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This guide explains how to solve Sudoku Puzzles. This has been written by the authors of Sudoku Solver. Sudoku Solver is a program that allows you to set Sudoku puzzles to be played on-line or printed. Sudoku Solver will also show you how to apply the techniques in this document to solve puzzles.
Techniques for reducing possibilities Summary, Removing Possibilities
## Solving Sudoku PuzzlesSolving Sudoku Puzzles is a logical process. Depending on the level of difficulty of a puzzle you will need to use different techniques to solve it. The basic techniques for solving puzzles are described below. If you use Sudoku Solver it will demonstrate these techniques to you and provide tools to help you apply them more effectively. ## Only Value for Cell
In some cases there is only one value that a particular cell can have as all other values are already in; the same row, the same column or the same square. In the above example the cell: row 3, column 5 must have the value 5. In the puzzle grid above with all the possibilities calculated and displayed you can easily see there are another 3 cells like this. When you do not have the possibilities calculated or displayed, cells where this technique can be applied can often be identified easily where there are already a lot of cells answered in the row, column or square. Without the possibilities calculated and displayed we could easily identify the cell: row 4, column 6, as a likely candidate. ## Only Cell For Value
In a given row, column or square there is often only one cell where a particular value can be placed. In the row above the cell in column 6 is the only cell in the row that can have the value 6.
Sudoku Solver provides a tool to help visualise this. Looking at the full puzzle above and only considering the answer 6 we can see where this answer occurs (highlighted in blue), and can identify row 1, column 6 as not only the only cell in row 1 that can have the answer 6 but also this is the only cell in square 2 that can have the answer 6. It is important to develop a technique for visualising this, it is often easy to identify answers using this technique where a number exists in two or three adjacent squares. You can visualise it as lines excluding the possibility from other cells as shown below.
## Summary of Basic TechniquesUsing the above two techniques you can solve many puzzles. Although the examples show the possibilities calculated, if you are solving a puzzle on paper you will find you can spot answers without going to the trouble of identifying all the possibilities for cells. ## Techniques for reducing possibilitiesTo solve more complex puzzles the possible values for a cell need to be determined and then techniques applied to logically reduce the possible values. These techniques are described below. ## Sets (Pairs, Triples, Quads)The simplest technique for reducing possibilities is to recognise where there are pairs of possibilities.
In the example above the two cells highlighted in blue both have the same possible values 1 and 7. If one cell has 1 the other must have a 7. We can remove 1 and 7 as possibilities for any other cells in this row.
In the example above, rather than a pair of values we have three cells which must share the three answers 3, 6 and 8. These values can be excluded from other cells in the row that contain them (first two). Note, as above all three values don’t have to be possible for all three cells, as long as there is a group of 3 cells which share the same 3 possible values you have a triple. Look out for combinations such as (1, 2), (2, 3) (3,1). This would be a triple even though each cell only has two possibilities. You can find groups of 4 cells that make a quad also. Mastering this technique will help to identify answers and allow you to solve all but the most complex puzzles. In the above example removing the values 3, 6 and 8 from the first two squares will expose another pair and provide the answer to cells 7 and then 9. ## Hidden SetsAnother technique similar to sets is hidden sets. Here a group of cells that must contain a set of values is identified and further possibilities in those cells can then be eliminated. These are generally much harder to spot.
In the above example the three values 2, 3 and 4 can only occur in the three highlighted cells. All other values in these cells can therefore be removed. This will then expose a pair providing us with an answer to a cell.
In the above example we could identify a hidden set of three values 1, 3 and 8 in the three highlighted cells however it would be easier to identify that the set of three cells in the middle column or that the answer 8 can only occur in the third cell and reduce the possibilities in the same way. Occasionally however the hidden set is the only way forward. ## Locked ValuesLocked values are where a given answer can only occur in one row or column in a square. This answer can be eliminated as a possibility from other cells in the same row or column.
In the example above, in the middle square, the answer 1 must be in the bottom row. As there must be a 1 in one of these cells the other cells in the same row can have the answer 1 removed as a possibility. ## Summary, Removing PossibilitiesWhen you have removed some possibilities using the more advanced techniques, re-check using the simpler techniques. ## Further Advanced TechniquesThe above techniques will solve most puzzles that you will see including those rated as difficult, tough etc. There are a number of more advanced techniques including X-Wing and Swordfish that can help to reduce possibilities further however these are rarely required and are therefore not included in this guide. |