The two main approaches to analysis are "candidate elimination" and "what-if".
In elimination, progress is made by successively eliminating candidate numbers from one or more cells to leave just one choice. After each answer has been achieved, another scan may be performed-usually checking to see the effect of the latest number. There are a number of elimination tactics, all of which are based on the simple rules given above, which have important and useful corollaries, including:
• A given set of n cells in any particular block, row, or column can only accommodate n different numbers. This is the basis for the "unmatched candidate deletion" technique, discussed below.
• Each set of candidate numbers, 1-9, must ultimately be in an independently self-consistent pattern. This is the basis for advanced analysis techniques that require inspection of the entire set of possibilities for a given candidate number. Only certain "closed circuit" or "n×n grid" possibilities exist (which have acquired peculiar names such as "X-wing" and "Swordfish", among others; see List of Sudoku terms and jargon for more information). If these patterns can be identified, elimination of candidate possibilities external to the grid framework can sometimes be achieved.
One of the most common elimination tactics is "unmatched candidate deletion". Cells with identical sets of candidate numbers are said to be matched if the quantity of candidate numbers in each is equal to the number of cells containing them; essentially, these are perfectly coincident contingencies. For example, cells are said to be matched within a particular row, column, or region (scope) if two cells contain the same pair of candidate numbers (p,q) and no others, or if three cells contain the same triplet of candidate numbers (p,q,r) and no others. The placement of these numbers anywhere else in the matching scope would make a solution for the matched cells impossible; thus, the candidate numbers (p,q,r) appearing in unmatched cells in the row, column or region scope can be deleted. This principle also works with candidate number subsets-if three cells have candidates (p,q,r), (p,q), and (q,r) or even just (p,r), (q,r), and (p,q), all of the set (p,q,r) elsewhere in the scope can be deleted. The principle is true for all quantities of candidate numbers.
A second related principle is also true - if each cell within a set of cells (in a row, column or region scope) contains the same set of candidate numbers, and if the number of cells is equal to the quantity of candidate numbers, the cells and numbers are matched and only those numbers can appear in matched cells. Other candidates in the matched cells can be eliminated. For example, if (p,q) can only appear in 2 cells (within a specific row, column, region scope), other candidates in the 2 cells can be eliminated.
The first principle is based on cells where only matched numbers appear. The second is based on numbers that appear only in matched cells. The validity of either principle is demonstrated by posing the question 'Would entering the eliminated number prevent completion of the other necessary placements?' If the answer to the question is 'Yes,' then the candidate number in question can be eliminated. Advanced techniques carry these concepts further to include multiple rows, columns, and blocks. (See "X-wing" and "Swordfish" above.)
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 a 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 reductio ad absurdum. Nishio is a limited form of this approach: for each candidate for a cell, the question is posed: will entering a particular number prevent completion of the other placements of that number? If the answer is yes, then that candidate can be eliminated. The what-if approach requires a pencil and eraser. This approach may be frowned on by logical purists as trial and error (and most published puzzles are built to ensure that it will never be necessary to resort to this tactic,) but it can arrive at solutions fairly rapidly.
Ideally one needs to find a combination of techniques which avoids some of the drawbacks of the above elements. The counting of regions, rows, and columns can feel boring. Writing candidate numbers into empty cells can be time-consuming. The what-if approach can be confusing unless you are well organised. The proverbial Holy Grail is to find a technique which minimises counting, marking up, and rubbing out.