remaining digit in it.

In other words, if only one digit can go in a cell, enter it!

Simplified Swinging Single

For each digit, do the following: Check in each row, column and box to see if the existing occurrences of the digit elsewhere in the puzzle exclude all but one of the empty cells in that row, column, or block. If so, enter the digit in that one cell.

In other words, if there is only one possibile cell for a digit (in a row column or box), enter it!

Lonely Loner

A Lonely Loner is a candidate list which consists of one single digit.

When a Lonely Loner has been found in a certain row, column and box, the digit can be removed from the candidate lists of all other cells in that row, column and box. And of course entered into the cell in question.

Swinging Single

If a digit only occurs in one candidate list in a row, column or box, this digit is a Swinging Single.

If a Swinging Single has been found, all other digits can be removed from that candidate list. And the digit can be entered into the cell in question.

Kolumn Killer

We have found a Kolumn Killer if, inside one box, a digit only occurs in the candidate lists of cells in one column (or row).

Since the digit defintely will occur in that column (or row) in the final solution, we know that that digit cannot occur in that column (or row) in any other box along the column (or row), and can therefore be removed from the candidate lists of those cells.

Example: Let us consider the center box of a Sudoku puzzle. Say digit 5 only can occur in the second row of this box. Then we can remove 5 from the candidate lists of all other cells in that row.

Boxing Box

If, in either a row or a column, a certain digit only occurs in the candidate lists of cells in the same box, that digit can be removed from all other candidate lists in that box.

Example: Let us consider the case when the digit 5 cannot occur in any of the last six cells in the first row. Then we know that the 5 has to go in one of the first three cells of that row. Since those three cells all are in the same box, we know that the 5 cannot be part of the candidate lists of any of the cells in the second and third row of that box.

Cozy Couple

A Cozy Couple (a.k.a. Naked Pair) is a pair of digits which occur alone in two cells in the same row, column or box.

When a Cozy Couple is found in a row, column or box, the digits in that Cozy Couple can therefore be deleted from the candidate lists of all other cells in the row, column or box.

Example: Say for example that the first and the second cell in a row have only two possibilities: digit 4 and digit 5. Since they are the only digits which can go in those two cells, we know that the 5 either has to go in the first or the second cell. (If it did not, the 4 would be left as the only option for both cells, an outcome which would violate the rules of Sudoku.) And similarly we know that the 4 has to be entered in either of those cells as well. Although we do not know in which one, this information is still helpful since it means that those two digits cannot go in any other cells in that row.

Partying Pair

A Partying Pair (a.k.a. Hidden Pair) are any two digits which

• only occur in the candidate lists of two cells in a row, column or box, and
• occur in the candidate lists of the same two cells.

When a Partying Pair has been found, all other digits can be removed from the candidate lists in those two cells.

Example: Let us assume that the digits 1, 2 and 3 are the members of the candidate list in the last cell in the last column, and that 1, 2, 3, 4, and 5 are the digits in the candidate list of the first cell in the last column. Let us also assume that 2 and 3 do not occur in any candidate lists elsewhere in that column.

Since we know that the 2 either must go in the last or the first cell of the last column, and that the 3 also must go in one of those two cells, there will be no possiblility for the 1, 4 and 5 to be entered in either of those cells. We can therefore remove the 1, 4 and 5 from the candidate list in the first cell of the last column, and the 1 from the candidate list of the last cell in that column.

Gathering Gang

A Gathering Gang is the extension of a Cozy Couple, in the same way as a Cozy Couple is the extension of a Lonely Loner. But in this case, we are looking for a group of size 3 or larger.

A Gathering Gang of size n is a set of n digits (n > 2) which occur alone in the candidate lists of n cells in the same row, column or box. Note that all n digits do not have to be present in all of the n candidate lists.

When a Gathering Gang has been found, the member digits can be deleted from all other candidate lists in that row, cell or box.

Example: Assume we have a box with the following candidate lists in three of its nine cells: {1, 2, 3}, {1, 2, 3}, and {1, 3}. The digits 1, 2 and 3 can then be deleted from the candidate lists in all other cells in that box, since they together constitute a Gathering Gang of size three.

Groovy Group

A Groovy Group is the extension of a Partying Pair, in the same way as a Partying Pair is the extension of a Swinging Single. But in this case, we are looking for a group of size 3 or larger.

A Groovy Group of size n is a set of n digits (n > 2) which occur in the candidate lists of only n cells in a row, column or box. Note that all n digits do not have to be present in all of the n candidate lists.

When a Groovy Group has been found, all digits other than the member digits can be deleted from the candidate lists of those n cells in that row, cell or box.

Example: Assume we have a box with the follwoing three candidate lists in three of its cells: {1, 2, 3, 4, 5}, {4, 5, 6, 7, 8, 9}, and {3, 4}.

Assuming that the digits 3, 4 and 5 do not occur in any other candidate lists in that box, the digits 3, 4, and 5, then constitute a Groovy Group of size 3. Digits 1 and 2, and 6, 7, 8, and 9 can then be removed from the candidate lists of these three cells, since the digits 3, 4, and 5 all will go there.

X-Wing

The X-Wing technique gets its name from the resemblance of an X, flapping its wings.

An X-Wing is four occurrences of the same digit in the candidate lists of four cells. The location of these cells are determined by the following requirements:

• The four cells must be the corners of an imaginary rectangle. This means that two of the four cells belong to the same row and that the other two belong to a second row. And that similarly the four cells also only occupy a total of two columns.
• There can be no occurrences of the digit in the candidate lists of the other cells in the two rows, to which the four cells belong.

Once an X-Wing has been found, the digit can be removed from all candidate lists of the other cells in the two columns which the X-Wing occupy.

In the above description of an X-Wing the word "row" can be replaced by "column" and vice versa. The digit can hence be removed from all other cells in the two rows if there were no other occurrences of the digit in the two columns.

Example: Let us assume that the first and second cell in rows three and four, all contain the digit 1 in their candidate lists. These four occurrences of the 1 fulfill the first requirement of an X-Wing. Let us also assume that there are no other candidate lists in the first two columns which contain the digit 1. This would mean that also the second requirement is fulfilled, and we can therefore remove the digit 1 from the candidate lists of all other cells in the third and fourth row.

Swordfish

A Swordfish is an extended version of an X-Wing. Instead of the digits being located in the corners of a rectangle, the digits in a Swordfish are located on a 3x3 grid.

A Swordfish contains between 6 and 9 occurrences of the same digit in the candidate lists of equally many cells. The location of these cells are determined by the following requirements:

• The cells must be located in an imaginary 3x3 grid, with at least two cells in each row and column.
• There can be no occurrences of the digit in the candidate lists of the other cells located on the same rows as the 3x3 grid.

Once a Swordfish has been found, the digit can be removed from all candidate lists of the other cells in the three columns of which the Swordfish consists.

In the above description of a Swordfish the word "row" can be replaced by "column" and vice versa. The digit can hence be removed from all cells on the three rows concerned, if there were no other occurrences of the digit in the three columns.

Example: Let us assume that the first, second and fourth cell in rows three, four and five all contain the digit 1 in their candidate lists. These nine occurrences of the 1 fulfill the first requirement of a Swordfish. Let us also assume that there are no other candidate lists in columns one, two and four, which contain the digit 1. This would mean that also the second requirement is fulfilled, and we can therefore remove the digit 1 from the candidate lists of all other cells in the third, fourth and fifth row.

The Names of the Techniques

We have tried to name the techniques so that they will be easier to remember, but the logic behind the names might still be lost without an explanation.

The second part of the names Lonely Loner, Swinging Single, Cozy Couple, Partying Pair, Gathering Gang and Groovy Group reflects the group size (one, two and three or more). The first part of the names Lonely Loner, Cozy Couple and Gathering Gang are chosen to try to reflect that the members of the group are alone (or rather that the group is alone in the cells). The first part of the names Swinging Single, Partying Pair and Groovy Group, on the other hand, are chosen to try to reflect that the members of the group are accompanied by many other digits in the cells.

Tip for Updating Candidate Lists

Whenever you have entered a digit in a cell, you need to remove that digit from all candidate lists in the corresponding row, column and box. Doing so might allow you to apply any of the above techniques again. It is however worth noting that the removal of a digit from a candidate list in a cell, only can create Lonely Loners, Cozy Couples and Gathering Gang where that digit is NOT included. However, if a Swinging Single, Partying Pair, Groovy Group, Kolumn Killer or Boxing Box is created, the removed digit will be involved. Knowing this will reduce the time it takes to check for new moves when updating the candidate lists.