Ten Sudoku techniques to help you solve faster

You don't have to follow the rules to play Sudoku but if you want to be good and solve the difficult puzzles, Thang Bom recommends you experiment with different techniques below. Find the ones that work the best for you.

1. Naked Single
This technique is also known as "sole candidate".

It is often the case that a cell can only possibly take a single value, when the contents of the other cells in the same row, column and block are considered. This is when, between them, the row, column and block use eight different digits, leaving only a single digit available for the cell.

For example, in the partial puzzle below, the marked cell can only be a 6. All other digits are excluded by the other contents of the row, column and block.

2. Hidden Single
If a cell is the only one in a row, column or block that can take a particular value, then it must have that value. This is because all rows, columns and blocks, must contain each of the digits 1 to 9.

For example, in the partial puzzle below, the marked cell is the only one in block five that can hold a 2, and so it must hold a 2.

With the notable exception of forcing chains, the remaining techniques are all about reducing the number of candidates for cells. The aim being to reduce the candidates to such an extent that the first two techniques can be used.

3. Block and Column / Row Interactions
Sometimes, when you examine a block, you can determine that a certain number must be in a specific row or column, even though you cannot determine exactly which cell in that row or column. This is enough information to remove that number from the candidate list for other cells in the same row or column, but outside the block.

For example, in the partial puzzle below, the 7 in block one can only occur in column two. This means we can eliminate 7 from the candidates for the marked cells.

4. Block / Block Interactions
Firstly, If a number appears as candidates for only two cells in two different blocks, but both cells are in the same column or row, it is possible to remove that number as a candidate for other cells in that column or row.

For example, in the partial puzzle below, the cells marked with * are the only cells in blocks two and five that can contain a 3. This means the 3 in column four must be in block two or five, as must the 3 in column five. As there can be no other 3s in columns four or five, 3 can be eliminated as a candidate for the cells in these columns for block eight.

Secondly, in the example below, the cells marked with * are the only cells in blocks four and six that can contain a 2. This means that 2 can be eliminated from the candidates for rows four and six in block five.

5. Naked Subset
This technique is known as "naked pair" if two candidates are involved, "naked triplet" if three, or "naked quad" if four. It is sometimes also called "disjoint subset".

If two cells in the same row, column or block have only the same two candidates, then those candidates can be removed from the candidates of the other cells in that row, column or block. This is because one of the cells must hold one of the candidates, and the other cell must hold the other candidate - so neither can go in any of the other cells.

This technique can be applied to more than two cells at once, but in all cases, the number of cells must be the same as the number of different candidates.
For example, consider a row that has the candidates:
{1, 7}, {6, 7, 9}, {1, 6, 7, 9}, {1, 7}, {1, 4, 7, 6}, {2, 3, 6, 7}, {3, 4, 6, 8, 9}, {2, 3, 4, 6, 8}, {5}

(The single {5} indicates that this cell already holds the value 5.) You can see that there are two cells that have the same two candidates 1 and 7. One of these cells must hold the 1, and the other cell must hold the 7, although we don't know which is which. So 1 and 7 can be removed from the candidates for the other cells. This reduces the candidates to:
{1, 7}, {6, 9}, {6, 9}, {1, 7}, {4, 6}, {2, 3, 6}, {3, 4, 6, 8, 9}, {2, 3, 4, 6, 8}, {5}

So now there are two cells that have 6 and 9 as the only candidates. Repeating the process for these numbers leaves:
{1, 7}, {6, 9}, {6, 9}, {1, 7}, {4}, {2, 3}, {3, 4, 8}, {2, 3, 4, 8}, {5}

Now we have a cell with a single candidate - i.e. we have reduced the candidates to the extent that we have determined the only value that can possibly go into this cell.

6. Hidden Subset
This technique is known as "hidden pair" if two candidates are involved, "hidden triplet" if three, or "hidden quad" if four. It is sometimes also called "unique subset".

This technique is very similar to naked subsets, but instead of affecting other cells with the same row, column or block, candidates are eliminated from the cells that hold the subset. If there are N cells, with N candidates between them that don't appear elsewhere in the same row, column or block, then any other candidates for those cells can be eliminated.

For example, consider a block that has the following candidates:
{4, 5, 6, 9}, {4, 9}, {5, 6, 9}, {2, 4}, {1, 2, 3, 4, 7}, {1, 2, 3, 7}, {2, 5, 6}, {1, 2, 7}, {8}

(The single {8} indicates that this cell already holds the value 8.) You can see that there are only three cells that have any of the candidates 1, 3 or 7. (These cells have other candidates too, but they're the ones that we can eliminate.) Three candidates with only three possible cells between them means that one of the candidates must be in each of the cells. So, obviously, these three cells cannot hold any other value, meaning we can eliminate any other candidates for these cells.

In this example, we're left with:
{4, 5, 6, 9}, {4, 9}, {5, 6, 9}, {2, 4}, {1, 3, 7}, {1, 3, 7}, {2, 5, 6}, {1, 7}, {8}

Naked subsets and hidden subsets are related - I usually describe them as being opposite sides of the same coin. If a naked subset is present, then so is a hidden one, although it may be longer and so harder to spot. The opposite is also true, if a hidden subset is present, so is a naked one. They obey the following relationship:

NumberOfDigitsInNakedSubset + NumberOfDigitsInHiddenSubset + NumberOfFilledCellsInUnit = 9

or to put it another way:

NumberOfDigitsInNakedSubset + NumberOfDigitsInHiddenSubset =
NumberOfEmptyCellsInUnit

7. X-Wing (Advanced)
In the partial puzzle below, the only cells in rows one and nine that can contain a 9 are those marked. (The other cells in these rows are either already occupied, or else cannot contain a 9 because there are already 9s in the same column.) Since there must be a 9 in both row one and row nine, but they cannot occupy the same column, it follows that either the top-left and bottom-right marked cells contain the 9s, or the bottom-left and top-right cells do. (It can't be the bottom-right and top-right, nor the bottom-left and top-left, as then there would be two 9s in the same column.

Similarly, it can't be top-left and top-right, nor bottom-left and bottom-right as then there would be two 9s in the same row.) So, we can't say whether the 9s are in top-left and bottom-right, or bottom-left and top-right, but either way, it excludes 9s from the other cells in both columns. The end result is that 9 can be eliminated from the candidates for other cells in both of the affected columns.

8. Forcing Chains (Advanced)
Forcing chains is a technique that allows you to deduce with certainty the content of a cell from considering the implications resulting from the placement of each of another cell's candidates. (This technique is also know as "double-implication chains".)Forcing Chains (Advanced)

For example, in the following puzzle:

(The numbers is curly brackets { } are the candidates for the cell.)
Consider r1c2. This has the two candidates, 2 and 7. We will consider the implications of each of these candidates in turn.

if r1c2 = 2, then r2c1 = 1, and r5c1 = 2
if r1c2 = 7, then r1c7 = 3, and r5c7 = 1, and r5c1 = 2

So whichever of the two possible values are placed into (1, 2), we've deduced that (5, 1) must hold a 2. In other words, whichever chain of cells we follow, a certain cell is forced to have a specific value.

Note: unless the puzzle has multiple solutions, one of the considered candidates must be incorrect. This means it must eventually lead to either a contradiction or a dead end. If, when considering a single candidate, you reach a dead end, or find two chains that lead to different conclusions, you can eliminate that candidate from the starting cell. This is verging onto trial-and-error, and SadMan Software Sudoku doesn't do this as part of the forcing chain strategy. However, it can be useful when solving manually.

9. Nishio
This is a limited form of trial and error. For each candidate for a cell, it asks the question:

If I put this number in this cell, will this prevent me from completing the other placements of this number? If the answer if yes, then that candidate can be eliminated.

10. Trial and Error
There are some that would argue trial and error is not a logical technique, and is no better than guessing. Although it's not a technique I like to use, I do consider it logical. When further moves seem impossible, trial and error may be the only way forward. Indeed, some puzzles cannot be completed without it.

The technique involves selecting one candidate for a cell - without any particular reason for that selection - and then seeing whether the puzzle can then be completed. If it can, well done (although, there could also be other solutions - test the other candidates too.) If not, the trial and error move, and any subsequent moves, are undone, and a different choice is made. For some puzzles, it may be necessary to use trial and error several times. For others, it may be required only once.

In order to better manage the complexity, it's usual, if possible, to choose a cell with only two candidates, but that doesn't have to be the case.

It's worth noting, that this technique alone will always generate a solution if the puzzle can be solved, no other technique can guarantee that. But when used alone, it becomes the equivalent of a brute-force attack.

Have you got another technique? Email it to us or share it with others in the forums.