Basic Solving Strategy

Before beginning I would like to say, if you are better at learning a concept from a book and by practice I would recommend The Pocket Idiot's Guide to Super Sudoku as a great book to begin with. The book covers basic and advanced solving strategies with diagrams and easy to follow instructions. For the more seasoned Sudoku veterans, I would recommend a book like Sudoku Genius, which discusses some of the more advanced strategies and contains more challenging puzzles to solve. Once you get the hang of them, Handheld Electronic Sudoku and Sudoku board games are a great ways to play Sudoku and to share them with your friends and family.

The basic process for solving a Sudoku puzzle is a combination of scanning, marking up, and analyzing. For reference, here is the Sudoku puzzle used in our examples:

Scanning
Scanning is used in the beginning and throughout the solving process between each analysis step. Scanning consists of two basic techniques:

• Cross-hatching is the process of scanning rows and columns to identify which regions may contain a certain number by the process of elimination.
• Counting is the process of counting from 1 through 9 in regions, rows and columns to identify missing numbers. In more difficult puzzles it may be easier to count the values that it cannot be to see which possibilities are left.

Marking up
After scanning and no further numbers can be discovered, it is necessary to start with logical analysis of the puzzle. Most people find it useful to write possible candidate numbers into the cells using a subscript notation. An alternative to this technique is to write in numbers that the cell cannot be.

Example of marking up our example Sudoku puzzle

While marking up some solutions may become apparent. For example if a number only appears once in the markups for a given region it is clear that the digit should be there.

Analysis
The main approaches to analysis are "candidate elimination" and "what-if"

Candidate elimination is the process of successively eliminating candidate numbers from one or more cells to leave just one choice. After each answer has been achieved, another scan should be performed to see if any more answers can be determined.

• The first method of candidate elimination works by identifying matched cells. Cells are said to be matched within a particular row, column, or region if two cells contain the same pair of candidate numbers and no others. The placement of these numbers anywhere else within that same scope would make a solution for the matched cells impossible; thus, the candidate numbers appearing in unmatched cells in that same row, column or region can be deleted.
• A second related principle is also true. If, within any set of cells (row, column or region), a set of candidate numbers can only appear within a number of cells equal to the quantity of candidate numbers, the cells and numbers are matched and only those numbers can appear in the matched cells. Other candidates in the matched cells can be eliminated.

In the "what-if" approach, a cell with only two candidate numbers is selected, and a guess is made. The steps above are repeated unless duplication is found or a cell is left with no possible candidate, in which case the alternative candidate is the solution. In logical terms, this is known as reduction to the impossible.

Ideally you need to find the combination of techniques that suit you. Scanning may be boring, marking up can be time consuming, and the what-if method may get confusing. The best technique is the one you are comfortable with and helps you solve the puzzle the fastest.

After you are done with the puzzle, the completed puzzle should look like this: