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The "naked single" solving technique also known as "singleton" or "lone number" is one of the simplest Sudoku solving techniques. Using this technique the candidate values of an empty cell are determined by examining the values of filled cells in the row, column and box to which the cell belongs. If the empty cell has just one single candidate value then this must be the value of the cell. In the example below, if you examine the values of the filled cells in the same row, column and box as the highlighted cell you can see that the highlighted cell has only a single candidate value since all other possible candidate values are already used. To illustrate this more clearly the same example with candidate values in empty cells is shown below, as you can see the highlighted cell has a single candidate value. The "naked pair" solving technique is an intermediate solving technique. In this technique the Sudoku is scanned for a pair of cells in a row, column or box containing only the same two candidates.

Using this technique the candidate values of an empty cell are determined by examining the values of filled cells in the row, column and box to which the cell belongs.

If the empty cell has just one single candidate value then this must be the value of the cell.

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In the example below, if you examine the values of the filled cells in the same row, column and box as the highlighted cell you can see that the highlighted cell has only a single candidate value since all other possible candidate values are already used.

To illustrate this more clearly the same example with candidate values in empty cells is shown below, as you can see the highlighted cell has a single candidate value.

The "naked pair" solving technique is an intermediate solving technique. In this technique the Sudoku is scanned for a pair of cells in a row, column or box containing only the same two candidates. Since these candidates must go in these cells, they can therefore be removed from the candidate lists of all other unsolved cells in that row, column or box. Reducing candidate lists may reveal a hidden or naked single in another unsolved cell, generally however the technique is a step to solving the next cell.

The highlighted cells in the example below show a naked pair containing the candidate values 2,3 in the third column.

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Since this is a naked pair the candidate values can be removed from all other unsolved cells in the column. This reveals a second naked pair containing the candidate values 7,9 in the same column. Reducing the candidate lists of other unsolved cells in the same box reveals a naked single.

The "naked triple" solving technique is similar to the naked pair solving technique described above. In a naked triple, three cells in a row, column or block contain some combination of the same three candidates. Each individual cell in the naked triple does not have to contain all three candidates however.

In fact it is perfectly legal for each individual cell to have only two of the three candidates.

A Naked Quad occurs when four cells in a group contain no candidates other that the same four candidates. In the example on the right, the candidates 2, 5, 7 & 9 in the 3 left most cells and bottom middle cell of a box form a Naked Quad. Therefore candidates . Naked Quad A Naked Quad occurs when exactly four digits are candidates in four cells of a house. You can eliminate these digits from the other cells in this house. In the sudoku diagram below, in the third row, only the four digits 3, 4, 6 and 8 can be placed in the R3C1, R3C2, R3C3 and R3C9 cells. Naked Quad. Naked quads are rare, but they can occur. They can be difficult to find, however, unless you specifically look for them. In the example above, the numbers in red form a naked quad. The four squares that contain the red numbers only contain numbers 2, 4, 8, and 9. That means that those four squares MUST contain those four numbers.

The highlighted cells in the example below show a naked triple containing the candidate values 2,4,7 in the centre box. These candidate values can be removed from all other unsolved cells in the box. This reveals a naked pair containing the candidate values 3,5 in the central row of the same box. Reducing the candidate lists of other unsolved cells in the same row reveals a naked single in that row. Start with the assumption it can have any digit or value between 1 and 9, and then remove all values which have already been assigned to other cells in its respective row, column and 3x3 box.

This leaves each blank cell with a list of candidates. Repeat the following logical steps until the puzzle is solved.

Only progress to more difficult steps when simpler steps neither reveal new values nor exclude candidates from blank cells. Singles: Any cells which have only one candidate can safely be assigned that value.

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It is very important whenever a value is assigned to a cell, that this value is also excluded as a candidate from all other blank cells sharing the same row, column and box. Programs like Simple Sudoku will do this laborious step automatically for you too.

Naked Quad

Hidden Singles: Very frequently, there is only one candidate for a given row, column or box, but it is hidden among other candidates. In the example on the right, the candidate 6 is only found in the middle right cell of the 3x3 box.

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Since every box must have a 6, this cell must be that 6. While the two steps above are the only ones which will directly assign a cell value, they will only solve the simplest puzzles.

That's fortunate, otherwise Sudoku wouldn't be as popular as it is today. The following steps in increasing complexity will reduce the number of candidates in blank cells so, sooner or later, a 'single' candidate or 'hidden single' candidate will appear.

Locked Candidates 1: Sometimes a candidate within a box is restricted to one row or column.

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Since one of these cells must contain that specific candidate, the candidate can safely be excluded from the remaining cells in that row or column outside of the box. In the example below, the right box only has candidate 2's in its bottom row.

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Since, one of those cells must be a 2, no cells in that row outside that box can be a 2. Therefore 2 can be excluded as a candidate from the highlighted cells. Locked Candidates 2: Sometimes a candidate within a row or column is restricted to one box.

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Since one of these cells must contain that specific candidate, the candidate can safely be excluded from the remaining cells in the box. In the example on the right, the left column has candidate 9's only in the middle box. Therefore, since one of these cells must be a 9 otherwise the column would be without a 99's can safely be excluded from all cells in this middle box except those in the left column.

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Naked Pairs: If two cells in a group contain an identical pair of candidates and only those two candidates, then no other cells in that group could be those values. These 2 candidates can be excluded from other cells in the group. A Naked Triple occurs when three cells in a group contain no candidates other that the same three candidates.

Naked Pairs, Naked Triples, Naked Quads Strategy

The cells which make up a Naked Triple don't have to contain every candidate of the triple. If these candidates are found in other cells in the group they can be excluded. A Naked Quad occurs when four cells in a group contain no candidates other that the same four candidates. Hidden Pairs: If two cells in a group contain a pair of candidates hidden amongst other candidates that are not found in any other cells in that group, then other candidates in those two cells can be excluded safely.

Hidden Triples: If three candidates are restricted to three cells in a given group, then all other candidates in those three cells can be excluded.

Naked Quad. A Naked Quad is a Naked Subset of size 4. It is formed by 4 cells that have candidates for only 4 digits and are collocated in the same house. Example. The following example shows a Naked Quad in the middle box. Naked Quads. A Naked Quad is rarer, especially in its full form but still useful if they can be spotted. The same logic from Naked Triples applies to combinations of numbers. This list is long but it starts And can be a tortuous foursome like this (12) (23) (34) (14) Here's an . Naked Quad. The "naked quad" solving technique is similar to the naked triple solving technique described above. In a naked quad, four cells in a row, column or box contain some combination of the same four candidates only. The highlighted cells in the example below show a naked quad containing the candidate values (1,7,8,9).

In the example below, the candidates 3, 6 and 7 are found only in column four, six and seven. Therefore, all other candidates can be excluded from those three cells. Hidden triples are generally extremely hard to spot but fortunately they're rarely required to solve a puzzle. Hidden Quads: If four candidates are restricted to four cells in a given group, then all other candidates in those four cells can be excluded. Hidden Quads are very rare, which is fortunate since they're almost impossible to spot even when you know they're there.

Try and spot the Hidden Quad in the row below.

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Move your mouse over the image to reveal the answer. The following techniques are no more difficult than those above but requires observation as to how specific candidates relate to each other in particular patterns beyond any given row, column or box. If a value has only 2 possible locations in a given row ie it has a candidate in only 2 cells in that rowthen it must be assigned to one of these 2 cells.

Therefore, it's not possible for any other cells in these two columns to contain candidate C. This same logic applies when a puzzle that has two columns where candidate C is restricted to exactly the same two rows.

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This will be understood more easily by examining the example below. Filtering has been added so only candidate 6's are visible.

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Cells which contain the filtered candidate are automatically highlighted pale green in Simple Sudoku unless subsequently recolored - which is the case for the blue cells this example.

The cells marked with a blue highlight form an "X-Wing" since rows one and nine both have only two cells with candidate 6's and these two cells share the same two columns.

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Therefore, other candidate 6's in columns six and nine highlighted yellow can safely be removed.

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