## SOLVING SUDOKU

The One Rule:
 Fill in all blank cells making sure that each row, column and 3 by 3 box contains the numbers 1 to 9.

The Basics:
 Firstly, it's impossible to get very far without carefully maintaining a list of 'possible values' or candidates for each blank cell. Doing this by hand is laborious and prone to error, and often detracts from the fun of solving these puzzles. Fortunately, programs like Simple Sudoku will do this for you, while leaving you with the fun of applying logic to solve each puzzle. If you don't have a program to help - systematically analyse at each blank cell. 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. 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.)

 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. Since every box must have a 6, this cell must be that 6.

Beyond the Basics:
 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. 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. 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. 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 9), 9's can safely be excluded from all cells in this middle box except those in the left column.

 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. In the example below, the candidates 6 & 8 in columns six and seven form a Naked Pair within the row. Therefore, since one of these cells must be the 6 and the other must be the 8, candidates 6 & 8 can be excluded from all other cells in the row (in this case just the highlighted cell).

 Naked Triples & Naked Quads: The same principle that applies to Naked Pairs applies to Naked Triples & Naked Quads. A Naked Triple occurs when three cells in a group contain no candidates other that the same three candidates. 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. In the example on the right, a Naked Triple is formed by the top left, bottom left & bottom right cells of a box since they only contain the candidates 1, 4 & 6. Therefore the candidates 1 & 4 in the highlighted cells can be excluded safely.

 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 5 & 7 that are in the highlighted cells can be excluded.

 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. In the example on the right, the candidates 1 & 9 are only located in two highlighted cells of a box, and therefore form a 'hidden' pair. All candidates except 1 & 9 can safely be excluded from these two cells as one cell must be the 1 while the other must be the 9.

 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. 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. (Move your mouse over the image to reveal the answer).